Calculus

• Area of an Ellipse [Dault, 11/4/1996]
  How do you find the area of an ellipse?

• Calculus Chain Rule [Stuart, 11/23/1997]
  Could you give me the chain rule in easy terms, not a formula?

• Coordinates of Relative Maxima or Minima [Lunceford, 12/31/1994]
  Given f(x) = 15 x^(2/3) + 5x, find coordinates of relative max. or min. For what x values is the function concave down?

• Counting Bacteria with Calculus [Peterson, 2/15/1996]
  The number of bacteria in a culture increases from 3000 to 9000 in 8 hours. If the rate of increase is proportional to the number present, estimate the number of bacteria at the end of 24 hours.

• Critical points [Wise, 12/19/1994]
  Find all critical points (if any) in the following problem:
  k(t)=1/the square root of (t-squared +1).

• Definite Integral Across Addition, Subtraction [Heat, 8/10/1996]
  What is the integral of 3(x^2)-5x+9 from 0 to 7?

• Definition of the Signum Function [Little, 05/31/2000]
  Can you give me a simple definition of the signum function, and any practical examples of its usage?

• Derivatives [Longhurst, 10/28/1996]
  What is a derivative?

• Differentiation [Ali, 05/17/1998]
  I'm trying to find a GOOD definition for "differentiation."

• Find the Value of a... [Burris, 1/9/1995]
  Find the value of "a"so that the circles with equations (x-a)^2 + y^2 = 2 and (x+a)^2 + y^2 = 2 intersect at points where their tangents are perpendicular.

• Finding the Limit [George, 1/25/1996]
  I have a question about the limit of a sequence ...

• Finding the Minimum Average Cost [Erin, 2/12/1996]
  The cost of producing x units of a certain product is given by C = 10,000 + 5x + (1/9)x^2. Find the value of x that gives the minimum average cost.

• Formal Definition of a Limit [Rebecca, 10/17/2001]
  Could you please explain the formal definition of a limit? I need help specifically with finding a delta for a given epsilon and using the epsilon-delta definiton of a limit.

• Functions Without a Second Derivative [Theta, 6/28/1996]
  What are some examples of functions of a real variable whose derivatives don't have derivatives?

• Head start on Calculus: Differentiation, Integration [Truesdale, 07/08/1997]
  I'm soon to be a Calculus student in high school. Could you please teach me some Calculus over the internet?

• How do you get the volume equations for a sphere and pyramid? [Ron C. Maggiano, 6/10/1996]
  I am now thoroughly confused: we just learned the formulas for volume of a sphere and volume of a pyramid, but, he wouldn't tell me how to do it.

• An Introduction to Derivatives [Erika, 05/17/1998]
  I need to find out about third derivatives for a project. Can you give me some information, including uses, of derivatives?

• Logarithms: History and Use [Temple, 7/12/1996]
  How do I explain why logarithms work to non-math-oriented people? Also, what is the history of the development of the concept of logarithm? Why is it called a "logarithm"?

• Mean Value Theorem [So, Fan, 11/18/1994]
  This Theorem states that "If f(X) is continuous in [a,b], then there is a 'z' in (a,b) such that the integral from a to b f(x)dx = f(z)(b-a). How does the Mean Value Theorem apply to life? Why do we need this theorem?

• Murdered Professor [Wilson, 06/26/1997]
  Your professor has been murdered. Using exponents and differential equations, prove that it did not happen while you were in his office.

• Product and Rules, How to Remember Them [Wozniak, 10/19/1995]
  I have forgotten the formula for the integral of f(x)*g(x) and f(x)/g(x). Can you help me?

• Proof of L'Hopital's Rule [Williams, 12/23/1998]
  Can you show me a proof of L'Hopital's Rule and say how it relates to the different versions of the Rule?

• Proof of Product and Quotient Rules [Theron, 02/07/2002]
  I would like to know how the product rule and the quotient rule came about so I can better understand calculus.

• Proving the Fundamental Theorems of Calculus [Wellington, 02/02/2001]
  Can you provide a proof of the first and second Fundamental Theorems of Calculus?

• Related Rates [Verrico, 11/26/1998]
  A ladder 4m long is leaning against the vertical wall of a house... how fast is the top of the ladder sliding down that wall...?

• Techniques of Integration - Change of Variables [Jenny, 02/17/1999]
  Solve: (integral sign) sin 2x/ sq rt (9-cos^4 x) dx .

• Tips for Integrating Functions [Cowan, 05/27/1998]
  Can you give me some tips to help me integrate functions?

• Two Ways to Differentiate [Ryul, 6/30/1996]
  How can it be that if y=x^2, then y'=2x, but if y = x+x+...+x (x*x) then y'=1+1+...+1 (1*x) = x?

• What is Calculus? [Doedtman, 05/06/1997]
  What is calculus and how does it work?

• What is a Derivative? [Max, 06/07/1998]
  What is the concise definition of a derivative? How do we use derivatives?

• What is the Purpose of Determining a Derivative? [Lay, 1/12/1995]
  What is the purpose of the derivative?

• Why is "e" so important? [Chang, 1/8/1995]
  Why is "e" so important? How significant is "e" compared with "pi?" How did it come about? How is it defined? Why is it taught only at higher level mathematics? Are there other numbers like "e"?

• Differentiating y = x^x [Antill, 11/3/1994]
  Please could you differentiate y=X^X (that's X to the power of X)?

• Sea Navigation: Great Circles [Kairis, 11/4/1994]
  In sea navigation, given two points A,B with LON and LAT coordinates, find the travel distance from A to B. The distance is actually the great circle passing A and B. The solution is needed in the form of a double integral where the first int covers the LONs and the second the LATs. I guess the difficult part is to form the vector.

• Area Between a Line and a Curve [Portley, 11/5/1994]
  A high school senior writes: "The horizontal line y=c intersects the curve y=2x-3x^3 in the first quadrant. Area A is bounded by y=c, x=0, and the curve. Area B is bounded above by the curve and below by y=c. Find c such that (area of A)=(area of B)." I can do this using integrals, but I don't want to solve any cubics. There must be an easier way to find c.

• Nondifferentiable Functions [Finin, 11/6/1994]
  In my calculus book, it mentions a function that is not differentiable at any point due to the fact that it is not smooth at any point. It does not go any farther, and I was interested in hearing more.

• Conjugates [Bryce, 11/8/1994]
  What's a conjugate? (I'm in calculus, I've heard the term before, I just had a mental malfunction...)

• Calculus Constants [Cott, 11/16/1994]
  If a "u substitution" is made for different parts of the function, I get different answers. However the back of my book says that only one of the answers is correct. Here is the problem: Integral of secant squared of 3x times tangent of 3x.

• Antiderivative [Fan, 11/17/1994]
  One of my friends has asked me to solve a problem for her. I arrived at the right answer using my method of doing it. Later, she showed me her method. It seems correct, too. I was just wondering if you can find a mistake for her. Question/answer: antiderivative of secant square 3X times tangent 3X= 1/6 (cos3X) to the negative 2 plus C.

• Vector calculus [Chandavarkar, 11/18/1994]
  I have been studying vector calculus, and I was interested in finding more about the principles of the divergence theorem and Stokes' theorem.

• 2nd Derivative [Mullen, 11/21/1994]
  I am a grade twelve student taking calculus and was wondering if you could help me with this problem: y=2x(6x+5)exp4 - solve to the second derivative.

• Area of a volume of revolution [Chandavarkar, 11/22/1994]
  I am a little confused about finding volumes when graphs are revolved around an axis - specifically about finding the volume of a disk and the volume of a washer.

• Range of rational functions [Mehrman, Nowotenski, 12/1/1994]
  Given the following rational function: f(x)= (x-1)/(x^2+3x-1), how do you find the range? Our book gives the answer of: (neg infinity to 0.12] U [0.65 to pos infinity).

• Cubic Equations [Vitolo, 12/8/1994]
  In calculus, we were looking for the solutions for a third degree polynomial equation. Using my TI-85 to find the solutions, I stumbled upon an interesting observation. Given a cubic of the form ax^3 + bx + c, the absolute value of the sum of any two zeros seems to be equal to the absolute value of the third. However, I have so far been unable to give a general proof.

• Slope of root x. . . [Burris, 12/14/1994]
  How do I find the slope of: root x = -ln (xy) at (4,2) ?

• Integral of (1+sec(x))^2 dx [Cott, 1/5/1995]
  Please show me the steps for solving this problem: integral of (1+sec(x))^2 dx.

• Two Somewhat Similar Problems [Burris, 1/9/1995]
  I need help with two similar limit of integral problems.

• Point on a Parabola [Burris, 1/9/1995]
  How do you find the point on the parabola y = x^2 nearest the point (-3,0)?

• Slope of the Tangent [Burris, 1/9/1995]
  Given y = A sinx + B tanx. Find A and B if the slope of the tangent to the curve at x1 = pi/4 is m1 = 4 + root 2, and at x = 0, m2 = 4.

• Why use the natural log...? [Chen, 1/9/1995]
  Why do we have to use the natural log for the antiderivative of 1/x? Why doesn't the power rule work on the antiderivative of 1/x?

• Function X is a variable... [Fan, 1/9/1995]
  Right now we are learning general exponential and logarithmic functions. In one of my homework problems....

• A Norman Window [Tienhaara, 1/16/1995]
  A Norman window is a window in the shape of a rectangle with a semicircle attached to the top. Assuming that the perimeter of the window is 12 feet, find the dimensions that allow the maximum amount of light to enter.

• Calculus Questions [Mares, 1/27/1995]
  A student from Riverdale has interesting integration questions.

• Integration by Parts [Courtois, 2/15/1995]
  I need the integral from 0 to 1 of X^4*exp(-x/2). Can you help?

• Integration Problem [Confused, 2/22/1995]
  How do you integrate the function: (tan 2x)^3 dx, in terms of x?

• Math Questions [Holton, 3/19/1995]
  What is the volume of the solid of revolution created by the function, y = cos(x) and y = (x+1)/3, when these are revolved about the x-axis?

• L'Hopital's Rule [Nabi, 4/23/1995]
  I can't figure out how to take the limit using L'Hopital's rule on this problem...

• Design a More Efficient Soda Can [Bullock, 4/23/1995]
  The problem is to design a more efficient soda can that holds the regular 12 oz. of liquid. The can needs to have the least possible surface area.

• Using the Chain Rule [Kutuphane, 5/2/1995]
  What are the integrals of sin(ln(4x+5)) and ln(sinx) ?

• Integral of e^sinx [Muhtar, 5/15/1995]
  Can e^sinx be integrated ?

• Integrating Exponentials [Portman, 5/27/1995]
  What is the integral of 5^x dx ?

