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Browse College Calculus

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Rate of Change of Clock Hands [08/11/1999]
The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o'clock?

A Reduction Formula and a Special Method [1/23/1996]
Asking for the integration of F(X).

Related Rates [10/16/2003]
A related rate problem involving a 2 cm long hair lying on a spherical balloon as the balloon is inflated.

Relative and Absolute Extrema of a Function [01/07/2004]
What is the difference between the absolute extrema and the relative extrema in calculus?

Relativity [05/22/2002]
How do you demonstrate algebraically that [the following expressions] are proportional to each other even when they describe 'events' that are not on the expanding wavefront of light?

Riemann Sums as Estimates of Definite Integral [10/11/2003]
The velocity of a car is given by v(t) = t^2 + 2t where v(t) is measured in ft/sec and t is measured in seconds. Estimate the distance traveled from t=1 to t=4 using 3 subdivisions. How accurate is this estimate?

Rocket Position and Velocity [04/12/2001]
A rocket is launched from the ground. Its acceleration is measured every 5 seconds. Find the velocity and position of rocket at t = 40 seconds, using the trapezoidal rule and Simpson's rule.

Rope between Two Poles [05/04/1999]
Show that the shortest length of rope occurs when angle PRQ = angle SRT.

Salt Concentration [01/23/1997]
One tank contains fresh water while another contains a salt solution. If the contents of each tank are pumped into the other tank, what is the concentration of salt in each tank at any time?

Shortest Distance Between Two Vectors [04/07/1998]
How do you find the shortest distance between two 3-dimensional lines?

Simpson's Rule for Cubics [10/23/2000]
Why is Simpson's rule exact for cubic polynomials?

Sinusoidal Output of Electronic Oscillators [04/28/2005]
Why does an electronic oscillator oscillate in a sinusoidal wave shape when electronic devices work in a linear region?

Slope of 3-Dimensional Equations [11/12/1997]
Is it possible to find the slope of three-dimensional equations?

Snail! [06/20/2002]
A snail is climbing a window-pane, beginning in the evening at a height of e minus 1 meter from the base. It loses 1 meter each night. On the second day, it doubles its altitude of the morning. On the third day, it triples the altitude of the morning, and so on. What will be its altitude on the 51st day at dawn?

Solving a Differential Equation [6/5/1996]
The integral looking somewhat like this: S ((e^u)/u) du has resisted every attack on my part.

Solving a Differential Equation by Series [06/13/1999]
How can I solve y' = 2xy by series?

Solving Double Integrals Using Symmetry [11/05/2004]
I'm very curious to learn how symmetry could be used to solve double integrals such as z = e^(x^2) - e^(y^2) over the domain [0,1] x [0,1]. I've heard that the method allows you to solve such problems quickly.

Solving Equations Numerically: Newton's Method, Bisection Method [11/05/2002]
Can you tell me methods to work out equations that can only be solved numerically and not algebraically, and suggest some good equations with which I could try out these methods on using graphs?

Solving for the Non-homogeneous Equation [10/21/2001]
I have the equation: x^2(d^2y/dx^2) - 3x(dy/dx) = 3x^3. How do I solve for the non-homogeneous equation?

Solving Functional Equations [08/11/2005]
How can I solve an equation like f(f(x)) + 5f(x) + 7 = x^2 + 6x?

Spread of a Virus Through a City [05/09/2002]
A Flu-like virus is spreading through a city of population 260000 at a rate proportional to the product of the number of people already infected and the number of people stilll uninfected.

Standard Deviation of Uniform Distribution [02/16/1998]
In the equation Std. deviation = (b-a)/ square root of 12, where did the square root of 12 come from?

Standard vs. Symmetric Derivative of Sin(x) [02/07/2006]
Can I find the derivative of f(x) = sin(x) by using the limit (sin(x+h) - sin(x-h))/2h as h->0 and then using a trig identity to simplify the difference of sines?

Stationary Point [10/27/1999]
How can I show that the curve y = x*ln(2x-3) has no stationary point in the region x greater than 2?

Strongest Possible Beam From a Log [07/11/2002]
Find the dimensions of the strongest beam that can be cut from a cylindrical log of radius 'a'.

Substituting y = vx in Differential Equations [02/16/2001]
How can I make the equation (y^2 - x^2)dy + 2xy*dx = 0 separable?

Summation Formulas for Trigonometric Functions [04/24/2001]
How do you find the area under the curve of a trigonometric function using the definition of a limit and not an approximation? Are there summation formulas for trigonometric functions?

Surface Area and Volume Derivative [10/30/2000]
For what 3D figures is the derivative of the volume formula equal to the formula for surface area? With respect to which variable would you need to differentiate?

Surface Area of an Ellipsoid [02/11/1997]
How do you calculate the surface area of an ellipsoid?

Surface Area of an n-dimensional Sphere [07/28/1997]
I was wondering how to calculate the surface area of a sphere in n dimensions.

The Surface Area of a Rotational Curve [04/17/1998]
I'm having difficulty understanding how my math professor proved the formula for the surface area of this curve...

Surface Area of a Sphere [10/03/1997]
How is the surface area of a sphere calculated, and why?

Surface Area of a Sphere [04/10/1998]
Can you derive the formula for the surface area of a sphere?

Surface Area of Solid of Revolution [05/21/2001]
I tried to derive the formula for the surface area of a cone by taking the integral the circumference of the solid of revolution, but it didn't work. What did I do wrong? Can the formula be derived using this method?

Surface Integrals of Vector Calculus [01/05/1999]
Find the SURFACE_INT(F dS) of the vector field F = (x^3, y^3, z^3) through the surface of the solid circular cylinder...

Tangent Lines and Odd Degree Polynomials [07/24/1998]
If p(x) is a polynomial of odd degree, determine whether every point in the plane lies on at least one line tangent to the curve y = p(x).

Taylor approximation of tan^2(x) [6/12/1996]
Just to check that I can't do this because f'(0) = infinity...

Taylor Series Expansion [11/24/2001]
A distance from A to B is 1000 meters. As one traverses it at 1 meter per second, the distance is instantaneously and uniformly stretched 1000 additional meters. How long does it take to get from A to B?

Tensor Calculus [03/06/1998]
Scalars, vectors, and tensors, and their components.

Tensor Calculus [10/10/2002]
Could you please remind me of the general definition of a determinant for a tensor with more than 2 indices, like T_m1_m2..._mk ?

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