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Browse College Calculus

Stars indicate particularly interesting answers or good places to begin browsing.



Finding the Critical Point [05/09/2000]
How can I find the critical point of f(x,y,z,t) = (x^2)*(y^2)*(z^2) + (t^2)*(x^2) + 3x?

Finding the Shape of a Funnel Curve [04/08/2001]
Find the equation of the curve of revolution of a coin leaving a chute.

Finding the Sum of Arithmetico-Geometric Series [09/13/2004]
Find the sum of the infinite series 1/7 + 4/(7^2) + 9/(7^3) + 16/(7^4) + ... I would also like to know if there is a general rule to find the sum of (n^2/p^n) for n = 1 to infinity.

Fixed-Point Theorem [05/02/2002]
An intuitive reason why the theorem is true, and a proof.

Formula for the Curvative of a Curve [10/10/2002]
Is there a formula or theorem I can use to convince someone who believes a wall has only side that there is another side to it?

Frobenius Series Solutions [05/01/2008]
Determine if the differential equation (x^2)y'' + xy' + (x^2-8)y = 0 has two linearly independent Frobenius series solutions.

Frustum of a Pyramid with a Rectangular Base [02/20/2002]
I am an engineer with a water treatment agency and need to figure the amount of water per foot of elevation in our reservoirs...

Fuel Left in an Ellipsoidal Tank [10/19/2002]
How much fuel is left in a tank at any depth, given that the tank is ellipsoidal?

Function problem [12/7/1994]
Find the slope of the tangent line for the function y2 = x2 - x4 at (0, 0), also explain what is going on at the origin.

Functions of a Complex Variable [10/05/2002]
Show that w=f(z*), where z*=x-iy and f is a differentiable function, is not an analytic function of z.

Gamma Function Division [12/20/2005]
Is there a formula for Gamma(n)/Gamma(m) for n and m having integer values >= 1 and m < n?

Generalized Cubic Equation [05/22/2000]
How do we find a generalized cubic rule for a family of cubics that cut through the point (1,1), have a gradient of 2 when x = 1, and have 1 stationary point?

General Solution of dy/dt + y = t [10/25/1999]
How can I find the general solution to the equation dy/dt + y = t?

General Solution to Differential Equation [07/13/2004]
How do I find the general solution of d^3x/dy^3 = 0?

Generating the Volume of an n-Sphere [02/07/2002]
Could you help me find an equation for the integration of a unit-ball in n-dimensions?

Gradient and Maximum Increase of a Function [07/19/2005]
Why does the gradient point in the direction of the maximum increase of a function? Several resources make the statement, but no one explains it. Can the dot product be used to justify it?

Gradients of Perpendicular Lines [01/16/2003]
Why is the product of the two gradients of perpendicular lines equal to -1?

Great Circle Parametric Equation [05/25/1998]
How can you calculate specific points of a great circle on a sphere? Can you help me find the parametric equation?

Green's Theorem with Parameterized Curves [10/18/2003]
I understand what Green's theorem states and how to use it, but I don't understand how to use it with parameterized curves.

Helix Lying on a Cylinder [11/26/1997]
I am trying to find a parametric equation for a helix (in 3 space) lying on the cylinder...

Homogeneous Linear Differential Equations [05/06/2001]
How would one solve y''-2y'+5y = 0 ? Are there general rules for solving such differential equations involving y'''', y''', y'', y', and y? How would you solve (y''')^2+(y'')^3-y = 0?

Hyperbola and Cone [03/04/1999]
Two questions on three-dimensional figures.

Ice Cream Cone Problem [02/27/2000]
What radius of a sphere of ice cream placed in a cone of height h and base angle of 30 degrees will give the most volume of ice cream inside the cone, as opposed to above the cone?

Implicit Functions [11/26/1997]
Please give me a definition and several examples of an implicit function.

The Impossibility of Integrating x^x [08/07/1998]
How would you test to see if the integral of x^x can be expressed in a finite number of elementary functions?

Impossible Integral [11/12/2001]
As our homework we were assigned an integral that no one has been able to answer: (sin y)/y dy (there are no limits).

Indefinite Integrals [07/04/2002]
Solve: Integral dx/(sin x)^5 + (cos x)^5; Integral dx/(sin x)^3 + (cos x)^5.

Infinite Series Involving Arithmetic and Geometric Means [11/27/2003]
Start with two numbers, find their arithmetic and geometric means, then find the means of the two results and continue this process indefinitely. Does the series converge?

Integral Calculus [08/18/1999]
Find the integrals of sin^10x and (1/x)*(1+x^4)^3/4.

Integral Notation - Missing Integrands [07/17/2003]
I have seen some integral notation used that I am not familiar with...

Integral of dx/(1+(tanx)^sqrt2) [06/28/2003]
Find the value of the integral that ranges from 0 to pi/2 of dx/(1+ (tanx)^S), where S is the square root of 2.

Integral of e^(e^x) [5/12/1995]
What is the integral of e to e to the x power?

Integral of Exponents, Natural Logs... [8/27/1996]
How do I integrate x^(-ln x).e^(-x.lnx + x).dx ?

Integral of sin(x)/x [8/29/1996]
Is there a symbolic (not a numerical method) solution to the definite integral of sin(x)/x from 0 to some value?

Integral of Trig Function, Exponent [8/6/1996]
What is the integral of f(x)=(tanx)^-1/2?

Integrals in Polar Coordinates [04/15/2003]
How can I evaluate an expression like (sin(theta)*(constant)) for theta = 0...2*Pi?

Integrating exp(-x^2) [11/09/2005]
It seems like all the ways of finding integral exp(-x^2) involve making it a double integral and using polar coordinates. I was wondering if there is any way to do it without making it a double integral?

Integrating sin(x)/x [02/16/2005]
I'm trying to find the integral from 1 to t of sin(x)/x so that I can calculate the limit as t goes to infinity.

Integrating the Standard Distribution Curve [06/25/1999]
I need to find the integral from infinity to negative infinity of (e^ ((- x^2)/2))/((2pi)^(1/2)).

Integrating with Modified Euler's Method [08/13/2007]
Using the modified Eulerís method, find the value of y at x = 0.2 with h = 0.1 where dy/dx = 1 - y with y(0) = 0.

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