• Proof of a Bounded Series using the Binomial Theorem [Barr, 6/6/1995]
  Prove that (1+1/n)^n is bounded from above.

• Analytic Integration Problem [yliem, 7/13/1995]
  Is it possible to integrate [exp(-zcosh(t)] dt analytically?

• Minimum Value Problem [Twynam, 7/14/1995]
  I need to find the minimum value of E for E = be^(-ar) - dr^(-6) where b, a, and d are constants and r is the variable.

• Limit Problem [Chavolla, 7/16/1995]
  Find the limit of (x^2 + 2x) / (5x - 5) as x tends to infinity.

• How Long is the Tape? [Dorsey, 7/31/1995]
  Jones has a roll of tape for which he knows the thickness of the tape, diameter of the core, and diameter of the entire roll of tape. Is there a formula for determining the length of the tape?

• Half-Life of a Radioactive Substance [Salaz, 8/4/1995]
  The half-life of a radioactive substance is d days. If you begin with a sample weighing b g, how long will it be until c g remains?

• Exponential Growth of Bacteria [Sharpe, 8/7/1995]
  A colony of N bacteria increases with time according to the formula dN/dT = kN. If initially there are N' bacteria, show that the number at any time is given by N = N'e^kt.

• L'Hopital's Rule and Limits [Sharpe, 8/7/1995]
  Find the limit of [sin 3x / tan (x/3)] as x goes to 0.

• Using the Definition of Derivative [White, 9/1/1995]
  Using the definition of derivative, find f'(x), in simplest form, for f(x)=x^(1/3). (I know through the "tricks" that the answer is (1/3)*x^(-2/3) but I can't get to that using the definition of derivative.) If f(x)= [x], prove or disprove that f(x) has a limit at x=1.

• Deriving the Quotient Rule [Swipnet, 9/5/1995]
  How do I prove this derivation: (f/g)' = fg'-fg'/g^2 ?

• Integral Problems [Strazzari, 9/14/1995]
  A student asks questions about convergence and divergence in integrals.

• Using the Power Rule to Find the Derivative [Barr, 9/24/1995]
  We are trying to prove that the "power rule" works for finding the derivative of x^(3/2) (where x>0).

• Finding the Domain of a Function [Carlson, 9/28/1995]
  What is the domain of f(x)=((2-x)/(x-2))^.5

• Finding Integrals Using Trigonometric Substitutions [Man, 10/2/1995]
  What is the integral of 1/(x(1-x))^(1/2)? How can I integrate this function using only the derivative of arcsin or arctan?

• Finding the Equation of a Tangent Line [Gutierrez, 10/6/1995]
  What is the equation of the tangent line to the graph of f(x)=x^(2/5) at the point (-32,4)?

• Finding the Limit [Bullock, 10/24/1995]
  For a limit to exist, the right side and left side have to equal each other. Say you have a limit such as "lim as x->4 for function (3x-4). To see if this function exists, you have to check it on both sides. Would you plug in 3 for the left and for the right?

• The Relationship of the Tangent to a Curve and the Area under a Curve [Jahn, 10/25/1995]
  The problem of drawing a tangent to a given curve at a given point is closely connected to the problem of finding areas ("areas under a curve"). What is the connection, and when and by whom was it discovered?

• Solving a Window Problem with Calculus [Pyne, 11/11/1995]
  My question is just like the Norman window question, but I have an isosceles triangle on top. Where do I start? I seem to have 3 unknowns. b, h and x.

• Finding the Width of a Corridor [Lattner, 11/13/1995]
  A steel girder 27 ft long is moved horizontally along a passageway 8 ft wide and into a corridor at right angles to the passageway. How wide must the corridor be for the girder to go around the corner? Neglect the horizontal width of the girder.

• Minimizing the Cost of a Box [Lyn, 11/14/1995]
  A closed box with a square base is to contain 252 cubic feet. The bottom costs $5 per square foot, the top costs $2 per square foot, and the sides cost $3 per square foot. Find the dimensions that will minimize the cost.

• The Area Under a Parabola and the Length of a Curve [John, 11/15/1995]
  Find the area under the parabola f(x) = x^2 + 1 for x = 0 to x = 2, and y > 0. Also find the length of the curve from x = 0 to x = 2.

• Simplifying and Solving a Natural Log Problem [Mmm, 11/15/1995]
  Integrate ln(r/a)dr - the limits are a to b. Is this possible, or is there some table for it?

• Minimizing the Cost of Phone Line Construction [Paul, 12/3/1995]
  A telephone company has to run a line from point A on one side of a river to another point B that is on the other side, 5km down from the point opposite A. The river is uniformly 12km wide. The company can run the line along the shoreline to a point C and then under the river to B. The cost of the line along the shore is $1000 per km and the cost under the river is twice as much. Where should point C be to minimize the cost?

• Maximizing the Volume of a Cylinder [Paul, 12/3/1995]
  Find the dimensions of the cylinder of maximum volume that can be inscribed in a cone having a diameter of 40 cm and a height of 30 cm. Show that the maximum area of the cylinder is 4/9 the volume of the cone.

• Maximum Area of an Arc [Bender, 12/4/1995]
  A string of length 25cm is to be formed into the arc of a circle so that the area of the segment formed will be a maximum.

• Integration by Parts [Meier, 12/8/1995]
  How do you Integrate: x^n/e^x ?

• Calculus - limits [Smith, 12/10/1995]
  Find all the discontinuities of the following function and label those that are removable...

• The Speed of Change of a Shadow [Phalcon, 12/12/1995]
  A street lamp whose light is 5 meters from the ground is 6 meters from a vertical brick wall. A person, 2 meters tall...

• Optimization [Donabedian, 12/13/1995]
  An AP Calculus student asks questions about his chapter test.

• Water Draining from a Conical Reservoir [Tony, 12/20/1995]
  A circular conical reservoir, vertex down, has depth 20 ft and radius of the top 10 ft. Water is leaking out so that the surface is falling at the rate of 1/2 ft/hr...

• Integral of a Trig Function [Ster, 12/20/1996]
  F(x)=Integral from -1 to 1 of 1/sin(x)

• Area between two Curves [Kramer, 12/24/1995]
  Find the area between the curve x = 1-y^2 and the y-axis?

• Shortest Distance between a Point and a Circle [Zelaz, 1/22/1996]
  Find the shortest distance between the point (4,5) and the circumference of x^2 + y^2 = 9.

• Finding the Maximum Area of a Rectangle [Zelaz, 1/28/1996]
  A rectangle is inscribed in a triangle (with certain dimensions). Find the maximum area of the rectangle.

• Finding the Indefinite Integral [Ford, 2/1/1996]
  I am trying to find the indefinite integral of the following function: f(x) = x^x.

• Integral of Cos^2 [Papenn, 2/6/1996]
  I can't seem to find the anti-derivative of (cos X)^2.

• Vector Calculus [wwslau, 2/8/1996]
  Find the total distance travelled by a particle along the path, sketch the path; find the equation of the tangent line at t1, sketch the curve and tangent at t1...

• Pre-calculus Function Question [Belter, 2/8/1996]
  Does the function f(x)=3De^(1/x) have a local minimum?

• Non-Linear Equation [Rachman, 2/13/1996]
  Solve 5x^2 + log(x) = 0 for x...

• Taylor Series (Calculus II) [Carew, 2/14/1996]
  I am having difficulty understanding the Taylor Series and Maclauren Series.

• Calculus Problem Using Taylor Series [Martin, 2/16/1996]
  Looking at the equation: 1 + x*2^(x-1)

• Washers and Discs, Graphs and Symmetry [Donabedian, 2/20/1996]
  Use the washer method to find the volumes of the solid generated by revolving the regions bounded by the lines and curves about the y-axis...

• Using the Chain Rule [Rashid, 2/26/1996]
  d/dx sin(sin (2cos(2x/3))) = ?

• When to use the Shell Method or Washer Method to Calculate Volume [TJ, 2/26/1996]
  We recently learned about the washer method and the shell method. I was wondering which method to use in different situations.

• The Relation of the Speed of a Boat to the Water Line [Weaz, 2/27/1996]
  What is the mathematical formula or equation for calculating the speed of a boat as it relates to the water line of the boat.?

• Water Rising in a Tank [Ocastill, 3/7/1996]
  Water is rising in a tank with a sloping floor. Find how far it has risen after a certain amount of time.

• Inscribed Cone [Tang, 3/11/1996]
  Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 'r'.

• Differentiating the Rate of Decay [Burns, 3/14/1996]
  A lump of radioactive subtance is disintegrating at time 't' days after it was first observed to have mass 10 grams and: (dm/dt) = -km (where k is a positive constant). Find the time, in days, for the substance to reduce to 1 gram in mass, given that its half life is 8 days.

• Integral using Substitution [Pearce, 4/11/1996]
  Evaluate: xdx / (x^2+1)^(1/2), given u^2 = x^2+1.

• Integral of Ln [Wwrr, 4/13/1996]
  I need to solve the integral from 0 to 1 of ln(1-x) / x symbolically, if possible.

• Sequences and Series [Froh, 4/27/1996]
  Determine the number of arithmetic means between two numbers whose sum is 5.5.

• Projectile Velocity and Vectors [Karen, 5/7/1996]
  If the initial velocity is 834 meter^2/sec at an angle of 60 degrees, and x is 21,000, at what time, t, is the object at x = 21000 meters?

• Solving Differential Equations [Alnoor, 5/8/1996]
  (2x^2 + 2y^2 - y)dx + (x^2y + y^3 + x)dy = 0

• Calculus (Minimizing Volume) [Wiliam, 5/13/1996]
  The arch y=sin x,0 less than or = to x less than or = to pi, is revolved around the line y=c to generate a solid. Find the value of c that minimizes the volume.

• Modelling Discrete Data with Exponential Functions [Wilson, 5/14/1996]
  How can I fit an exponential function to a set of three discrete data points?

• Calculus (Area of a Plane Region) [Ecnesa, 5/20/1996]
  Problem: y = 4-x2 ; x axis - a) Draw a figure showing the region and a rectangular element of area; b) express the area of the region as the limit of a Riemann sum; c) find the limit in part b by evaluating a definite integral by the second fundamental theorem of the calculus.

• Operating a Tour Service [Christy, 5/21/1996]
  How many people does it take to maximize your profit?

• Where to Land on the Beach? [Christy, 5/22/1996]
  An island is 6 miles offshore... where should you land in order to go from the island to a store on shore in the least possible time?

• Integration [Mings, 5/23/1996]
  Integrate the following function: x^2 * e^(-2x^2)

• Integration by Parts [Anthony, 5/24/1996]
  How do you integrate x^x?

