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Selected answers to common questions:
Maximizing the volume of a box.
Maximizing the volume of a cylinder.
Volume of a tank.
What is a derivative?
- 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
- 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
- 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
- Infinite Product [12/01/1997]
How do I find the infinite product of 3^1/3 * 9^1/9 * 27^1/27 * .....
- 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
- 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
- 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
- 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?
- Integral of Ln [4/13/1996]
I need to solve the integral from 0 to 1 of ln(1-x) / x symbolically, if