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 TOPICS This page:   calculus    Search   Dr. Math See also the Internet Library:   calculus HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry Browse High School Calculus Stars indicate particularly interesting answers or good places to begin browsing. Selected answers to common questions:     Chain rule.     Maximizing the volume of a box.     Maximizing the volume of a cylinder.     Volume of a tank.     What is a derivative? Tipped and Partially Filled Frustum [12/14/2003] A vessel in a plant where I work is the frustum of a cone on its side. A liquid is contained in this section and pours out the end of the cone section, therefore the liquid only takes up a certain portion of the cone's volume. How can I compute the volume of the liquid? Triangle Area [05/24/1999] How do you find the maximum area of a right triangle? Trigonometric Functions and Euler's Identity [04/10/1998] We need to prove three trigonometric functions in various ways. Trigonometry and the Chain Rule [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? A Trigonometry Integral Requiring Two Substitutions [05/03/1998] Substituting twice and using trig to integrate sqrt(1 + sin(x)). Triple Integrals [12/07/1997] Determine the limits... The Troublesome Endpoints of a Trigonometric Series [04/10/2011] A student has doubts about the radius of convergence assumed in a proof that involves integrating a power series. Doctor Vogler confirms his suspicion about the behavior of endpoints of integrated series, and introduces Abel's Theorem. Trying to Integrate f(x) = exp(-ax^2) [11/23/2001] How would you integrate a function of the form f(x) = exp(-ax^2)? Tuning a Piano [05/28/2002] Each piano has a unique tuning curve that can be approximated through 3-point quadratic interpolation. Is there a formula that will allow me to measure three different notes on the piano and find the tuning curve in between these notes? Two Points Moving... [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... Two Somewhat Similar Problems [1/9/1995] I need help with two similar limit of integral problems. Two Ways to Find a Formula [04/14/1998] I need to show that Sigma(rx^r) = (x-(n+1)x^(n+1)+nx^(n+2))/(1-x)^2. Two Ways to Integrate [04/15/1998] How do you integrate: Int[sin^4(x)*cos^6(x)]dx? Undefined and Indeterminable ... at the Same Time? [09/05/2010] A student wonders whether the labels "undefined" and "indeterminate form" could apply to one and the same expression. Doctor Vogler considers several expressions, functions, and limits to distinguish the different contexts that call for such terminology. Understanding the Need for Limits [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? Unit and Basis Vectors in Three Dimensions [05/09/1998] Explanations and uses of unit vectors and basis vectors. Using Differentiation to Find Derivatives [11/16/2002] Our class would greatly appreciate it if you could simplify the whole idea of using differentiation to find derivatives, and give a real-world example. Using L'Hopital's Rule [01/22/1999] Prove that e^(-1/3) is the limit for y =(arctan(x)/x)^(1/(x^2)) when x -> 0. Using Newton's Method to Solve an Implicit Equation [10/14/2007] How can I solve the equation (sin(X))/X = 0.7031? My book gives the answer but does not explain how to find it. Using the Chain Rule: Define Functions [5/2/1995] What are the integrals of sin(ln(4x+5)) and ln(sinx) ? Using the Chain Rule: Finding a Derivative [2/26/1996] d/dx sin(sin (2cos(2x/3))) = ? Using the Definition of Derivative [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. Using the Power Rule to Find the Derivative [9/24/1995] We are trying to prove that the "power rule" works for finding the derivative of x^(3/2) (where x>0). Vector Analysis [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... Vector Calculus [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... Vector calculus [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. Vector Calculus [02/10/1999] Prove the following identity: a . (grad(a . v) - curl (v x a)) = div v... Vector Equations Of Lines [6/28/1996] A boat leaves a harbour, O, position vector (0i+0j) at 9 a.m... at what time will it arrive? Visualizing 1 + 1/x [10/10/2003] Show that the sum of a positive number and its reciprocal is at least 2. Volume of a Cone or Pyramid [03/30/1998] Proofs that the volume of a cone or pyramid is (1/3)b*h. Volume of an Elliptical Cone [10/15/1998] Can you help me on find the volume for an elliptical cone by using a triple integration? Volume of a Right Circular Cone [10/07/2001] Using calculus, derive the formula for the volume of a right circular cone with a radius of r and height h. Volume of a Rotated Region [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. Volume of a Shape [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). Volume of a Solid [10/02/1997] The base of a solid is the region inside the circle x^2 + y^2 = 4... Volume of a Sphere [04/21/1998] Can you help me derive and prove the formula for the volume of a sphere? Volume of a Torus [04/22/1999] I would like to know how to find the volume of a torus using integrals. Volume of Intersecting Pipes [10/27/2000] How can you calculate the volume of the intersection of two perpendicular pipes of the same radius? Volume of Spherical Cap [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. Volume Using Cross Sections [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. Page: []

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