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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?

Harmonic Series [04/28/2003]
Prove that if in the sum 1 + 1/2 + 1/3 + ... + 1/n we throw out each term that contains 9 as a digit in its denominator, then the sum of the remaining terms is less than 80.

Heat Supplied and Heat Loss of a Furnace [05/17/2000]
Heat is supplied by a furnace at a constant rate, but is lost at a rate determined by the difference between its temperature and the temperature of its surroundings...

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...

History and Applications of the Natural Logarithm [03/02/1998]
I'm so surprised at how often the number e comes up. Where did it come from? Who first derived it? Why is it so common in the field of biology?

How a Calculator Determines Values of Trig Functions [03/19/2005]
I'm interested in how my calculator finds the answer of sin(pi/4) or tan(pi/6), for example. Does it use power series?

How Fast is the Lighthouse Beam Moving? [10/13/2002]
A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km away from P?

How Long is the Tape? [7/31/1995]
Jones has a roll of tape for which he knows the thickness of the tape, diameter of the core, and diameter of the entire roll of tape. Is there a formula for determining the length of the tape?

How Should the Pipe be Laid? [03/25/1999]
Oil from an offshore rig is to be pumped to a location on the edge of the shore... determine how the pipe should be laid to minimize cost.

How to Find the Range of a Function [02/25/1998]
How do you find the range of a function like g(x) = (x+1)/(x^2-1)?

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

Implicit Differentiation [10/21/1996]
How do I find dy/dx for ((xy)^(1/2)) + ((x + 2y)^(1/2)) = 4?

Implicit Differentiation [11/1/1996]
Given that (x^p)*(y^q) = (x+y)^(p+q), prove that: dy/dx = y/x.

Implicit Differentiation [11/4/1996]
Find the slope of the tangent line at the point (3,4) on the circle x^2+y^2 = 25.

Implicit Differentiation [11/03/1998]
How do you differentiate an equation that cannot be expressed in the form y = f(x)?

Implicit Differentiation [10/15/2001]
We have not been able to show that (1+y)/(1-x) = 0... Where have we gone wrong?

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

Implicit vs. Explicit Differentiation [01/04/2001]
What is the difference between implicit differentiation and explicit differentiation?

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?

Income from Winning the Lottery [7/30/1996]
How much money would I need to win in the lottery to have an income of $75,000.00 year until I die? That should be in about 30 more years.

Indefinite and Definite Integrals of Zero [09/12/2004]
If the integral of 0 is a constant, and the integral also calculates the area under a curve, how can the area under f(x) = 0 be a constant? Shouldn't it be 0?

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

Infinite Product [12/01/1997]
How do I find the infinite product of 3^1/3 * 9^1/9 * 27^1/27 * ..... (3^n)^1/3^n ?

Infinite Sequence [09/28/1997]
From f:n ---> 3 + (-1/2)^n where n belongs to the set of natural numbers, describe a strip that contains all but a finite number of points of the graph of f.

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?

Infinite Sums of Arctan [04/15/2002]
The sum as n = 0 to n = infinity of arctan ((1)/(n^2 +n +1)): I know it equals pi/2 but I don't see how.

Infinity a Concept, Not a Number [03/16/2003]
If 1 chocolate bar is divided among an infinite number of people, no one gets anything. Where did the chocolate bar go? Doesn't it imply that 1/infinity = infinitesimally small?

Infinity in a Fraction [05/13/1997]
Why does lim n --> oo (n+1)^.5/(n-1)^.5 = 1?

Inner Product and L-2 Distance [11/14/2001]
Why is "distance between two functions" calculated by multiplying f(x) and g(x) and then integrating with respect to x within the defined domain?

Inscribed Cone [3/11/1996]
Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 'r'.

Integral Calculus [09/18/1997]
How do you integrate the following function: sqrt(1-x^2) from 0-1?

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

Integral of (1+sec(x))^2 dx [1/5/1995]
Please show me the steps for solving this problem: integral of (1+sec(x))^2 dx.

The Integral of a Negative Function, and the Negative Integral of a Function [08/27/2016]
A teen struggles to justify a basic property of definite integrals. With Riemann sums, Doctor Ali breaks it down — point by point, interval by interval.

Integral of a Power [9/8/1996]
Why does the integral from -1 to 2 of x^2/3 = x^3/9 |from -1 to 2?

Integral of a Trig Function [12/20/1996]
F(x)=Integral from -1 to 1 of 1/sin(x)

Integral of Cos^2 [2/6/1996]
I can't seem to find the anti-derivative of (cos X)^2.

Integral of e^sinx [5/15/1995]
Can e^sinx be integrated ?

The Integral of e^(x^2)dx [03/26/1998]
How do you solve the integral of e^(x^2)dx?

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

Integral of e^xsinx [01/23/2001]
I always get e^xsinx or e^xcosx to integrate, and it just keeps going. Could you show me the steps to this indefinite integral?

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