• Iterated Limits [Mendez, 5/25/1996]
  I don't know how to do the following problems...

• Integration [Wong, 5/29/1996]
  What does integral of (cos(x).x(cub).sinh(x)square).dx mean, and what's the answer?

• Derivation of Geometric Formulas [Swimduck, 5/29/1996]
  What are the formulas for the surface area, total surface area, and volume of a sphere, and volume of a pyramid and cone?

• Exponential Growth & Compound Interest [kkay, 6/2/1996]
  (1) The cost of goods and services in an urban area increased 1.5% last month. If the rate continues, what will be the annual rate of increase? (2) Suppose a certain population increases by 30% every 10 years. By what percent does it increase each year?

• Solving a Differential Equation [ar443, 6/5/1996]
  The integral looking somewhat like this: S ((e^u)/u) du has resisted every attack on my part.

• Integral tan(x)? [mbanister, 6/10/1996]
  What is the antiderivative of the function tan(x)?

• Rate of Change of a Function [Jhsu, 6/10/1996]
  Show that the expression for the rate of change can also be given by -cos x cot^2 x.

• Primitive function [Tamas, 6/11/1996]
  Give a primitive function of (sin x) / (ln (2 + x^2)).

• Taylor approximation of tan^2(x) [Stribblehill, 6/12/1996]
  Just to check that I can't do this because f'(0) = infinity...

• Quadratic Function and Diagonals of a Diamond [Medalves, 6/14/1996]
  The sum of the two lozenges (diagonals of a diamond) are maximum when the angle between the longest lozenge and the side of the diamond is 30 degrees, diamond side length = sqrt 2...

• Big O: 2 Functions Map from N to N+ [Turner, 6/25/1996]
  If f, g: N -> N+, then either f(x) is O(g(x)) or g(x) is O(f(x)). Why?

• Cubic Polynomial [Lanaya, 6/27/1996]
  Show that the graph of every cubic polynomial has a point of symmetry.

• Minimum, Maximum Value of a Two-variable Function [Brasher, 6/27/1996]
  Find f min and f max where x and y are real numbers and f(x,y) = 2(sin x)(cos y)+3(sin x)(siny)+6(cos x).

• Vector Equations Of Lines [Roberts, 6/28/1996]
  A boat leaves a harbour, O, position vector (0i+0j) at 9 a.m... at what time will it arrive?

• Solving a Logarithm [Tesiong, 6/29/1996]
  log a = -1.3

• Find Polynomial G(x) (Impossibility) [Geldermann, 7/1/1996]
  For all real x, F(x) = x+3 and G(x) is a polynomial of degree two such that G(F(x)) = x+2. Find G(x).

• Vector Analysis [Ivan, 7/6/1996]
  (1) Given three points A,B,C, find the angle between R(AB) and R(AC); the (scalar) area of triangle ABC; a unit vector perpendicular to ABC...

• Ptolemy's Theorem and Double-angle Formulas [Ayu, 7/11/1996]
  Is it possible to show the formula Sin(A-B), Cos(A+B) or Cos(A-B) using Ptolemy's Theorem?

• Limit Proof [Moore, 7/13/1996]
  If lim(An/n) = L and L>0, how do you show that lim(An) = + infinity?

• Learning Differential Equations [Mdubinko, 7/15/1996]
  What resources can I use to learn about differential equations?

• Proof of Tangent Double Angle Identity [Penelope, 7/16/1996]
  I would like to see the proof of the tangent double angle identity.

• Proof of Absolute Value, Division [Chu, 7/16/1996]
  If y does not equal to 0, prove that |x / y| = |x|/|y|.

• Two Points Moving... [Buchanan, 7/18/1996]
  Point P is moving in the positive direction along the y axis at a constant velocity of 10 m/s. Point Q is always moving with a constant speed of 11 m/s directly towards point P...

• Arc Length of Sine Curve [Leeper, 7/18/1996]
  What is the arc length of sin x from x = [0, 2pi]?

• Cardan's Cubic Method [Messer, 7/19/1996]
  Cardan's well known formula solution of cubic equation x^3 - 3x^2 - x - Sqrt[2] = 0 gives us a single "monstrous" algebraic real root which appears to defy further simplification....

• Maximizing the Volume of a Cylinder [Lisa, 7/24/1996]
  I need to maximize the volume of a right-circular cylinder that fits inside a sphere of radius 1 m.

• Setting Interest Rate to Maximize Profit [Lisa, 7/25/1996]
  A loan company is limited to charging maximum 18 percent interest on a loan. The amount of money available for loans is proportional to the interest rate the company will pay its investors...

• Algebra, Limit and Derivative Problems [Suryadie, 7/28/1996]
  Let x+y=1 and x^3+y^3=19. What is the value of x^2 + y^2?... f(x) = sqrt(2x+4), find f'(x)... d(8x^3+8x^2+6x+1)/d(2x) = ...?

• Subdividing a Field; Falling Water Level [Sunnie, 7/28/1996]
  What's the maximum area I can fence with a thousand yards? A water tank has a shape of a right circular cone... how fast is the water level falling?

• Lifting an Object of Changing Mass [Ponte, 7/28/1996]
  Suppose you grab the end of a chain that weights 3 lb/ft and lift it straight up off the floor at a constant speed of 2 ft/s: determine the force as a function of height; how much work do you do in lifting the top of the chain 4 feet?

• Area, Circumference of an Ellipse [Kroon, 7/29/1996]
  How do I calculate the area and circumference of a given ellipse?

• Income from Winning the Lottery [Medlin, 7/30/1996]
  How much money would I need to win in the lottery to have an income of $75,000.00 year until I die? That should be in about 30 more years.

• Methods of Approximation [Nicorgua, 7/31/1996]
  Is there an alternative to Newton's method of approximation to solve y^y = 25?

• Equation of Parabola with Given Vertex, Point [Everette, 8/4/1996]
  How do I find the equations for all parabolas containing the origin whose vertex is at (2,1)?

• Integral of Trig Function, Exponent [Sarmiento, 8/6/1996]
  What is the integral of f(x)=(tanx)^-1/2?

• Water in a Horizontal Tank [Michael, 8/7/1996]
  What's the volume of water in a cylindrical tank 72" long and 36" in diameter, filled only to 4.25"?

• Solving for Rate [Bradie, 8/8/1996]
  In the equation P=V*R/(1-(1+R)^(-n)), how do you solve for "R"?

• Integral of Triangular Surface [Suresh, 8/9/1996]
  Is it possible to numerically integrate S { 1/3 (x^3 i + y^3 j + 0 k) . n} dS where n is the unit normal to the surface S, a triangle in a plane?

• L'Hopital's Rule [Archie, 8/11/1996]
  How can I solve the following problem: limit as x-> infinity of x^(1/ x)?

• Rate of Change of Angle [Dave, 8/12/1996]
  Find the rate at which angle BXA is changing in radians per second...

• Integration Trick [fys95sig, 8/13/1996]
  If f(x)=1/sin(x), what is the integral of f(x)?

• Min, Max of 3-Variable Function [Alan, 8/16/1996]
  Find all the critical points and determine their nature for the function z = x^3 - 6xy + y^3.

• Halving and Halving Again - Zeno's Paradox [Chris, 8/22/1996]
  Since we can never really get to zero by reducing something by halves, does that mean that we are floating on air?

• Raabe's Theorem [Aciego, 8/23/1996]
  Regarding Raabe's result for the convergence of a numerical series of non-negative terms - I am looking for is a convergent series having m = 1.

• AP Calculus Test [alicole, 8/24/1996]
  Where can I find a site with information about the AP calculus test?

• Integral of Exponents, Natural Logs... [Piruk, 8/27/1996]
  How do I integrate x^(-ln x).e^(-x.lnx + x).dx ?

• Integral of sin(x)/x [Dbm, 8/29/1996]
  Is there a symbolic (not a numerical method) solution to the definite integral of sin(x)/x from 0 to some value?

• Find Curve Given Slope, Point [Bill, 8/29/1996]
  Find the curve whose slope at (x,y) is 3x^2 and which passes through (1,-1).

• Continuity [Hayes, 9/1/1996]
  Is the absolute function continuous?

• Chain Rule [Ramsey, 9/5/1996]
  How can I apply the chain rule to find du/dx for dy/dx=(nu^n-1)*(du/dx)?

• Limit As n Approaches Infinity [Eaton, 9/7/1996]
  How do I find the limit (n-> -infin.) of (sqrt(2n^2+1))/(2n-1)?

• Integral of a Power [Cummings, 9/8/1996]
  Why does the integral from -1 to 2 of x^2/3 = x^3/9 |from -1 to 2?

• Position Function of a Moving Particle [Alcal, 9/9/1996]
  A particle moves on a straight line with a velocity v(t)=sin(w)cos^2w; f(0)=0; w is constant; find f(t).

• Solving for Arc Length [Garver, 9/10/1996]
  How do I use the formula for arc length solve for the length of the curve?

• Method to Integrate sqrt(1-x^2) [Mpalizzi, 9/11/1996]
  What is the integral of [sqrt(1-x^2)]dx from [-1,1]?

• Meaning of Derivative [Tsang, 10/12/1996]
  What's a plain English meaning of the derivative?

• Navigation Formula [Jost, 10/14/1996]
  Given the starting latitude, longitude, distance, and course of a plane, what formula gives its destination latitude and longitude?

• Tangents to a Parabola [Spoelhof, 10/21/1996]
  Find the slopes of two lines tangent to the parabola y = x^2 that pass through the point (2,1).

• Implicit Differentiation [Zielinski, 10/21/1996]
  How do I find dy/dx for ((xy)^(1/2)) + ((x + 2y)^(1/2)) = 4?

• Chain Rule [Robinson, 10/23/1996]
  I have a question on the "chain rule" when finding the derivatives of polynomials.

• A Limit Problem [Oler, 10/24/1996]
  What is the limit of (x^2 -4) / (2x) as x approaches infinity?

• Finding the Extrema of a Function [Huinink, 10/25/1996]
  Find the extrema of f(x)=(ln(1+exp(-a-bx-cx^2)))/((x+d)^2)...

• Integrating X^x, Closed Form [Judes, 10/31/1996]
  How do you express the equation y = xcosx in terms of y?

• Implicit Differentiation [Brenchi, 11/1/1996]
  Given that (x^p)*(y^q) = (x+y)^(p+q), prove that: dy/dx = y/x.

• Integration Over a Complicated Region [Brenchi, 11/1/1996]
  Given a region defined by five relations simultaneously, find its area.

