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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?
- Integration Over a Complicated Region [11/1/1996]
Given a region defined by five relations simultaneously, find its area.
- Integration Problem [2/22/1995]
How do you integrate the function: (tan 2x)^3 dx, in terms of x?
- Integration Questions [03/09/2001]
Calculate the area of the finite region bounded by the curves y=x^2 and
- Integration Trick [8/13/1996]
If f(x)=1/sin(x), what is the integral of f(x)?
- An Integration with Substitution [06/06/1998]
I'm having trouble with the following integration. Is substitution needed
- Intriguing Limit [04/30/1997]
Why is the following true: lim x --> oo (1-1/x)^x = 1/e?
- Introduction to Line Integrals [02/08/2004]
C is the broken path line that goes from (1,1,2) to (2,0,-1) by
following a line parallel to the x axis from (1,1,2) to (2,1,2) then
parallel to the y axis from (2,1,2) to (2,0,2) then parallel to the z
axis from (2,0,2) to (2,0,-1). Find int_C [ f(x,y,z)*dx + g(x,y,z)*dy
+ h(x,y,z)*dz ].
- Inverse of a Multivariate Function [05/30/2002]
Let f:NxN -> N such that f(x,y) = 2^x(2y + 1) - 1 for all natural
numbers x, y. Let the inverse of f, g be given by g:N -> NxN. Find the
inverse of the function g.
- Inverting Functions [07/19/2002]
To find the inverse of a function y=f(x), do I interchange the
variables x and y, or do I solve for x in terms of y?
- Is Infinity a Number ... in Inversive Geometry? [01/15/2010]
A reader attempts to demonstrate infinity as concretely measurable in
an inversive geometric construction. Doctor Tom explains analyzes the
argument, weighing the pros and cons of the axioms of non-Euclidean
geometries, and going on to expose an apparent paradox.
- Is pi Squared Rational or Irrational? [01/07/2011]
Doctor Ali provides a proof by contradiction.
- Is the Function Invertible? [09/22/1997]
For the following functions f(x) decide if the function is invertible as a
function from R to R...
- Iterated Limits [5/25/1996]
I don't know how to do the following problems...
- Jerk - Derivative of Acceleration [03/16/2001]
I need a mathematical term that begins with the letter J, its definition,
and a high-school-level explanation of the term.
- kth Derivative of x^n [03/04/2003]
What is a quick way to find the nth derivative of a function? Example:
the 7th derivative of x^8.
- LaGrange Error for a Taylor Polynomial [05/28/2000]
How can you find the LaGrange error for a Taylor polynomial?
- Lagrange Multipliers [01/08/1998]
I have a problem with Lagrange Multipliers - can you help?
- Lagrange Multipliers [01/28/2001]
The temperature of a point(x,y,z) on the unit sphere is given by
T(x,y,z)=xy+yz. Using Lagrange multipliers, find the temperature of the
hottest point on the sphere.
- Lagrange Multipliers and Constraints [11/24/1998]
When using the Lagrange Multiplier method, how do you determine which of
the two equations is the constraint?
- Laplace Transform [01/25/2001]
What are Laplace transforms, and what are their applications?
- Latus Rectum [08/08/2002]
I'm trying to find the definition, an explanation, a formula, or
anything that will help me to better understand what latus rectum is.
- Learning Differential Equations [7/15/1996]
What resources can I use to learn about differential equations?
- Lengthening Shadow [10/13/2003]
A man 6 feet tall walks at 4 miles an hour directly away from a
lampost 18 feet tall. Why does his shadow lengthen at a constant rate?
- Length of a Cubic Curve [12/10/1997]
I need to calculate the true length of a cubic curve.
- Let f(x) = 1 + 1/2 + 1/3 + ... + 1/[(2^n)-1] [05/15/1999]
Which of the following inequalities are correct?
- L'Hopital's Rule [4/23/1995]
I can't figure out how to take the limit using L'Hopital's rule on this
- L'Hopital's Rule [8/11/1996]
How can I solve the following problem: limit as x-> infinity of x^(1/ x)?
- L'Hopital's Rule and Limits [8/7/1995]
Find the limit of [sin 3x / tan (x/3)] as x goes to 0.
- Lifting an Object of Changing Mass [7/28/1996]
Suppose you grab the end of a chain that weights 3 lb/ft and lift it
straight up off the floor at a constant speed of 2 ft/s: determine the
force as a function of height; how much work do you do in lifting the top
of the chain 4 feet?
- Limit As n Approaches Infinity [9/7/1996]
How do I find the limit (n-> -infin.) of (sqrt(2n^2+1))/(2n-1)?
- Limit Intuitions [10/01/1998]
Can you explain the intuition behind the formal definition of a limit?
derive the equation of the tangent line of a function at a given point?
- Limit of a Factorial and Exponent in x [05/21/2014]
A calculus teacher wonders how to prove a limit involving the factorial of a variable raised
to a power of the same unknown. Seeing an opportunity to prove a simple version of Stirling's Approximation, Doctor Vogler introduces it with integral calculus and the Squeeze Theorem.
- Limit of n^2/2^n as n Tends to Infinity [12/05/2002]
Demonstrate that the limit as n tends to infinity of the fraction n^2
/ 2^n = 0.
- The Limit of Sin(1/x) [10/02/1998]
Why does the sine of 1/x have no limit as x approaches 0?
- Limit of x sin(1/x) [04/23/2002]
I assumed from the graph that the function had a limit at x=0 of 0,
but since it involves sin(1/0) I can not prove this using the basic
trigonometric limits (sin x/x and (1-cos x)/x), L'Hopital's
rule, or by rearranging the equation. Can you help?
- Limit Problem [7/16/1995]
Find the limit of (x^2 + 2x) / (5x - 5) as x tends to infinity.
- A Limit Problem [10/24/1996]
What is the limit of (x^2 -4) / (2x) as x approaches infinity?
- Limit Problems [08/08/1998]
What is the limit of (sqrt(2-t) - sqrt(2))/t as t->0?
- Limit Proof [7/13/1996]
If lim(An/n) = L and L>0, how do you show that lim(An) = + infinity?
- Limit Proofs with L'Hopital's Rule [05/27/1998]
How would you prove the following three limits? ...