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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?
- 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]
- 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.
- Equation [04/03/1997]
Solve for x...
- Equation for a Curved Line [01/19/1997]
Is there a formula into which you can enter a starting point, a starting
slope, an ending point, and an ending slope that gives the equation of
the curved line between these points?
- Equation of a Circle [08/25/1997]
Write the equation of the specified circle in general form...
- Equation of Parabola with Given Vertex, Point [8/4/1996]
How do I find the equations for all parabolas containing the origin whose
vertex is at (2,1)?
- Equations for the Paths of Four Lady Bugs [12/02/1998]
Four ladybugs travel from the corners of a square towards the center.
Find the equations that describe the path.
- Escaping L'Hopital's Loop [10/14/2003]
I'm trying to apply L'Hopital's rule, but I keep ending up with 0/0
after each application. What am I missing?
- Euler Equation and DeMoivre's Theorem [05/18/1999]
Do you have a proof of the equation e^(i*Pi) + 1 = 0?
- Evaluate the Limit [12/03/1997]
In Calculus today we were trying to evalulate the limit of (1-cos x)/
(x^2) as X->0.
- Evaluating Two Definite Integrals [02/28/1998]
Evaluating a definite integral and determining an enclosed area.
- Exact? Area under a Curve [03/13/1998]
Would you please explain why the area under a curve is exact and not an
- Examples of Cooling, Growth and Decay [06/03/1999]
Can you give some real life problems involving Newton's cooling method,
logistic growth, and exponential growth and decay?
- Explaining the Intermediate Value Theorem [02/18/2004]
I want to know how to explain the Intermediate Value Theorem in its
most simple form: If f is a continuous function on the closed interval
[a,b] and N is any number between f(a) and f(b), there must be a
number c in (a,b) such that f(c)= N.
- Explanation of L'Hopital's Rule [02/06/2001]
I understand the method, and I somewhat understand the proof, but can you
give me an explanation of why L'Hopital's Rule actually works?
- Exploring the Distance from (0,0) to (1,1) with Limits [10/15/2006]
Any route traveling from (0,0) to (1,1) going only north and east will
cover a total distance of 2 units. But the straight line distance from
(0,0) to (1,1) is sqrt(2) units. It seems that if I think of a
staircase connecting the two points and let the stairs become
infinitely small, the limit of the north/east route distance should
converge to sqrt(2). But it doesn't! What's going on here?
- Exponential Growth & Compound Interest [6/2/1996]
(1) The cost of goods and services in an urban area increased 1.5% last
month. If the rate continues, what will be the annual rate of increase?
(2) Suppose a certain population increases by 30% every 10 years. By what
percent does it increase each year?
- Exponential Growth of a Population [05/03/1998]
A population P(t) is increasing at a rate directly proportional to 800 -
P(t), where the constant of proportionality is k...
- Exponential Growth of Bacteria [8/7/1995]
A colony of N bacteria increases with time according to the formula dN/dT
= kN. If initially there are N' bacteria, show that the number at any
time is given by N = N'e^kt.
- Factoring Polynomials [02/12/1999]
What are some real-life applications of factoring polynomials?
- The Figure of Maximum Area and Given Perimeter [06/02/1998]
Can you help me show, with and without calculus, that the geometric figure of a maximum area and given perimeter is a circle? What are the dimensions of a triangle with perimeter p that encloses the maximum area?
- Find a Function, Integrals [12/17/1997]
Suppose the graph of f has the formula f(x)=-x+1 for 0<=x<=1; x-1
for 1<x<=2 ...
- Find Curve Given Slope, Point [8/29/1996]
Find the curve whose slope at (x,y) is 3x^2 and which passes through (1,-
- Finding a Definite Integral with Natural Log and Sine [05/14/2008]
I'm trying to evaluate the definite integral of ln(sin(x)) dx from 0
to pi. I've tried integration by parts but am not getting anywhere.
- Finding a Limit [02/28/2002]
My calculus teacher gave my class the answer to this limit; however, we
cannot figure out how to get the answer that is given.
- Finding a Vertical Asymptote [05/11/2000]
How can I find the vertical asymptote of the equation xy^2 - x^3y = 6?
- Finding First Derivatives by Definition [10/30/1998]
Can you show me an easy example of how to find the first derivative of a
function using the definition?
- Finding Integrals Using Trigonometric Substitutions [10/2/1995]
What is the integral of 1/(x(1-x))^(1/2)? How can I integrate this
function using only the derivative of arcsin or arctan?
- Finding Limits Using Natural Logs and L'Hopital's Rule [05/03/1998]
I think I need to use l'Hopital's Rule and the natural logarithm function
to find lim (x goes to 0) [(cos (2x))^(3/(x^2))].
- Finding Maxima and Minima: Turning Points [05/10/1998]
Can you help me find all of the maxima and minima of the function: 5e^(-
- Finding Tangents to Curves Using Derivatives [09/29/1999]
Three problems that involve finding the tangents of a curve using
- Finding the Area of Overlap between Two Circles [04/16/2008]
Suppose I know the centers and radii of two circles that overlap. How
can I find the area of that overlapping region?
- Finding the Derivative [07/27/1999]
Find dy/dN if y= A*(1 - e(-cN))* (1 - e(-cM)).
- Finding the Domain of a Function [9/28/1995]
What is the domain of f(x)=((2-x)/(x-2))^.5
- Finding the Equation of a Tangent Line [10/6/1995]
What is the equation of the tangent line to the graph of f(x)=x^(2/5) at
the point (-32,4)?
- Finding the Equation of a Tangent Line [11/24/1998]
I only have where the tangent intersects the y axis...
- Finding the Extrema of a Function [10/25/1996]
Find the extrema of f(x)=(ln(1+exp(-a-bx-cx^2)))/((x+d)^2)...
- Finding the Factorial of Non-Integers [01/06/2005]
I thought the factorial function was only defined for positive
integers, but I notice that my calulator returns a value for
non-integers like 3.5! What's going on?