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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?
- Minimizing the Cost of Phone Line Construction [12/3/1995]
A telephone company has to run a line from point A on one side of a river
to another point B that is on the other side, 5km down from the point
opposite A. The river is uniformly 12km wide. The company can run the
line along the shoreline to a point C and then under the river to B. The
cost of the line along the shore is $1000 per km and the cost under the
river is twice as much. Where should point C be to minimize the cost?
- Minimizing the Length of a Crease [11/11/1999]
How can I minimize the the length (L) of the crease if one of the corners
of an 8.5" x 11" sheet of paper is lifted up and placed on the opposite
longer edge, and then the paper is smoothed flat?
- Minimizing the Sums of Squares [06/12/1998]
Find two numbers such that their sum is 20, and the sum of their squares
is as small as possible.
- Minimizing the Surface Area of a Can [05/22/2000]
What coke can dimensions would use the least amount of aluminum while
still holding 375 ml?
- Minimizing the Time to Rescue [11/29/1999]
A man is drowning in the water 300 meters down the beach and 100 meters
out from where I'm standing. I can run 5 meters/second and swim 3
- Minimum and Maximum Pollution [04/26/2000]
Find the points of minimum and maximum pollution between two plants
releasing 60 and 240 ppm of pollution...
- Minimum, Maximum Value of a Two-variable Function [6/27/1996]
Find f min and f max where x and y are real numbers and f(x,y) = 2(sin
x)(cos y)+3(sin x)(siny)+6(cos x).
- Minimum Value Problem [7/14/1995]
I need to find the minimum value of E for E = be^(-ar) - dr^(-6) where b,
a, and d are constants and r is the variable.
- Min, Max of 3-Variable Function [8/16/1996]
Find all the critical points and determine their nature for the function
z = x^3 - 6xy + y^3.
- Modelling Discrete Data with Exponential Functions [5/14/1996]
How can I fit an exponential function to a set of three discrete data
- Monkey Climbing a Chain [09/22/1999]
A 20-lb. monkey is attached to a 50-ft. chain that weighs 0.5 lb. per
- The Monotone Convergence Theorem [10/02/1998]
Could you please explain the meaning and purpose of the monotone
- Moving a Circle on a Polar Graph [05/04/2001]
How do you move a circle so the center is not (0,0), but to where the
center can be (r,[theta])?
- Moving Particle [07/14/1999]
Assume a particle moves on the x-axis according to the formula x = t^ 3-
6t^2+9t+5. Find: the velocity when t = 3...
- The Moving Shadow [03/24/1999]
A man walks toward a light... at what rate does the tip of his shadow
move and at what rate does its length change?
- Multidimensional Calculus and Vector Geometry [02/09/1999]
The depth of iron ore can be approximated by a plane...
- Navigation Formula [10/14/1996]
Given the starting latitude, longitude, distance, and course of a plane,
what formula gives its destination latitude and longitude?
- Newton-Raphson Method [02/28/2000]
How can I find the 1st, 2nd, 3rd iterations and the parameters of x^3 -
13.1x^2 + 48.48x - 46.62 using the Newton-Raphson method?
- Newton-Raphson Method [06/24/2009]
Are there any equations that cannot be solved using the Newton-Raphson
method, regardless of the initial estimate?
- Newton's Method and Square Roots [12/04/1998]
Can you explain how finding square roots by hand relates to Newton's
method for approximating the zero of a function?
- Non-Constant Functions [02/12/2001]
Are there any non-constant functions f and g such that (f*g)' = f'*g'?
- Nondifferentiable Functions [11/6/1994]
In my calculus book, it mentions a function that is not differentiable
at any point due to the fact that it is not smooth at any point. It does
not go any farther, and I was interested in hearing more.
- Non-Linear Equation [2/13/1996]
Solve 5x^2 + log(x) = 0 for x...
- Nonvertical Asymptotes [10/16/2001]
Does this problem have slant asymptotes? f(x)=(x^(4/3)+x^(1/3)-2)/(x^
(4/3)-16). I'm also having trouble taking the derivative of this problem:
- A Norman Window [1/16/1995]
A Norman window is a window in the shape of a rectangle with a semicircle
attached to the top. Assuming that the perimeter of the window is 12
feet, find the dimensions that allow the maximum amount of light to
- The N Targets Problem [11/15/2001]
Given n targets with equal probability of being hit by a cannon, that is,
1/n. Let X be the number of firings required to hit all the targets. What
is the expectation E(X) of X?
- Numeric Derivatives Using TI-83, TI-92 [10/17/2003]
My calculator says that the numeric derivative of sin(x) is
sin(2x)/(2x). Shouldn't it be cos(x)?
- Oblique Asymptotes [04/03/2002]
I'm stuck on how to calculate oblique limits, for example, x^2/(x-3).
- Odd Bernoulli Numbers Must Be 0 [02/16/2003]
Take the second derivative of t/exp(t) -1 and find that it is an even
function; show that an even function, when expressed as a Taylor
Series, has only even powers of t; draw the conclusion that the odd
Bernoulli numbers have to be 0.
- ODEs and Integration with the Chain Rule [05/30/1998]
What is an ordinary differential equation? How do you integrate f'(x) =
- Oil Can Dimensions [12/11/2001]
What are the dimensions of an oil can with a one-liter capacity that uses
the least amount of tin?
- Operating a Tour Service [5/21/1996]
How many people does it take to maximize your profit?
- Optimization [12/13/1995]
An AP Calculus student asks questions about his chapter test.
- Optimization [11/26/1996]
To make a funnel, we take a circular piece of metal, cut out a sector,
and connect the two radial edges together to make an open cone. What
should the angle of the sector be to maximize the volume of the cone?
- Optimization: Minimum Area [11/07/1997]
How do you fold a piece of paper (rect. with width a and unlimited
length) so one corner just reaches the righthand side for minimum area?
- Optimization (Min-Max) [01/21/2001]
If a piece of string of fixed length is made to enclose a rectangle, show
that the enclosed area is greatest when the rectangle is a square.
- Optimizing Advertising [10/15/2003]
Doubling the amount spent on advertising increases total income by 20
percent. What is the optimum level of advertising?
- An Ordinary Differential Equation in Everyday Life? [03/26/2011]
A student wonders about the real world applicability of ordinary differential equations.
Doctor Jordan steps through an example.
- Ordinary Differential Equation, Second Order [10/26/2002]
I've been looking for solutions to the differential equation y'' = x^2
- Parametric Equations [07/21/1998]
A curve is given parametrically by the equations: x = (3-2k)^2 and y
= (2+k)^2. Find dy/dx, the x-intercept, the cartesian equation...