• Finding the Roots of a Function [Thompson, 11/4/1996]
  How do you find the positive roots of x^9+3x^8-5x^3+4x+6=0? Show that x^n- 1=0 has exactly 2 roots if n is even, and only 1 real root if n is odd

• Polar Coordinates [Thompson, 11/4/1996]
  Find the polar equation of x^3+y^3=3axy with x+y+a=0 as an asymptote.

• Implicit Differentiation [Thompson, 11/4/1996]
  Find the slope of the tangent line at the point (3,4) on the circle x^2+y^2 = 25.

• Maximizing the Volume of a Box [Timpl, 11/05/1996]
  What dimensions should a rectangular piece of paper have to maximize the volume of the box made by cutting the corners out and folding?

• Why Can't Some Functions be Integrated? [Shenoy, 11/06/1996]
  How do I evaluate Integral[x tanx dx]?

• Proof that INT(1/x)dx = lnx [Faulkner, 11/08/1996]
  How do you integrate (1/x)dx?

• Integration by Partial Fractions [Tang, 11/14/1996]
  How do you integrate f(x) = x^2/[(x^4) + 1]?

• Optimization [Sanec, 11/26/1996]
  To make a funnel, we take a circular piece of metal, cut out a sector, and connect the two radial edges together to make an open cone. What should the angle of the sector be to maximize the volume of the cone?

• Raytrace of a Star Sapphire [Charles, 12/11/1996]
  How do I find the rotations necessary to create a raytrace of a star sapphire so that the star always faces the camera?

• Brackets or Parentheses? [McDonald, 01/07/1997]
  When using interval notation to describe when a function is increasing and decreasing, how do I know whether to use brackets or parentheses?

• Integration by Parts [Agustinus, 01/07/1997]
  How do you solve Int [sin (ln x)] dx?

• Area Between Curves [Joe, 01/11/1997]
  Find the area bounded on the right by x+y = 2, on the left by y = x^2, and below by the x-axis.

• Equation for a Curved Line [Hardinger, 01/19/1997]
  Is there a formula into which you can enter a starting point, a starting slope, an ending point, and an ending slope that gives the equation of the curved line between these points?

• Domain/Range of a Function [Jennifer, 01/22/1997]
  How do you find the domain and range of the function f(x) = 2x^2-3x+1? (Both with and without calculus.)

• Polar Coordinates for Velocity and Acceleration of a Particle [Manville, 03/11/1997]
  I know that velocity components are dr/dt and rdo/dt and that components of acceleration are d^2r/dt^2 - r(do/dt)^2, etc. I would appreciate any help at all in understanding how these are generated and/or how to use them.

• Stokes-Greens-Gauss Theorems [Mullen, 03/25/1997]
  Are the Stokes-Greens-Gauss) theorems related? What is their significance?

• Calculus Word Problems [Bacchus, 04/03/1997]
  Minimizing functions and costs.

• The Bay of Fundy [Bacchus, 04/03/1997]
  The Bay of Fundy in Canada is reputed to have the largest tides in the world, with the difference between low and high water level being as much as 15 meters...

• Equation [Bacchus, 04/03/1997]
  Solve for x...

• Rational Function [Bacchus, 04/03/1997]
  Given the rational function y=r(x)=(4x^6-x^4)/(x^5+5) describe its end behavior...

• Cube Root Algorithm [Walters, 04/04/1997]
  Is there an algorithm for working out the cube root of numbers without a calculator?

• Simpson's Rule [Adam, 04/12/1997]
  Find the volume of the solid formed if the curve y = cos -1 x is rotated about the x-axis between x=0 and x=1.

• Maximum Volume of a Box [Paul, 04/13/1997]
  A rectangular sheet of cardboard measures 16cm by 6cm. Equal squares are cut out of each corner and the sides are turned up to form an open rectangular box. What is the maximum volume of the box?

• Intriguing Limit [Oliver, 04/30/1997]
  Why is the following true: lim x --> oo (1-1/x)^x = 1/e?

• Maximizing Area [Guerrero, 05/02/1997]
  A wire is cut into two pieces. One piece is shaped into a square, the other into an equilateral triangle. How should the wire be cut to maximize the area enclosed by the the two pieces?

• Definite Integrals [Padlo, 05/04/1997]
  What is the difference between the area under the curve from a to b and the definite integral from a to b?

• Creasing Paper [Tommi, 05/11/1997]
  Find the shortest crease when folding a piece of paper 6 units wide by 25 units long.

• Infinity in a Fraction [Kerr, 05/13/1997]
  Why does lim n --> oo (n+1)^.5/(n-1)^.5 = 1?

• Point of Inflection [Reniers, 05/17/1997]
  The cubic polynomial h(x) = x^3 - 3bx^2 + 3cx + d has a local maximum and a local minimum...

• Analyzing a Function [Jessica, 05/19/1997]
  Find the domain, intercepts, asymptotes, critical points, points of inflection, and graph of the function: y = 3x^4-4x^3-12x^2.

• Solve It Numerically [Riviera, 05/21/1997]
  How do you solve the equation x^x = 100?

• Integration by Parts [Ajit, 05/27/1997]
  What is the integral of sec^3(x)?

• The Gradient [Fresco, 06/03/1997]
  What is the geometrical interpretation of the gradient?

• Partial Fractions [Duke, 06/08/1997]
  How do you break 1/x^2*(x+2) into partial fractions?

• Substitution [Kevin, 06/18/1997]
  Integrate sqrt(x^3-1)/x dx.

• Conical Drinking Cup [Gammill, 06/20/1997]
  A conical drinking cup is made from a circular piece of paper of radius r by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup.

• Area Under a Parabola [Matthew, 06/27/1997]
  Why is the area under the parabola y = x^2 between the y-axis and the line x = b equal to b^3/3?

• Calculus: Rate of Change in Volume [Kim, 07/27/1997]
  The radius of a right circular cylinder is decreasing at the rate of 4 feet per minute, while the height is increasing at the rate of 2 feet per minute. Find the rate of change in the volume when the radius is 2 feet and the height is 6 feet.

• Surface Area of an n-dimensional Sphere [Bollinger, 07/28/1997]
  I was wondering how to calculate the surface area of a sphere in n dimensions.

• Checkerboard Problem [Redhot, 07/30/1997]
  Is there an exponential pattern for how many squares there are on a checkboard?

• Related Rates [Ferguson, 08/05/1997]
  A tank contains 150 litres of brine solution. The concentration is 0.7 kg of salt per litre... Find the time when the salt content will be 90 kg.

• Picard's Method [Lam, 08/10/1997]
  Can you tell me about Picard's iteration method of solving differential equations?

• Calculus - Shell or Washer Method? [Tracy, 08/23/1997]
  I have a question regarding how to set up integrals for the volume of revolution.

• Equation of a Circle [Slawson, 08/25/1997]
  Write the equation of the specified circle in general form...

• Applied Problems in Maxima and Minima [Piggie, 08/30/1997]
  At what first quadrant point on the parabola y = 4 - x^2 does the tangent together with the coordinate axes determine a triangle of minimum area?

• Points of Intersection [Wang, 08/31/1997]
  Estimate all points of intersection of the graphs f(x) = x + sin x and g(x) = x^3 by using a calculator or computer. How do you know you have found them all?

• Area of a Parabola [Globetrotters, 09/05/1997]
  How do you find the area of a parabola? (I just finished Algebra 2.)

• Integral Calculus [Jane, 09/18/1997]
  How do you integrate the following function: sqrt(1-x^2) from 0-1?

• Simpson's Rule - Alternating Pattern [Holm, 09/20/1997]
  I'm trying to apply Simpson's rule to a computer program, but I don't understand it well enough - can you tell me how it works?

• Is the Function Invertible? [Lanka, 09/22/1997]
  For the following functions f(x) decide if the funcion is invertible as a function from R to R...

• Integral of N^-1 [Mimi, 09/26/1997]
  How do I find the integral of a number raised to the -1st power?

• Infinite Sequence [Leone, 09/28/1997]
  From f:n ---> 3 + (-1/2)^n where n belongs to the set of natural numbers, describe a strip that contains all but a finite number of points of the graph of f.

• What is Nonlinear Math? [Jones, 10/02/1997]
  What exactly is nonlinear math, and what is it used for?

• Surface Area of a Sphere [Kaszuba, 10/03/1997]
  How is the surface area of a sphere calculated, and why?

• Where is f Continuous? [Stone, 10/07/1997]
  Graph the following... Besides at x = 3, where is f continuous?

• Limits - Indeterminate Forms [Mary, 10/12/1997]
  I cannot do a problem where I need to convert into the form 0/0 and then use L'Hopital's Rule...

• Calculating Any Root [Tat, 10/13/1997]
  I need to find an algorithm to determine any root of a number. I was told I could determine the estimated value by using Newton's Method...

• Applications of Imaginary Numbers [Zakrzewski, 10/14/1997]
  Where are imaginary numbers used today in real life, as in the work force or other areas that use math?

• Calculating the Shortest Flight Path [Perry, 10/30/1997]
  Is this spherical trigonometry or can it be solved using simpler geometric equations?

• Optimization: Minimum Area [Yamashita, 11/07/1997]
  How do you fold a piece of paper (rect. with width a and unlimited length) so one corner just reaches the righthand side for minimum area?

• Integration of Sin(x^2) [Joshua, 11/10/1997]
  I have been given the solution in the form of Frensel's Sin, but it explains nothing about how it was integrated. I am not looking for an equation, I am looking for a reason!

• Volume of a Shape [Friendly, 11/11/1997]
  What is the volume of the shape formed by rotating the parabola y=x^2 around the line y=x? (From x = 0 to 1).

• Slope of 3-Dimensional Equations [Keenan, 11/12/1997]
  Is it possible to find the slope of three-dimensional equations?

• Where did e^(i*pi)+1 = 0 come from? [Jill, 11/12/1997]
  Is there a way to solve the equation e^(i*pi)+1 = 0 without using or using very little calculus?

• Differentiate Twice [James, 11/14/1997]
  If y = xsin(3x) prove that y''+9y = 6cos(3x), where y'' is its second derivative.

• Convergence of Cos(x^2) or Sin(x^2) [Underwood, 11/15/1997]
  How would you prove that the integral from n=0 to infinite of sin(x^ 2)dx or cos(x^2)dx converges?

• Differentiation Problem [Confer, 11/15/1997]
  A light shines from the top of a pole 50 ft. high. A ball is dropped from the same height at a point 30 ft. away from the light...

• Alternating Harmonic Series [Campbell, 11/18/1997]
  I am trying to find the proof for the sum of the alternating harmonic series. I did find out that it is ln(2), but please tell me why?

• Cylinder Problem [Straalen, 11/22/1997]
  Is it possible to construct two cans with different volumes but the same surface area?

