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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?

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 meters/second...

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 points?

Monkey Climbing a Chain [09/22/1999]
A 20-lb. monkey is attached to a 50-ft. chain that weighs 0.5 lb. per (linear) foot...

The Monotone Convergence Theorem [10/02/1998]
Could you please explain the meaning and purpose of the monotone convergence theorem?

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: f(x)=((x^2(x+1))-(1-x))^(1/2).

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 enter.

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) = (x-1)^4?

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 / y^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...

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