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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?
 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 + (35x)^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 selfsame 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^23x+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 xy/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?
 EpsilonDelta Proofs [09/28/2004]

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