• Calculus: Limits [Batchilder, 11/24/1997]
  Why does my textbook define the limit of (1/x) as x approaches zero as infinity?

• Implicit Functions [Norris, 11/26/1997]
  Please give me a definition and several examples of an implicit function.

• Helix Lying on a Cylinder [Alvarez, 11/26/1997]
  I am trying to find a parametric equation for a helix (in 3 space) lying on the cylinder...

• Art Gallery Problem [Cloutier, 11/30/1997]
  A picture is two meters high and is hanging so the bottom of the picture is one meter above eye level. How far from the wall should you stand so the angle of vision occupied by the picture is a maximum?

• Infinite Product [Nguyen, 12/01/1997]
  How do I find the infinite product of 3^1/3 * 9^1/9 * 27^1/27 * ..... (3^n)^1/3^n ?

• Rectification of a Quartic [Sharkey, 12/02/1997]
  I've been told that some 4th order equations can be rectified. Are there any Cassinian ovals whose perimeter can be calculated without looking up tables?

• Evaluate the Limit [Yackoski, 12/03/1997]
  In Calculus today we were trying to evalulate the limit of (1-cos x)/ (x^2) as X->0.

• Triple Integrals [Sussman, 12/07/1997]
  Determine the limits...

• Length of a Cubic Curve [Gardiner, 12/10/1997]
  I need to calculate the true length of a cubic curve.

• Finding Volumes: Washer/Disc Methods [Nagle, 01/05/1998]
  I am having difficulties in the washer and disc methods of finding volumes, as in the problem y=x^2, x=3 about the y axis.

• Integration [BLT, 01/08/1998]
  What is the integrated area if the anti-derivative is undefined?

• Lagrange Multipliers [Aleja, 01/08/1998]
  I have a problem with Lagrange Multipliers - can you help?

• Differentiating [Hack, 01/14/1998]
  Please help me differentiate (y^2) * cos (1/y) = 2x + 2y.

• Find x in Degrees, not Radians [Ronny, 01/14/1998]
  I need to find the derivative of sin(x) when x is in degrees, not radians.

• Cow Grazing Half the Circle: Newton-Raphson Method [Julus, 01/18/1998]
  Assume a perfect circle filled with grass and a cow tied with a rope to the fence around it...

• Cycloid [Chris, 01/30/1998]
  What is a cycloid and what does it do?

• Simpson's Rule/Trapezoid Rule [Riddell, 02/01/1998]
  How is Simpson's rule derived and why it is a better approximation of the integral than the midpoint (rule?) or trapezoidal method?

• Volume of Spherical Cap [Arkin, 02/06/1998]
  I am trying to find the volume of a cap of a sphere with radius of 5. The cap has a height of 3 - it is as if the top of the sphere, 3 meters from the top, was severed from the rest of the sphere.

• A Calculus Question [Nash, 02/09/1998]
  I figured this worked out to -1/2: lim (x - (x^2 + x)^(1/2)) (x going to infinity).

• Derivative Interceptions [Arroa, 02/10/1998]
  I'm trying to figure out if there is a way to determine the points at which a derivative intersects a function.

• Algebraic Integration of Standard Normal Distribution Function [Keites, 02/11/1998]
  I've tried inspection and partial integration and don't seem to get the right results - I wonder if this is a function whose integral cannot be solved algebraically?

• Integration of Natural Logs [Jaeger, 02/12/1998]
  Why does the natural log of x equal the integral of 1/t dt from 1 to x? Why is INT[(1/t)dt] from 1 to x the natural log of x, or why was it defined this way?

• Chain Rule and Product Rule: Differentiating [Kelly, 02/19/1998]
  Why isn't the result what I expect when I use the Chain Rule and the Product Rule to differentiate?

• Determining Tangent to an Ellipse, Minimizing Area [Huesser, 02/20/1998]
  Find the equation of the tangent to the ellipse that forms, with the coordinate axes, the triangle of smallest possible area.

• How to Find the Range of a Function [Hunt, 02/25/1998]
  How do you find the range of a function like g(x) = (x+1)/(x^2-1)?

• Second Derivative of a Function With Absolute Values [Smith, 02/27/1998]
  Let f(x) = |x|^3. Find f"(0).

• Evaluating Two Definite Integrals [Katie, 02/28/1998]
  Evaluating a definite integral and determining an enclosed area.

• Area of A Sector of An Ellipse [Brady, 02/28/1998]
  Finding the area of a sector of an ellipse, given the semiminor and major axes and the angles of the 2 vectors bounding the sector.

• History and Applications of the Natural Logarithm [Fornwalt, 03/02/1998]
  I'm so surprised at how often the number e comes up. Where did it come from? Who first derived it? Why is it so common in the field of biology?

• Zero Laws and L'Hopital's Rule [Sheryl, 03/04/1998]
  Is zero divided by zero: a) zero, b) undefined, or c) one?

• Why Differentiability Implies Continuity [Noorduyn, 03/06/1998]
  Why must a function that is differentiable in an interval be continuous in that interval?

• What is a continued fraction? [Wollak, 03/06/1998]
  What is a continued fraction and what makes it different from the types of fractions or ratios I'm used to?

• Related Rates [Armstrong, 03/11/1998]
  The rate of change of the edges of a cube, given the rate of change of its volume.

• Exact? Area under a Curve [Riddle, 03/13/1998]
  Would you please explain why the area under a curve is exact and not an approximation?

• Integration of Trigonometric Function by Substitution [Haas, 03/18/1998]
  Integrate [tan 2x dx] by u-substitution.

• The Critical Number [MacDonald, 03/23/1998]
  Find the critical number of y = 2 cos x + sin 2x where x is greater than or equal to 0 and less than or equal to 2 Pi.

• Dominant Terms [Faerber, 03/25/1998]
  What are dominant terms, and how do you obtain their values?

• The Integral of e^(x^2)dx [Martinez, 03/26/1998]
  How do you solve the integral of e^(x^2)dx?

• The Logistic Model for Population Growth [Duchemin, 03/29/1998]
  I'm trying to solve the logistic model of population growth for P(t).

• Volume of a Cone or Pyramid [Swan, 03/30/1998]
  Proofs that the volume of a cone or pyramid is (1/3)b*h.

• e as a Series and a Limit [Lee, 03/30/1998]
  Why does e = 1 + 1/2! + 1/3! + 1/4! + ... and lim (1 + 1/n) ^ n, as n --> infinity?

• Distance From a Point to a Plane [Larock, 03/31/1998]
  Can you show me the proof of the formula for the distance between a point and a plane?

• Logistic Growth of A Rumor Spreading [Ghani, 04/11/1998]
  Assuming logistic growth, find how many people know the rumor after two weeks.

• Conditions for Real-valued Roots [Lehto, 04/01/1998]
  Using derivatives to find the conditions that ensure that an equation of degree 3 has 3 distinct real-valued roots.

• Surface Area of a Sphere [Chong, 04/10/1998]
  Can you derive the formula for the surface area of a sphere?

• Trigonometric Functions and Euler's Identity [Jung-bok, 04/10/1998]
  We need to prove three trigonometric functions in various ways.

• Two Ways to Integrate [Lin, 04/15/1998]
  How do you integrate: Int[sin^4(x)*cos^6(x)]dx?

• Two Ways to Find a Formula [Mon, 04/14/1998]
  I need to show that Sigma(rx^r) = (x-(n+1)x^(n+1)+nx^(n+2))/(1-x)^2.

• The Surface Area of a Rotational Curve [Shannon, 04/17/1998]
  I'm having difficulty understanding how my math professor proved the formula for the surface area of this curve...

• Volume of a Sphere [Fishman, 04/21/1998]
  Can you help me derive and prove the formula for the volume of a sphere?

• Maximum Points and the Second Derivative [Sue, 04/24/1998]
  When the second derivative of a point is a negative, is the point said to be a maximum?

• Exponential Growth of a Population [Caren, 05/03/1998]
  A population P(t) is increasing at a rate directly proportional to 800 - P(t), where the constant of proportionality is k...

• Continuity and Differentiability of Piecewise Function [Hopson, 05/04/1998]
  What values of "a" and "b" will make this piecewise function continuous and differentiable?

• Finding Limits Using Natural Logs and L'Hopital's Rule [Bailey, 05/03/1998]
  I think I need to use l'Hopital's Rule and the natural logarithm function to find lim (x goes to 0) [(cos (2x))^(3/(x^2))].

• A Trigonometry Integral Requiring Two Substitutions [Paul, 05/03/1998]
  Substituting twice and using trig to integrate sqrt(1 + sin(x)).

• Why Are 1^infinity, infinity^0, and 0^0 Indeterminate Forms? [Hanna, 05/08/1998]
  Using limits to prove that 1^infinity, infinity^0, and 0^0 are indeterminate forms.

• Differentiating and Integrating the Formula for Area of Circle [Bohanon, 05/11/1998]
  The formula for a circle's circumference is the derivative of the formula for its area. What is the significance of this?

• Differentiability, Intervals, Inflection Points of Piecewise Function [Emineth, 05/13/1998]
  For what values of k and p will the function be continuous and differentiable? On what interval will it be increasing? Find all points of inflection.

• Absolute Values and e in an Integrand [Raymond, 05/10/1998]
  Integrating INT(e^(-|x|)) dx by treating x > 0 and x < 0 as separate cases.

• Finding Maxima and Minima: Turning Points [James, 05/10/1998]
  Can you help me find all of the maxima and minima of the function: 5e^(-x/2)sin(2x)?

• Great Circle Parametric Equation [Lotter, 05/25/1998]
  How can you calculate specific points of a great circle on a sphere? Can you help me find the parametric equation?

• The Gamma Function and Its Derivative [Geib, 05/26/1998]
  Does n! = the integral from 0 to infinity of (x^n)(e^-x)dx hold true for all real numbers? If so, can we find the derivative of n!?

• Limit Proofs with L'Hopital's Rule [Sheth, 05/27/1998]
  How would you prove the following three limits? ...

• ODEs and Integration With the Chain Rule [Smith, 05/30/1998]
  What is an ordinary differential equation? How do you integrate f'(x) = (x-1)^4?

• Max/Min Applications of Derivatives [Beth, 05/31/1998]
  I need help with these max/min applications of derivatives. If 40 passengers hire a train...

• The Figure of Maximum Area and Given Perimeter [Morris, 06/02/1998]
  Can you help me show, with and without calculus, that the geometric figure of a maximum area and given perimeter is a circle?

• An Integration with Substitution [Abecina, 06/06/1998]
  I'm having trouble with the following integration. Is substitution needed here?

