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High School Archive

Browse High School Calculus
Stars indicate particularly interesting answers or good places to begin browsing.

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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Deriving Simpson's Rule [06/07/1998]

Can you show me a derivation of Simpson's Rule?

Deriving the Area of a Sphere [10/21/2003]

I know the area of a sphere is 4phi(r^2), but I'm wondering how to derive that formula. I know it should be done in cylindrical coordinates, and I'm thinking that the arc of a circle is defined as rd(theta) and it's multiplied with rd(phi) to get (r^2)d(theta)d(phi). Could you please help explain this?

Deriving the Gamma Function [12/15/2000]

How can you prove that sqrt(pi)/2 = (1/2)!, and what is a fractional factorial like that equal to?

Deriving the Integral for the Surface Area of a Sphere [06/28/2001]

I need to find the cost per square foot of steel that makes up a tank, so I need a way to derive the surface area of part of a sphere with a given diameter...

Deriving the Quotient Rule [9/5/1995]

How do I prove this derivation: (f/g)' = fg'-fg'/g^2 ?

Deriving Trig Functions; Taylor Series [05/01/2001]

How would I find, from first principles - no tables, no calculator - for example, 32 degrees? If I use a formula, how is it derived?

Descartes' Method for Tangents [09/02/1998]

Can you help me find the equation of the tangent of y^2 = 2x using Descartes' method...?

Design a More Efficient Soda Can [4/23/1995]

The problem is to design a more efficient soda can that holds the regular 12 oz. of liquid. The can needs to have the least possible surface area.

Determining Tangent to an Ellipse, Minimizing Area [02/20/1998]

Find the equation of the tangent to the ellipse that forms, with the coordinate axes, the triangle of smallest possible area.

Determining the Length of a Coil of Ribbon [08/31/2001]

How will calculus give me the same answer I found with algebra and geometry?

Determining Whether a Function is Continuous [06/10/2000]

How can you tell whether or not a function is continuous?

Differentiability, Intervals, Inflection Points of Piecewise Function [05/13/1998]

For what values of k and p will the function be continuous and differentiable? On what interval will it be increasing? Find all points of inflection.

Differential Equations and Flow Rate [02/14/1999]

A solution containing 3 lb/gal of dye flows into a 100 gal tank at 5 gal/min ...

Differential Notation [04/16/2009]

Why is the second derivative written as d^2y/dx^2?

Differentials [09/18/1998]

What is the difference between finding the derivatives of a function (dy/dx), and finding its differentials (dy,dx)?

Differentiate Twice [11/14/1997]

If y = xsin(3x) prove that y''+9y = 6cos(3x), where y'' is its second derivative.

Differentiating [01/14/1998]

Please help me differentiate (y^2) * cos (1/y) = 2x + 2y.

Differentiating and Integrating the Formula for Area of Circle [05/11/1998]

The formula for a circle's circumference is the derivative of the formula for its area. What is the significance of this?

Differentiating a Polynomial [04/25/1999]

Differentiate with respect to x: f(x) = x^8 + 3x^5 + (3-5x)^4.

Differentiating the Ceiling Function [10/16/2000]

How do I find dD/dp for an equation involving the ceiling function? How do I minimize D as a function of P?

Differentiating the Rate of Decay [3/14/1996]

A lump of radioactive subtance is disintegrating at time 't' days after it was first observed to have mass 10 grams and: (dm/dt) = -km (where k is a positive constant). Find the time, in days, for the substance to reduce to 1 gram in mass, given that its half life is 8 days.

Differentiating Under the Integral Sign [01/11/2001]

Can you give me an example of the integration method called "differentiating under the integral sign"?

Differentiating y with Respect to ... y? [11/05/2010]

A student familiar with differentiation struggles to take the derivative of a function with respect to the self-same variable. Doctor Ali puts the student back on track with a clearer substitution.

Differentiating y = x^x [11/3/1994]

Please could you differentiate y=X^X (that's X to the power of X)?

Differentiation Problem [11/15/1997]

A light shines from the top of a pole 50 ft. high. A ball is dropped from the same height at a point 30 ft. away from the light...

Distance From a Point to a Plane [03/31/1998]

Can you show me the proof of the formula for the distance between a point and a plane?

Distance to the Sun [04/16/1999]

Find the distance from Earth to the sun when t = 90 days...

Does f Have a Local Extrema at x = 0? [07/10/2003]

f(x) = (x^3)/6 + (x^2)/2 + cosx - 1. Does f have a local extremum at x = 0?

Domain, Asymptotes, Intercepts of a Function [04/01/2003]

What is the domain of this function? What asymptotes does it have? What are the x and y intercepts? Etc...

Domain/Range of a Function [01/22/1997]

How do you find the domain and range of the function f(x) = 2x^2-3x+1? (Both with and without calculus.)

Dominant Terms [03/25/1998]

What are dominant terms, and how do you obtain their values?

Donkey Grazing Half a Field [08/08/1997]

A donkey is attached by a rope to a point on the perimeter of a circular field. How long should the rope be so that the donkey can graze exactly half the field?

Double Integration in Polar Coordinates [03/27/2003]

Evaluate double integral x-y/x*2+y*2 over x*2+y*2 equal to or less than 1.

e as a Series and a Limit [03/30/1998]

Why does e = 1 + 1/2! + 1/3! + 1/4! + ... and lim (1 + 1/n) ^ n, as n -- > infinity?

An Easy Definition of Calculus [9/4/1995]

What is calculus?

e^(e^x) = 2 [01/28/2002]

I have been trying to solve e^(e^x) = 2. Help!

An Ellipse Or A Circle? - Parametric Equations [12/05/1998]

Is this parametric equation elliptical or a circle?... And how do I compute the slopes at points 0, pi/4, pi/2, 3pi/2,and 2pi?

An Elliptic Integral [01/05/2003]

Integrate Sqrt[Sin[x]].

Epsilon/Delta Definition of Limits [08/26/1999]

Can you explain how to use the epsilon/delta definition of limits?

Epsilon-Delta Proofs [09/28/2004]

An explanation of the thinking behind two epsilon-delta proofs, one from a calculus textbook and one from an answer in our archives.

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