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High School Archive

Browse High School Calculus
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Selected answers to common questions:
    Chain rule.
    Maximizing the volume of a box.
    Maximizing the volume of a cylinder.
    Volume of a tank.
    What is a derivative?

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Polar Coordinates [11/4/1996]

Find the polar equation of x^3+y^3=3axy with x+y+a=0 as an asymptote.

Polar Coordinates for Velocity and Acceleration of a Particle [03/11/1997]

I know that velocity components are dr/dt and rdo/dt and that components of acceleration are d^2r/dt^2 - r(do/dt)^2, etc. I would appreciate any help at all in understanding how these are generated and/or how to use them.

Position Function of a Moving Particle [9/9/1996]

A particle moves on a straight line with a velocity v(t)=sin(w)cos^2w; f(0)=0; w is constant; find f(t).

Possible Proof That 1 + 1 Does Not Equal 2 [04/19/2001]

I just want verification... would this proof work?

Power Series for Sine and Cosine [10/12/2000]

Can you explain, without using calculus, how to get the power series for sine and cosine?

Pre-calculus Function Question [2/8/1996]

Does the function f(x)=3De^(1/x) have a local minimum?

Primitive function [6/11/1996]

Give a primitive function of (sin x) / (ln (2 + x^2)).

Projectile Velocity and Vectors [5/7/1996]

If the initial velocity is 834 meter^2/sec at an angle of 60 degrees, and x is 21,000, at what time, t, is the object at x = 21000 meters?

Proof by First Principles [11/16/2001]

How can I prove by first principles that the derivative of tan(x) is sec^2(x)?

Proof for the Radius of Curvature [08/17/2003]

The equation of a circle of radius R centered at the origin is x^2+y^2 = R^2. Demonstrate that the radius of curvature is equal to R.

Proof of a Bounded Series using the Binomial Theorem [6/6/1995]

Prove that (1+1/n)^n is bounded from above.

Proof of Absolute Value, Division [7/16/1996]

If y does not equal to 0, prove that |x / y| = |x|/|y|.

Proof of Cosine of 36 Degrees [11/26/2001]

Prove: cosine {(pi)/5} = {1+5^(1/2)}/4.

Proof of Derivative for Function f(x) = ax^n [01/14/2004]

Is there a way to prove or show why the derivative of ax^n turns out to be anx^(n-1)?

Proof of Integration by Parts [02/06/2001]

How do you derive the formula for integration by parts?

Proof of Series ln(1+x) [11/15/2001]

I need to show that the series ln(1+x) equals x-x^2/2+x^3/3-x^4/4, and so on, whenever x is between -1 and 1.

Proof of Stirling's Approximation [03/09/2006]

Can you prove that lim ((e^n)(n!)) / ((n^n)(n)^1/2 = (2pi)^1/2 ?

Proof of Tangent Double Angle Identity [7/16/1996]

I would like to see the proof of the tangent double angle identity.

Proof of the Chain Rule [09/06/2004]

I am using the Chain Rule (dy/dx = dy/du * du/dx) in my math class, and I would like to see it proved, which we don't do in class.

Proofs of e [03/21/2002]

I found out that the definition of e is: e = lim, as n approaches infinity (1+1/n)^n. Is there a proof for this or for 1+2(e-1/n)?

Proof that 2 Equals 1 Using Derivatives [05/10/2000]

If d(x^2)/dx = d(x+x+...+x)/dx, then 2x = 1+1+...+1 = x. What is the error in this "proof"?

Proof that INT(1/x)dx = lnx [11/08/1996]

How do you integrate (1/x)dx?

Proving Limits at Infinity Using the Formal Definition [04/11/2004]

We are required to find a limit (informally), then prove that our answer is correct by using the formal definition of limits. I'm not sure I'm on the right track.

Proving Limits Formally [01/04/1999]

Given a sequence, I can guess what the limit it. How do I actually prove my guess is the limit?

Proving That d(e^x)/dx = e^x [06/30/1999]

How can I find the derivative of y = e^x, using the definition of the derivative?

Proving the Derivative of e^x Is e^x [02/20/2006]

What is the delta x proof of (dy/dx)e^x = e^x?

Ptolemy's Theorem and Double-angle Formulas [7/11/1996]

Is it possible to show the formula Sin(A-B), Cos(A+B) or Cos(A-B) using Ptolemy's Theorem?

Quadratic Extrema [07/20/2002]

Can you tell me how to find the minimum or maximum of a quadratic function?

Quadratic Function and Diagonals of a Diamond [6/14/1996]

The sum of the two lozenges (diagonals of a diamond) are maximum when the angle between the longest lozenge and the side of the diamond is 30 degrees, diamond side length = sqrt 2...

Quotient Function Mnemonic [03/11/2002]

Help me to differentiate the following: x/((x^2)-4).

Raabe's Theorem [8/23/1996]

Regarding Raabe's result for the convergence of a numerical series of non-negative terms - I am looking for is a convergent series having m = 1.

Radian vs. Degree, Sin Derivatives [10/23/2001]

Why do you get a different answer when you ask your calculator for the derivative of sinx in degree rather than radian mode?

Radius of Curvature [07/23/2003]

I have several ellipses whose major and minor diameters I know, but I have no information about their foci.

Range of rational functions [12/1/1994]

Given the following rational function: f(x)= (x-1)/(x^2+3x-1), how do you find the range? Our book gives the answer of: (neg infinity to 0.12] U [0.65 to pos infinity).

Rate of Change Constant? [03/20/2002]

The rate of change of e^x is e^x. Does this mean that the rate of change is constant? Why are sinx,cosx,... and sinhx,coshx,.... similar?

Rate of Change of a Function [6/10/1996]

Show that the expression for the rate of change can also be given by -cos x cot^2 x.

Rate of Change of Angle [8/12/1996]

Find the rate at which angle BXA is changing in radians per second...

Rate of Change of Radius of a Cone [03/07/1999]

Using the measured rate of change in volume of water in a cone, calculate the rate at which the circular area of water reduces when the radius is r centimetres.

Ratioing Errors [01/07/2002]

How do you calculate the error bars for ratios?

Rational Function [04/03/1997]

Given the rational function y=r(x)=(4x^6-x^4)/(x^5+5) describe its end behavior...

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