• Deriving Simpson's Rule [Carol, 06/07/1998]
  Can you show me a derivation of Simpson's Rule?

• Minimizing the Sums of Squares [Hrycenko, 06/12/1998]
  Find two numbers such that their sum is 20, and the sum of their squares is as small as possible.

• When to Use the Chain and Quotient Rules [Matthew, 06/12/1998]
  In the following problem, would you use the chain rule or the quotient rule first to differentiate? Problem: (2x+1)^2/(2x+4).

• Sketching a Function [MatherWu, 07/06/1998]
  Can you help me piece a function together so that the following hold? It is increasing and concave up on (-infinity, 1) ...

• Parametric Equations [Li, 07/21/1998]
  A curve is given parametrically by the equations: x = (3-2k)^2 and y = (2+k)^2. Find dy/dx, the x-intercept, the cartesian equation...

• Limits of the Natural Logarithm [Andrew, 07/22/1998]
  Can you help us find the limit of x (ln x)^n as x goes to infinity, for all n? Do we use L'Hopital's Rule?

• Arc Length Formula [Sue, 07/28/1998]
  Find the arc length of a curve in symbolic form and decimal approximation.

• The Impossibility of Integrating x^x [Black, 08/07/1998]
  How would you test to see if the integral of x^x can be expressed in a finite number of elementary functions?

• Limit Problems [Josephine, 08/08/1998]
  What is the limit of (sqrt(2-t) - sqrt(2))/t as t->0?

• Descartes' Method for Tangents [Sara, 09/02/1998]
  Can you help me find the equation of the tangent of y^2 = 2x using Descartes' method...?

• Delta-Epsilon Limit Proofs [Angeles, 09/05/1998]
  Can you help me with some delta-epsilon proofs? How do you choose delta?

• Yacht Distances and Vectors [Stokes, 09/05/1998]
  Can you help me find the smallest distance between two yachts using vectors?

• Differentials [Sanabria, 09/18/1998]
  What is the difference between finding the derivatives of a function (dy/dx), and finding its differentials (dy,dx)?

• The Derivative of f(x)^g(x) [Bandel, 09/26/1998]
  Is there a general formula to find the derivative of f(x)^g(x)?

• Limit Intuitions [Bryan, 10/01/1998]
  Can you explain the intuition behind the formal definition of a limit? derive the equation of the tangent line of a function at a given point?

• The Limit of Sin(1/x) [Johnson, 10/02/1998]
  Why does the sine of 1/x have no limit as x approaches 0?

• The Squeeze Theorem [Shin, 10/12/1998]
  How do you use the Squeeze Theorem to find the limit of f(x) = x * sin(1/x) as x approaches 0?

• The Monotone Convergence Theorem [Flarity, 10/02/1998]
  Could you please explain the meaning and purpose of the monotone convergence theorem?

• Volume of an Elliptical Cone [Francesca, 10/15/1998]
  Can you help me on find the volume for an elliptical cone by using a triple integration?

• Integrating with the Arc Length Formula [Dinh, 10/19/1998]
  Can you help me integrate functions using the arclength formula, e.g. x^(2/3)+y^(2/3)=1 and y=e^-x?

• Shakespearean Differential Equations [Brown, 10/27/1998]
  Can you help me solve these differential equations: dG/dt=aJ-bG and dJ/ dt=cG?

• Finding First Derivatives by Definition [Lisa, 10/30/1998]
  Can you show me an easy example of how to find the first derivative of a function using the definition?

• Volume of a Rotated Region [Marianne, 11/02/1998]
  Find the volume of a solid formed by revolving the region bounded by y=x^2+1, y=0, x=0, and x=1, about the y-axis.

• Implicit Differentiation [Mong, 11/03/1998]
  How do you diffrentiate an equation that cannot be expressed in the form y = f(x)?

• Related Rates and a Decreasing Radius [Bartel, 11/18/1998]
  A nitroglycerin lump is decreasing at a rate of 2 cu.cm/sec. Use related rates to find how fast the lump's radius decreases at 9pi/4 cu.cm.

• Finding the Equation of a Tangent Line [Ng, 11/24/1998]
  I only have where the tangent intersects the y axis...

• Lagrange Multipliers and Constraints [Sam, 11/24/1998]
  When using the Lagrange Multiplier method, how do you determine which of the two equations is the constraint?

• Trigonometry and the Chain Rule [Amanda, 12/01/1998]
  I need help differentiating expressions such as: y = 2 csc^3(sqrt(x)) and y = x/2 - (sin (2x))/4. Where does the chain rule fit in?

• Equations for the Paths of Four Lady Bugs [Morcombe, 12/02/1998]
  Four ladybugs travel from the corners of a square towards the center. Find the equations that describe the path.

• Newton's Method and Square Roots [Hendricks, 12/04/1998]
  Can you explain how finding square roots by hand relates to Newton's method for approximating the zero of a function?

• Derivatives [Rakesh, 12/07/1998]
  Can you write x^2 = x+x+...+x (x times) if x is not a positive integer?

• What is 1^infinity? [Hiscock, 12/10/1998]
  Why is 1^infinity indeterminate? How can you show this using limits? Why is it not equal to 1, as intuition leads us to believe?

• A Sphere's Surface Area and Volume [Matt, 12/17/1998]
  What is the relation between a sphere's surface area and its volume? How does their ratio change?

• AP Calculus - Minimizing Time Traveled [Mergenthaler, 12/22/1998]
  A person on a boat 9 km from the shore must go 12 km down the shoreline in the shortest time possible. Find a function for where the person should hit land...

• AP Calculus - Minimums and Maximums [Guppy, 12/23/1998]
  Hints for finding specified x-coordinates...

• Proving Limits Formally [Glover, 01/04/1999]
  Given a sequence, I can guess what the limit it. How do I actually prove my guess is the limit?

• Barycentric Calculus [Noble, 01/06/1999]
  How does barycentric calculus compare with trilinear or cartesian calculus?

• General and Particular Solutions [Abbas, 01/08/1999]
  What is the difference between a general solution and a particular solution of a differential equation?

• Related Rates Problems [Sze, 01/15/1999]
  A plane flying horizontally at an altitude of 1 mi and a speed of 500 miles/hour...

• Using L'Hopital's Rule [Mirza, 01/22/1999]
  Prove that e^(-1/3) is the limit for y =(arctan(x)/x)^(1/(x^2)) when x -> 0.

• Multidimensional Calculus and Vector Geometry [Thomssen, 02/09/1999]
  The depth of iron ore can be approximated by a plane...

• Vector Calculus [Denduyver, 02/10/1999]
  Prove the following identity: a . (grad(a . v) - curl (v x a)) = div v...

• Surface Area of a Football [Brian, 02/12/1999]
  Calculate the surface area of a football with a circumference of 22 inches and an arc length of 14 inches.

• Factoring Polynomials [Valentino, 02/12/1999]
  What are some real-life applications of factoring polynomials?

• Differential Equations and Flow Rate [Stephen, 02/14/1999]
  A solution containing 3 lb/gal of dye flows into a 100 gal tank at 5 gal/min ...

• When Did It Start Snowing? [Chu, 02/17/1999]
  Convert to a differential equation: A snowblower throws 30 cu. ft. of snow per minute...

• Integration [Crys, 02/17/1999]
  Integrate x*tan^2 x with respect to x.

• Surface Area and Volume: Related Rates [Jack, 02/22/1999]
  The surface area of a cube is changing at a rate of 8 in./s^2. How fast is the volume changing when the surface area is 60 square inches?

• Center of Mass [Genevieve, 02/22/1999]
  Find the center of mass of a thin plate of constant density covering the region bounded by the parabolas y = 2x^2-4x and y = 2x-x^2.

• Integration Hints [Scott, 02/23/1999]
  Can you help me integrate (cos[x])^4?

• Related Rates [Brian, 03/01/1999]
  A circle is inscribed in a square tangent to each side of the square. The circumference of the circle is increasing at a constant rate of 6 inches per second...

• Hyperbola and Cone [Mikael, 03/04/1999]
  Two questions on three-dimensional figures.

• Rate of Change of Radius of a Cone [Andrew, 03/07/1999]
  Using the measured rate of change in volume of water in a cone, calculate the rate at which the circular area of water reduces when the radius is r centimetres.

• Degrees of Generality [Pete, 03/13/1999]
  Different types of integrals: Riemann, Riemann-Stieltjes, and Lebesque.

• Integration [Jason, 03/30/1999]
  Integrate (2 - x^2)^4x dx.

• Catenary Curve [Stephen, 03/30/1999]
  Find the vertex of a catenary curve.

• The Moving Shadow [Isaac, 03/24/1999]
  A man walks toward a light... at what rate does the tip of his shadow move and at what rate does its length change?

• What is the Integral of (tan x)^1/2 ? [Neal, 03/07/1999]
  My first substitution was u^2 = tan x ...

• How Should the Pipe be Laid? [Babin, 03/25/1999]
  Oil from an offshore rig is to be pumped to a location on the edge of the shore... determine how the pipe should be laid to minimize cost.

• A Rumor Spreads [Majithia, 04/01/1999]
  A rumor spreads through a community at the rate dy/dt = 2y(1-y), where y is the proportion of the population that has heard the rumor at time t...

• Derivative of a Function [Yang, 04/01/1999]
  What is the formula for calculating the Kth order derivative of the function: f(x) = exp(A(x))?

• What is the Longest Pole? [marinsek, 04/04/1999]
  A metal pole L ft long is pushed on the floor from one corridor into another corridor at a right angle...

• Related Rates - Clock Hands [Perry, 04/05/1999]
  How fast is the distance between the tips of the hands on a watch changing at one o'clock?

• Integration by Parts [Auge, 04/08/1999]
  How does one find the integral of [(sec x)^3 dx]?

• Cylinder Volume and Surface Area [Nguyen, 04/14/1999]
  How does one make a cylinder that holds the greatest volume if the entire surface area is 600 cm squared?

• Distance to the Sun [Buchholz, 04/16/1999]
  Find the distance from Earth to the sun when t = 90 days...

• Mean Value Theorem [Porter, 04/18/1999]
  Which number satisfies the conclusion of the Mean Value Theorem for f(x) = sin (x/2)?

• Chain Rule [Eric, 04/21/1999]
  Taking the derivative of a function of a function.

• Volume of a Torus [Michale, 04/22/1999]
  I would like to know how to find the volume of a torus using integrals.

• Differentiating a Polynomial [Loader, 04/25/1999]
  Differentiate with respect to x: f(x) = x^8 + 3x^5 + (3-5x)^4.

• Integration by Parts [Brown, 05/02/1999]
  Find the general integral of x*ln(x)-1.

