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Browse College Calculus

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Calculus Problem [11/29/1994]
I'm a senior in a college calculus class. The following problem was assigned, but the next day, after no one had completed it, our professor attempted to do it but quit after 25 minutes. Problem: Find the equation of the line tangent to the ellipse b^2*x^2 + a^2*y^2 = a^2*b^2 in the first quadrant that forms with the coordinate axes the triangle of smallest possible area (a & b are positive constants).

Counting Extrema [10/23/1996]
Find the number of minima and maxima of the function [Sin Na/Sin a]^2 on the interval [0, pi].

Dirac Delta Function [05/15/2000]
What is Dirac's delta function, and how is it used?

Elliptic Integrals [12/13/2000]
Exactly what are elliptic integrals?

Gabriel's Horn [2/15/1996]
How is it that the surface area of the horn is infinite while the volume is finite?

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

Vector Calculus: When Should the Enterprise Fire? [7/21/1996]
If F represents the focus of the ellipse and P is the position of the starship Enterprise on the ellipse... you must fire the engines when the ratio vector T/vector N is equal to the eccentricity of the ellipse.

Volume of Liquid in the Bottom of a Tank [08/11/2003]
I need to be able to calculate the volume of stored liquid in the parabolic bowl of a cylindrical vertical tank based on the liquid level.

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

0 Raised to a Power [04/25/2002]
What is the limit of 0^n as n approaches infinity? Can you explain why it equals zero or why it can't equal zero, or if it's undefined?

Applied Max/Min Problems [2/11/1996]
Find the largest possible volume of a right circular cylinder that is inscribed in a sphere of radius r.

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

Area and Center of Gravity for a Crescent [9/5/1995]
I need to calculate the area and center of gravity of a crescent: take a 10.00" diameter circle center at x,y 0,0 and add a 9.00" diameter circle center at x,y 2,0. The crescent is formed in 2 areas that are not overlapping.

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

Area of an Irregular Polygon [03/10/1997]
Given the measures of the angles and sides of an irregular polygon, how do you find its area?

Area Under the Curve y = x^0 [02/14/2005]
Sketch the graph of y = x^0, where x e [0,1], and calculate the exact area.

Arithmetic and Geometric Means [11/05/2000]
How can I use Lagrange multipliers to prove that the geometric mean of three positive numbers is less than or equal to the arithmetic mean?

Arterial Branching Calculus [05/28/2003]
In the surgical connection of a small artery to an existing larger artery, attention must be paid to minimising the viscous resistance to the blood flow... Investigate the distance from B of the join and hence the angle of attachment that minimises the resistance to flow.

Assigning a Piecewise Function in a Proof [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?

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

Bernoulli's Equation [12/7/1995]
I'm trying to solve the following equation: y' = a*y-b*y*y*y, y(0) = y_0

Boundary Value Problem [11/02/2002]
A boundary value problem in second-order linear homogeneous differential equations.

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

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

Calculus (Area of a Plane Region) [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.

Calculus of Piecewise Functions [06/07/2003]
Can I take the integral or derivative of a piecewise function like the floor function [u] or the absolute value function |u| and still notate it in concise form, |U| or [U]?

Can dy/dx Be Treated as a Fraction? [08/26/2004]
I learned that dy/dx was a notation that implied 'derivative of y with respect to x.' But in some calculus work, dy and dx seem to have individual meanings and dy/dx is treated as a fraction. How is this possible?

Cardan's Cubic Method [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....

Cases Where the Newton-Raphson Method Fails [06/30/2005]
Why does the Newton-Raphson method work for some functions but not for others?

Catching a Pig [05/09/1997]
A farmer wants to catch a pig. They start running at the same time. The pig runs straight ahead and the farmer always runs towards the pig. How long does it take the farmer to catch the pig?

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

Cauchy Principal Value [05/23/2000]
Can you tell me what the Cauchy Principal Value is?

Cauchy Principal Value Integral [09/17/2002]
Why isn't an integral from negative infinity to positive infinity defined as the limit as t (or any other variable) approaches infinity of the integral from -t to t?

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

Chain Rule Applied to Exponential Functions [10/11/2004]
How do I use the Chain Rule to find the derivative of a function such as f(t) = 27 e^(-.14t)?

Challenging Trig Expression Integration [04/12/2008]
How can I integrate the function (sinx)^3/[(sinx)^3 + (cosx)^3]?

Changing Angle of a Tank [06/11/2003]
Points A and B represent pressure sensors in fixed positions on the base of a round tank. The chord through CD represents the water level in the tank. Lines a and b are the heights of water registered by each sensor...

Choosing Factors When Integrating by Parts [10/08/2003]
When integrating by parts, how do I decide what factors to use?

Classifying Cubics into Three Basic Types [08/11/2002]
How can I classify each of y=x^3+x^2, y=x^3-2x^2, y=x^3-2x^2+x, y=x^3- 2x^2+2x, and y=-x^3+x^2-x into one of three basic types without actually drawing their graphs?

Closest Point to Origin in a Plane [11/04/2004]
In the plane x + y + 3z = 7, my book says we find the closest point to the origin by [d/[n]^2] * n. I don't understand WHY this operation gives us the closest point. I'd appreciate it if you can help explain how it works.

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