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Browse College Calculus

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Compactness in Town Planning [07/22/1999]
I need to find the area in terms of the longest chord of an n-gon with an odd number of sides, and must prove that my formula produces a maximum using second derivatives.

Comparing x^y and y^x [10/17/2002]
Given x greater than y greater than e, is y^x greater than x^y or is x^y greater than y^x (or neither)?

Considering a Perturbed Function F_eps(x) [11/20/2009]
Doctor Vogler outlines how to estimate a(eps) - 5 by using a Taylor approximation, differentiation, and some strategic factoring.

Constant Function Zero and Orthogonal Functions [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?

Continuous but Not Differentiable? [12/27/2003]
I have heard that Weierstrauss found a function that is continuous everywhere, but differentiable nowhere. How is that possible?

Convergence of Cos(x^2) or Sin(x^2) [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?

Cost of Fencing a Yard [02/24/1997]
Find the minimum cost to fence a yard that has an area of 500 square meters.

Curve Fitting Algorithm [6/20/1996]
Can you explain "least squares approximation"?

Definition of Cartesian Tensor [12/18/1998]
What is the precise definition of a Cartesian tensor?

Definition of the Limit [11/05/2002]
I need some help with the definition of the limit, particularly choosing delta for a given epsilon.

Degree of a Rational Function [06/19/2003]
Given: f(x) = p(x)/q(x). Prove that if the highest degree of p(x) is smaller than the highest degree of q(x), the horizontal is y=0.

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

Delta-Epsilon Proofs and Arbitrary Epsilon Choice [11/05/2002]
What is the nature of the special relationship between a function's behavior near a point, the choice of epsilon, and a corresponding 'good' delta ?

Derivation of Geometric Formulas [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?

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

Derivative of e^x by Limit Definition [05/21/2002]
Show that the derivative of e^x is e^x, using the limit definition but not the Taylor series expansion.

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

Deriving the Area of a Sphere [10/21/2003]
I know the area of a sphere is 4phi(r^2), but I'm wondering how to derive that formula. I know it should be done in cylindrical coordinates, and I'm thinking that the arc of a circle is defined as rd(theta) and it's multiplied with rd(phi) to get (r^2)d(theta)d(phi). Could you please help explain this?

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

Deriving the Integral for the Surface Area of a Sphere [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...

Determining and Checking Saddle Point for Autonomous System [06/18/2004]
Given an autonomous system defined by a pair of differential equations, I know how to find critical points and the Jacobian matrix, but how do I determine if the critical point is a saddle point?

Differential equation [12/14/1994]
I am trying to solve the differential equation: dy/dx=2y(root of (y^2-1)) at x=0 and y=2

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

Differential of an Equation [01/02/2003]
How do I calculate the differential of an equation, to be used in finding the minimum value of the curve the equation describes?

Differentiating a Nested Function [05/21/2000]
If g'(8) = 5, how can I find d/dx g(x^3) at x = 2?

Differentiating the Ceiling Function [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?

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

Double Integration in Polar Coordinates [03/27/2003]
Evaluate double integral x-y/x*2+y*2 over x*2+y*2 equal to or less than 1.

Ellipsoids [1/12/1995]
I am looking for an analytical method, or a proof that an analytical method does not exist, for the following: 1. Partial Surface Area of an Ellipsoid, 2. Partial Volume of an Ellipsoid, 3. The Intersection Volume of Two or More Ellipsoids.

Elliptical Arc Length [11/17/1999]
Is there a series approximation or some other numerical method for calculating the arc length of an ellipse?

Endpoints of Intervals Where a Function is Increasing or Decreasing [05/01/2009]
Why do some calculus books include the endpoints when determining the intervals in which the graph of a function increases or decrease while others do not? I feel that the intervals should be open.

Escaping L'Hopital's Loop [10/14/2003]
I'm trying to apply L'Hopital's rule, but I keep ending up with 0/0 after each application. What am I missing?

Exactness Criterion for Differentials [06/14/1999]
What "exactness criterion" would be applied to the coefficients of the differentials dT and do in the equation dS = C(dT/T) + a(do)?

Example of the Runge-Kutta Method [02/10/2001]
Using the Runge-Kutta method to solve ordinary differential equations of the form y'(x) = f(x,y).

Find a Function, Integrals [12/17/1997]
Suppose the graph of f has the formula f(x)=-x+1 for 0<=x<=1; x-1 for 1<x<=2 ...

Finding a Definite Integral with Natural Log and Sine [05/14/2008]
I'm trying to evaluate the definite integral of ln(sin(x)) dx from 0 to pi. I've tried integration by parts but am not getting anywhere.

Finding a Function to Split the Area between Curves [01/28/2006]
What value of m would cause the function g(x) = mx to divide the region bounded by f(x) = x - x^2 and the x-axis in the first quadrant into two regions of equal area?

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

Finding Extreme Values for Polynomials [7/6/1995]
Find the smallest possible value of the quantity X^2 + Y^2 under the restriction that 2X + 3Y = 6.

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

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