• Cola Consumption [Pirrone, 05/03/1999]
  The rate of consumption of Cola in the U.S. is S(t)= Ce^kt where S is measured in billions of gallons per year and t is measured in years from the beginning of 1980...

• Let f(x) = 1 + 1/2 + 1/3 + ... + 1/[(2^n)-1] [Manghat, 05/15/1999]
  Which of the following inequalities are correct?

• Solving an Equation using Power Series [Heinz, 05/18/1999]
  How does one solve the differential equation (t^4)*x'' + x = 0 for x?

• Euler Equation and DeMoivre's Theorem [Anthony, 05/18/1999]
  Do you have a proof of the equation e^(i*Pi) + 1 = 0?

• Curl of a Vector Field [Arumainathan, 05/22/1999]
  What does the term curl mean in math, and how do we define it in terms of fluid mechanics?

• Triangle Area [Dermot, 05/24/1999]
  How do you find the maximum area of a right triangle?

• Integration of Trigonometric Functions [Corenne, 05/28/1999]
  How can I integrate sin(x)sin(2x).dx ?

• Minimizing an Enclosed Area [Miller, 06/02/1999]
  How can I determine the position of a point P on the line y = 2 such that the line segments OP and PQ and arc OQ of the curve x = y^2 enclose minimum total area?

• Examples of Cooling, Growth and Decay [rangwalla, 06/03/1999]
  Can you give some real life problems involving Newton's cooling method, logistic growth, and exponential growth and decay?

• Center of Mass of a Semicircle [Abdou, 06/14/1999]
  Is there a standard formula I can use to know where the center of mass of a semicircle is?

• Plane Flying Parametrically [Jason, 06/28/1999]
  How can I find out when a plane, whose position approaching an airport is described parametrically by P_t = (1000,500,900)+ t[-100,-50,-90], will be closest to the traffic control center, located at (24,11,13)?

• Proving That d(e^x)/dx = e^x [Kissane, 06/30/1999]
  How can I find the derivative of y = e^x, using the definition of the derivative?

• Maximizing the Volume of a Box [Donald, 07/09/1999]
  If I have a rectangular card of size AxB, how can I find the size of the square cutout that maximizes the volume of the box produced when the edges are folded up?

• Riemann Sums and Antiderivatives [Kneezel, 07/11/1999]
  Are there any other ways of illustrating the link between Riemann sums and antiderivatives?

• Moving Particle [Peterson, 07/14/1999]
  Assume a particle moves on the x-axis according to the formula x = t^ 3-6t^2+9t+5. Find: the velocity when t = 3...

• Permutation and Combination Equality [Carlos, 07/18/1999]
  Prove that (nC0)^2 + (nC1)^2 + (nC2)^2 + ... + (nCn)^2 = (2nCn), where nCi = n!/((n-i)!*i!).

• Finding the Derivative [Kayla , 07/27/1999]
  Find dy/dN if y= A*(1 - e(-cN))* (1 - e(-cM)).

• Formula for the Trapezoid Rule [Lester, 07/31/1999]
  I am writing a program to find the moment area of any shape...

• Work Needed to Lift a Leaky Bucket [Allison, 08/01/1999]
  How can you calculate the amount of work needed to lift a bucket of water to the top of a well if there is a hole in the bucket causing water to leak out at a rate proportional to the height of the water in the bucket?

• Epsilon/Delta Definition of Limits [Barton, 08/26/1999]
  Can you explain how to use the epsilon/delta definition of limits?

• Monkey Climbing a Chain [Huma, 09/22/1999]
  A 20-lb. monkey is attached to a 50-ft. chain that weighs 0.5 lb. per (linear) foot...

• Finding Tangents to Curves Using Derivatives [Eshaque, 09/29/1999]
  Three problems that involve finding the tangents of a curve using calculus.

• Maximum Volume of a Box [Leahy, 10/21/1999]
  I am making a box out of a sheet of paper by cutting squares out of the corners. How can I show that the largest possible volume occurs when the side of the square is 1/6 the length of the sheet of paper?

• General Solution of dy/dt + y = t [Wilson, 10/25/1999]
  How can I find the general solution to the equation dy/dt + y = t?

• Stationary Point [Lee, 10/27/1999]
  How can I show that the curve y = x*ln(2x-3) has no stationary point in the region x greater than 2?

• Derivative of e^x [Reb, 10/28/1999]
  How can I prove that the derivative of e^x is e^x?

• Minimizing the Length of a Crease [Johnston, 11/11/1999]
  How can I minimize the the length (L) of the crease if one of the corners of an 8.5" x 11" sheet of paper is lifted up and placed on the opposite longer edge, and then the paper is smoothed flat?

• Minimizing the Time to Rescue [Palmquist, 11/29/1999]
  A man is drowning in the water 300 meters down the beach and 100 meters out from where I'm standing. I can run 5 meters/second and swim 3 meters/second...

• Newton-Raphson Method [Dyer, 02/28/2000]
  How can I find the 1st, 2nd, 3rd iterations and the parameters of x^3 - 13.1x^2 + 48.48x - 46.62 using the Newton-Raphson method?

• Minimum and Maximum Pollution [Dailey, 04/26/2000]
  Find the points of minimum and maximum pollution between two plants releasing 60 and 240 ppm of pollution...

• Proof that 2 Equals 1 Using Derivatives [Aldiyyate, 05/10/2000]
  If d(x^2)/dx = d(x+x+...+x)/dx, then 2x = 1+1+...+1 = x. What is the error in this "proof"?

• Finding a Vertical Asymptote [Connor, 05/11/2000]
  How can I find the vertical asymptote of the equation xy^2 - x^3y = 6?

• Carbon Dating the Shroud of Turin [Sullivan, 05/16/2000]
  In 1988, scientists found that the Shroud of Turin contained 91% of the amount of carbon-14 contained in freshly made cloth of the same material. How old is the Shroud according to the data?

• Heat Supplied and Heat Loss of a Furnace [Lehman, 05/17/2000]
  Heat is supplied by a furnace at a constant rate, but is lost at a rate determined by the difference between its temperature and the temperature of its surroundings...

• Minimizing the Surface Area of a Can [Donges, 05/22/2000]
  What coke can dimensions would use the least amount of aluminum while still holding 375 ml?

• Generalized Cubic Equation [Samantha, 05/22/2000]
  How do we find a generalized cubic rule for a family of cubics that cut through the point (1,1), have a gradient of 2 when x = 1, and have 1 stationary point?

• LaGrange Error for a Taylor Polynomial [Garfield, 05/28/2000]
  How can you find the LaGrange error for a Taylor polynomial?

• Determining Whether a Function is Continuous [Keenan, 06/10/2000]
  How can you tell whether or not a function is continuous?

• Power Series for Sine and Cosine [Mark, 10/12/2000]
  Can you explain, without using calculus, how to get the power series for sine and cosine?

• Differentiating the Ceiling Function [Raymond, 10/16/2000]
  How do I find dD/dp for an equation involving the ceiling function? How do I minimize D as a function of P?

• Simpson's Rule for Cubics [Stratis, 10/23/2000]
  Why is Simpson's rule exact for cubic polynomials?

• Surface Area and Volume Derivative [Mike, 10/30/2000]
  For what 3D figures is the derivative of the volume formula equal to the formula for surface area? With respect to which variable would you need to differentiate?

• Deriving the Gamma Function [Garcioparra, 12/15/2000]
  How can you prove that sqrt(pi)/2 = (1/2)!, and what is a fractional factorial like that equal to?

• Implicit vs. Explicit Differentiation [Fraser, 01/04/2001]
  What is the difference between implicit differentiation and explicit differentiation?

• Assigning a Piecewise Function in a Proof [Bhat, 01/11/2001]
  Can you explain how we can arbitrarily assign e(u) = 0 when u = g(c) in this chain rule proof of composite functions?

• Differentiating Under the Integral Sign [Mecholsky, 01/11/2001]
  Can you give me an example of the integration method called "differentiating under the integral sign"?

• A Brief Overview of Calculus [Pat, 01/16/2001]
  I am required to take a calculus course, but I have no experience with it. What should I concentrate on?

• Optimization (Min-Max) [Shah, 01/21/2001]
  If a piece of string of fixed length is made to enclose a rectangle, show that the enclosed area is greatest when the rectangle is a square.

• Integral of e^xsinx [George, 01/23/2001]
  I always get e^xsinx or e^xcosx to integrate, and it just keeps going. Could you show me the steps to this indefinite integral?

• Laplace Transform [Ronald, 01/25/2001]
  What are Laplace transforms, and what are their applications?

• Lagrange Multipliers [Li, 01/28/2001]
  The temperature of a point(x,y,z) on the unit sphere is given by T(x,y,z)=xy+yz. Using Lagrange multipliers, find the temperature of the hottest point on the sphere.

• Explanation of L'Hopital's Rule [Ward, 02/06/2001]
  I understand the method, and I somewhat understand the proof, but can you give me an explanation of why L'Hopital's Rule actually works?

• Proof of Integration by Parts [Kamrul, 02/06/2001]
  How do you derive the formula for integration by parts?

• The Riemann Sum of Sin(x) [Kaldoo, 02/07/2001]
  How do you evaluate the integral of sin(x) from 0 to pi by Riemann sum?

• Non-Constant Functions [Bishop, 02/12/2001]
  Are there any non-constant functions f and g such that (f*g)' = f'*g'?

• Understanding the Need for Limits [Josh, 02/19/2001]
  Can you explain limits in a way that makes the need for them clearer? Do limits apply to all functions, or are they only useful for certain ones?

• Integrals versus Antiderivatives [Orseske, 02/24/2001]
  What is the difference between an integral and an antiderivative?

• Limits of Sequences [Kapoor, 02/25/2001]
  Is the limit of [(1 + 1/sqrt(n))^(1.5n)], as n goes to infinity, e? What is the limit as n goes to infinity of [(1 + a/n)^n], where a is not equal to 0?

• Why Should I Study Calculus? [Dallek, 03/04/2001]
  I am starting an AP calculus class next year and I would like to know why I should study calculus.

• Integration Questions [Michelle, 03/09/2001]
  Calculate the area of the finite region bounded by the curves y=x^2 and y=sqrt(x)...

• Jerk - Derivative of Acceleration [Liz, 03/16/2001]
  I need a mathematical term that begins with the letter J, its definition, and a high-school-level explanation of the term.

• Partial Derivatives [Neeraj, 03/24/2001]
  What sense do partial derivatives make in the case where u is given as a function of two variables, say x and y...?

• Derivative of Tan x [Tong, 03/31/2001]
  Can you show me how to get the derivative of tan x using this definition of derivative...?

