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Browse College Calculus

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Integration by Partial Fraction Decomposition [06/24/1999]
How can I find the integral of dx/(x^4+x^2) or the integral of xdx/(x^2+2x+1)?

Integration Methods Beyond 3 Dimensions [11/12/1996]
How would you calculate the volume of a hypercube or a hypersphere?

Integration of a Trigonometric Function [06/10/1999]
How to integrate the function arctan(sqrt(1-x^2)).

Integration of ... Given dx/dt [7/5/1996]
How do I integrate da/dt = k.(a)^m.(1-a)^n where (a) is a fraction....?

Integration of Sin(x^2) [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!

Integration of y = e^(-x) [06/01/1999]
How can I show that the point (n+1,0) lies tangent to the curve y = e^(- x)? And how can I find the area of a region under that curve?

Interesting Differential Equation [06/18/2004]
Is there a nontrivial function f(x) such that df/dx = f(f(x)) ?

Interpreting Lagrange Multipliers [04/13/2001]
What does it mean when I get different values for lambda when using Lagrange multipliers to find the maximum and minimum values of the function f(x,y) = 6x^2 + 9y^2 subject to constraint x^2 + y^2 = 1?

Inverse Laplace Transforms [08/21/1997]
Are there any websites on this topic?

Inverse of a Multivariate Function [05/30/2002]
Let f:NxN -> N such that f(x,y) = 2^x(2y + 1) - 1 for all natural numbers x, y. Let the inverse of f, g be given by g:N -> NxN. Find the inverse of the function g.

Inverting Functions [07/19/2002]
To find the inverse of a function y=f(x), do I interchange the variables x and y, or do I solve for x in terms of y?

Jacobian Matrices in Transformations [08/05/1999]
Why do we need to set up the Jacobian matrix when doing transforms? What does the Jacobian determinant r mean geometrically?

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

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

Lagrange Multipliers [07/25/1999]
How can I find the extreme values of F(x,y) = xy + yz + xz using Lagrange multipliers?

Lagrange Multipliers [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.

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

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

Laplace Transforms [10/15/1997]
Can you explain me the overall concept of changing from Time-Domain to Frequency-Domain with a Laplace or a Z Transform?

Laplace Transforms [05/28/1997]
How do you solve L-1 {2(s+1)/(s^2+2s+10)} and L-1 {(9s^2+4s-10)/s(s- 1)(s+2)}?

Laplace Transforms [06/21/1997]
Can you explain the linearity of the Laplace transform? What's the point of this theorem?

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

Least Cost for Laying a Pipeline [06/21/1999]
How can I find the minimum-cost path for laying a pipeline from an off- shore oil platform to an on-shore refinery given the relative costs of underwater and on-land sections?

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

L'Hopital's Rule Explained in German [12/10/1999]
Can you tell me in German what the "Grenzwert" of 2^(n+1)-3/2^(n-1) is?

Limit of x sin(1/x) [04/23/2002]
I assumed from the graph that the function had a limit at x=0 of 0, but since it involves sin(1/0) I can not prove this using the basic trigonometric limits (sin x/x and (1-cos x)/x), L'Hopital's rule, or by rearranging the equation. Can you help?

Limits of Limits and Patterns in Higher Derivatives [11/11/2005]
By applying the derivative limit to the generic function f(x) several times in a row, I found some interesting patterns in higher derivatives, including an appearance of the binomial coefficients. Is my work correct?

Linear Interpolation Methods to Approximate Roots [12/25/2000]
What is the best criterion to use for ending the iteration process of linear interpolation methods for approximating non-linear equation roots? Also, which is the better explanation of the Newton-Raphson method; the geometric method, or using Taylor's expansion?

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.

Manipulating Limits [10/13/2003]
How do I use the limit definition to find the derivative at a point where the function becomes undefined?

Maximization of f(x,y) over a Constraint [03/04/1999]
Maximize f(x,y) = xy^2 over the ellipse, x^2/a^2 + y^2/b^2 = 1.

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 Quadrilateral Area [05/15/2001]
Given a quadrilateral with sides of lengths a,b,c,d, prove that its area is maximized when opposite angles are supplementary.

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?

The Meaning of 'dx' in an Integral [02/22/2002]
What meaning is attached to the 'dx' at the end of an integral?

Minimization [10/29/1996]
Ship A is 100 miles due east of ship B. At noon the ships set sail at different speeds in different directions. When will the ships be nearest to each other?

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

Minimum Distance Between a Line and an Ellipse [05/30/2000]
How can I find the extremum from a point on the line x+y=4 to a point on the ellipse x^2+4y^2=4? Is there a general way to solve such problems in a 3-dimensional space?

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