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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? Limits - Indeterminate Forms [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... Limits of Multi-Variable Functions [11/12/2004] Find the limit, if it exists, or show that the limit doesn't exist: lim (x*y*cos(y))/(3*x^2 + y^2) as (x,y) ==> (0,0). Limits of Sequences [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? Limits of the Natural Logarithm [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? Linearity and Concavity [12/07/2003] Why does a linear function have no concavity? Line Tangent to an Ellipse [03/29/2003] Find the equation of the tangent to the ellipse x^2 + y^2 = 76 at each of the given points: (8,2),(-7,3),(1,-5). Write your answers in the form y = mx + b. Logistic Growth of A Rumor Spreading [04/11/1998] Assuming logistic growth, find how many people know the rumor after two weeks. The Logistic Model for Population Growth [03/29/1998] I'm trying to solve the logistic model of population growth for P(t). Longest Ladder [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? Manipulating Limits [10/13/2003] How do I use the limit definition to find the derivative at a point where the function becomes undefined? Marginal Meanings [12/08/2016] Having calculated step-wise differences in output of the cost function, an adult wonders why those values disagree with output of the derivative of the same function. Acknowledging the ambiguities, Doctor Peterson makes sense of the approximations. Math Questions [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? Maximize Area of Trapezoid [07/28/2003] Given an isosceles trapezoid with three of its sides of length 10 cm, find the fourth side so that the area is maximized. Maximizing Area [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? Maximizing Oranges [06/29/2002] An orchard has 800 orange trees, each of which yields 120 oranges per season. For each new tree that is added to the orchard, the output per individual tree decreases by 2 oranges per season. Determine the number of trees that would maximize the orchard's output. Maximizing Revenues [07/29/2002] For every increase of \$1 in ticket price, 200 fewer people will attend each game. What ticket price will maximize revenues? Maximizing the Volume of a Box [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? Maximizing the Volume of a Box: Find Size of Square Cutout [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? Maximizing the Volume of a Cylinder [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. Maximizing the Volume of a Cylinder [7/24/1996] I need to maximize the volume of a right-circular cylinder that fits inside a sphere of radius 1 m. Maximizing the Volume of a Rain Gutter [10/21/2003] A rain gutter is to be constructed from a metal sheet of width 30cm, by bending up one-third of the sheet on each side by an angle (theta) from the horizontal (theta = zero represents the unbent sheet). Determine what theta should be chosen so that the gutter will carry the most water when it is full. Maximizing Window Area [02/24/1997] Maximize the area of a Norman window (rectangular with a semicircle on top) while minimizing the length of the perimeter. Maximum Area for Irrigation Channel [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? Maximum Area of an Arc [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. Maximum Area of Inscribed Triangle [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. Maximum Points and the Second Derivative [04/24/1998] When the second derivative of a point is a negative, is the point said to be a maximum? Maximum Surface Area for Total Edge Length [07/14/2002] A piece of wire of total length L units is used to form the nine edges of a prism whose ends are equilateral triangles and whose other faces are rectangles. What is the maximum surface area of this prism? Maximum Volume: Making a Box from a Sheet of Paper [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? Maximum Volume of a Box [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? Max/Min Applications of Derivatives [05/31/1998] I need help with these max/min applications of derivatives. If 40 passengers hire a train... Max/Min Problems with 3-D Shapes [03/23/2009] I am trying to find the dimensions of a cylindrical can that will use the minimal surface area for a given volume. I've done maximum and minimum problems with two-dimensional shapes. Does the process work the same way with three-dimensional objects? Meaning of Derivative [10/12/1996] What's a plain English meaning of the derivative? The Meaning of 'dx' in an Integral [02/22/2002] What meaning is attached to the 'dx' at the end of an integral? Mean Value Theorem [04/18/1999] Which number satisfies the conclusion of the Mean Value Theorem for f(x) = sin (x/2)? Melting a Cube [12/20/2001] How long does it take for an ice cube to melt? Methods of Approximation [7/31/1996] Is there an alternative to Newton's method of approximation to solve y^y = 25? Method to Integrate sqrt(1-x^2) [9/11/1996] What is the integral of [sqrt(1-x^2)]dx from [-1,1]? Minimizing an Enclosed Area [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? Minimizing the Cost of a Box [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. Minimizing the Cost of Phone Line Construction [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? Page: []

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