• Finding the Shape of a Funnel Curve [James, 04/08/2001]
  Find the equation of the curve of revolution of a coin leaving a chute.

• Possible Proof That 1 + 1 Does Not Equal 2 [Desmond, 04/19/2001]
  I just want verification... would this proof work?

• Summation Formulas for Trigonometric Functions [Aaron, 04/24/2001]
  How do you find the area under the curve of a trigonometric function using the definition of a limit and not an approximation? Are there summation formulas for trigonometric functions?

• Deriving Trig Functions; Taylor Series [Harold, 05/01/2001]
  How would I find, from first principles - no tables, no calculator - for example, 32 degrees? If I use a formula, how is it derived?

• Moving a Circle on a Polar Graph [Mike, 05/04/2001]
  How do you move a circle so the center is not (0,0), but to where the center can be (r,[theta])?

• Volume Using Cross Sections [Purvi, 05/28/2001]
  Find the volume of the region between y = |x| and y = -|x|+6 with cross sections that are equilateral triangles and perpendicular to the x-axis.

• Chain Rule [David, 05/30/2001]
  Prove the quotient where u/v is uv^-1 using the product rule and the chain rule for (dv^-1/dx...

• Goat and Silo [David, 05/31/2001]
  A goat is roped to a point A next to a silo 30 ft. in diameter. The rope is 15pi feet long. How much area can goat graze?

• 1 = 0 Fallacy [Dick, 06/22/2001]
  Here is a 'proof' we ran up against during high school in the Netherlands that makes use of integral calculus.

• Tangent without a Calculator [Martin, 06/24/2001]
  How do I find a tangent value if I have only an angle, and can't use a calculator?

• Deriving the Integral for the Surface Area of a Sphere [Chad, 06/28/2001]
  I need to find the cost per square foot of steel that makes up a tank, so I need a way to derive the surface area of part of a sphere with a given diameter...

• Area of an Ellipse [Tate, 08/03/2001]
  Show that the area enclosed by the ellipse x^2/a^2 + y^2/b^2 = 1 is pi*a*b, where a is the longer horizontal part and b is the shorter vertical part of an ellipse.

• Integrate Cos 2a [Katherine, 08/15/2001]
  The answer given in the book for integrating cos 2a is sin 2a/2. Does this follow a pattern?

• What is dx? [Michael, 08/25/2001]
  What does dx mean and where does it come from?

• Determining the Length of a Coil of Ribbon [George, 08/31/2001]
  How will calculus give me the same answer I found with algebra and geometry?

• What is an Integral? [Natasha, 09/06/2001]
  I've never really understood integrals and how exactly they work. Could you please explain?

• Graphing Limits [Justin, 09/12/2001]
  I was hoping you could explain the concept of "limits" and how to read them from graphs.

• Derivative of a Function [Komol, 10/05/2001]
  I am to find the derivative of the function (x-6)(x+1)/(x-6) at x=6. I simplified the function to a linear function x+1 with a "hole" at x=6. Then I tried to take the derivative.... but my calculator says division by zero, infinite result...

• Volume of a Right Circular Cone [Jeffrey, 10/07/2001]
  Using calculus, derive the formula for the volume of a right circular cone with a radius of r and height h.

• Alternate Forms of the Derivative Formula [Dina, 10/06/2001]
  Why does f'(a) = limit as x approaches a of [f(x)-f(a)]/(x-a), and how do you use this formula to find the derivative? I get confused about when to plug in x for a or a for x.

• Floor and Mod Functions Discontinuous [Alkhaled, 10/10/2001]
  What is f(x) = x mod b for some fixed b?

• Implicit Differentiation [Gary, 10/15/2001]
  We have not been able to show that (1+y)/(1-x) = 0... Where have we gone wrong?

• Nonvertical Asymptotes [Amrita, 10/16/2001]
  Does this problem have slant asymptotes? f(x)=(x^(4/3)+x^(1/3)-2)/(x^ (4/3)-16). I'm also having trouble taking the derivative of this problem: f(x)=((x^2(x+1))-(1-x))^(1/2).

• Find the Path and the Time Taken [Mike, 10/17/2001]
  A man wants to cross a river 500m wide. His rowing speed relative to the water is 3000 m/hr. The river flows at a speed of 1200 m/hr...

• Radian vs. Degree, Sin Derivatives [Liz, 10/23/2001]
  Why do you get a different answer when you ask your calculator for the derivative of sinx in degree rather than radian mode?

• Calculus Starting Points [Robert, 10/23/2001]
  I want to learn calculus because I have heard that it is important for applied math in engineering and I want to be an engineer. What should I learn, and in what order?

• Derivatives and Antiderivatives [Robert, 10/31/2001]
  When I run the formula int(sqrt(cos(x^2)),x=0..1) in Maple, it just returns the formula. Why?

• Second Order Differential Equation [Roger, 11/04/2001]
  By letting p = dy/dx, solve y(d2y/dx2) = 2 (dy/dx)2.

• Inner Product and L-2 Distance [Queenie, 11/14/2001]
  Why is "distance between two functions" calculated by multiplying f(x) and g(x) and then integrating with respect to x within the defined domain?

• Proof of Series ln(1+x) [Colleen, 11/15/2001]
  I need to show that the series ln(1+x) equals x-x^2/2+x^3/3-x^4/4, and so on, whenever x is between -1 and 1.

• Integrating 1/(1+x^n) [Varghese, 11/14/2001]
  Are there methods of integrating functions of the form 1/(1+x^n)? I have solved 1/(1+x^4) and am wondering about solutions for higher powers of x.

• The N Targets Problem [Jocelyn, 11/15/2001]
  Given n targets with equal probability of being hit by a cannon, that is, 1/n. Let X be the number of firings required to hit all the targets. What is the expectation E(X) of X?

• Proof by First Principles [Ewart, 11/16/2001]
  How can I prove by first principles that the derivative of tan(x) is sec^2(x)?

• Trying to Integrate f(x) = exp(-ax^2) [Craig, 11/23/2001]
  How would you integrate a function of the form f(x) = exp(-ax^2)?

• Taylor Series Expansion [Don, 11/24/2001]
  A distance from A to B is 1000 meters. As one traverses it at 1 meter per second, the distance is instantaneously and uniformly stretched 1000 additional meters. How long does it take to get from A to B?

• Proof of Cosine of 36 Degrees [John, 11/26/2001]
  Prove: cosine {(pi)/5} = {1+5^(1/2)}/4.

• Longest Ladder [Michelle, 11/30/2001]
  Two hallways, one 8 ft. wide and other 4 ft. wide, meet to form a right angle. What is the longest ladder that can go around the corner where the hallways meet?

• Oil Can Dimensions [Shannon, 12/11/2001]
  What are the dimensions of an oil can with a one-liter capacity that uses the least amount of tin?

• Maximum Area of Inscribed Triangle [Shawn, 12/10/2001]
  An isosceles triangle is inscribed in a circle of radius R. Find the value of Theta that maximizes the area of the triangle.

• Melting a Cube [Asma, 12/20/2001]
  How long does it take for an ice cube to melt?

• Ratioing Errors [Rich, 01/07/2002]
  How do you calculate the error bars for ratios?

• Find the Density Function [Tanya, 01/24/2002]
  Suppose X1, X2, X3 are independent and uniform on(0,1). Find the density function of S = X1 + X2 + X3.

• e^(e^x)) = 2 [Elizabeth, 01/28/2002]
  I have been trying to solve e^(e^x)) = 2. Help!

• Why Is dx Necessary? [Lee, 01/31/2002]
  In integral notation where you have one variable after an integral sign you must always end with the change in your variable (i.e.: INT[] x^2 dx). Why is the 'dx' necessary?

• Generating the Volume of an n-Sphere [Elara, 02/07/2002]
  Could you help me find an equation for the integration of a unit-ball in n-dimensions?

• Constant Function Zero and Orthogonal Functions [Neeraj, 02/08/2002]
  The constant function f(x) = 0 will give zero no matter what function g(x) it is integrated with. Does this mean that the constant function zero is orthogonal to all functions? Also, what could be the geometrical interpretation of orthogonal functions?

• Maximum Area for Irrigation Channel [Ingrid, 02/08/2002]
  An irrigation channel made of concrete is to have a cross section in the form of an isosceles trapezoid, three of whose sides are 4 feet long. How should it be shaped to have a maximum area?

• Limits Approaching Infinity [Miranda, 02/16/2002]
  Is the only way to answer them by memorizing all the general questions you may come across?

• Best Fit Quadratic Curve [Avin, 02/21/2002]
  Given a scatter plot of X and Y cordinates, find the best-fitting quadratic curve.

• Frustum of a Pyramid with a Rectangular Base [Brad, 02/20/2002]
  I am an engineer with a water treatment agency and need to figure the amount of water per foot of elevation in our reservoirs...

• Derivative of an Odd Function [Sheri, 02/27/2002]
  How do you show that the derivative of an odd function is even?

• Finding a Limit [Corrie, 02/28/2002]
  My calculus teacher gave my class the answer to this limit; however, we cannot figure out how to get the answer that is given.

• Integrating 1/(sinx)^6 dx [Ehab, 02/25/2002]
  What is the integration of 1/(sinx)^6 dx?

• Tan pi/8 [Stepheny, 03/16/2002]
  I must find the exact value of tan pi/8.

• Quotient Function Mnemonic [Ryan, 03/11/2002]
  Help me to differentiate the following: x/((x^2)-4).

• Centroid of an Arc [Derrick, 03/20/2002]
  How do I find the centroid (or centre of gravity) of an arc of a circle?

• Balance Point between Converging and Diverging Infinite Series [Jim, 03/20/2002]
  Is there a way to find the balance point between convergence and divergence for any type of series - that point where the n(th)term gets smaller JUST fast enough for the series to converge?

• Deriving Arctan without Using Numbers [Matt, 03/22/2002]
  How do you do you derive arctan without using numbers, or how do you prove (arctan)' = 1/1+x^2?

• Proofs of e [Christine, 03/21/2002]
  I found out that the definition of e is: e = lim, as n approaches infinity (1+1/n)^n. Is there a proof for this or for 1+2(e-1/n)?

• Rate of Change Constant? [Mariam, 03/20/2002]
  The rate of change of e^x is e^x. Does this mean that the rate of change is constant? Why are sinx,cosx,... and sinhx,coshx,.... similar?

• Fresnel Integral of sin(x^2) [Robert, 03/28/2002]
  I can not see how this integral from 0 to infinity can have any limit at all. If we used infinite series, we would not get a value for x = infinity.

• Oblique Asymptotes [Jo, 04/03/2002]
  I'm stuck on how to calculate oblique limits, for example, x^2/(x-3).