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High School Archive

Browse High School Calculus
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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?

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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?

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...

Partial Derivatives [03/24/2001]

What sense do partial derivatives make in the case where u is given as a function of two variables, say x and y...?

Partial Fractions [06/08/1997]

How do you break 1/x^2*(x+2) into partial fractions?

Partial Fractions [11/23/2003]

I need to put this fraction into power series formation: (3x^2 - x)/(x^3 - x^2 - x + 1). I've tried to use partial fractions but keep getting stuck.

Percentage of Numbers Divisible by 6, 8 [03/18/2003]

What percentage of numbers is divisible by 6? by 8?

Permutation and Combination Equality [07/18/1999]

Prove that (nC0)^2 + (nC1)^2 + (nC2)^2 + ... + (nCn)^2 = (2nCn), where nCi = n!/((n-i)!*i!).

Picard's Method [08/10/1997]

Can you tell me about Picard's iteration method of solving differential equations?

Plane Flying Parametrically [06/28/1999]

How can I find out when a plane, whose position approaching an airport is described parametrically by P_t = (1000,500,900)+ t[-100,-50,-90], will be closest to the traffic control center, located at (24,11,13)?

Point of Inflection [05/17/1997]

The cubic polynomial h(x) = x^3 - 3bx^2 + 3cx + d has a local maximum and a local minimum...

Point on a Parabola [1/9/1995]

How do you find the point on the parabola y = x^2 nearest the point (- 3,0)?

Points of Intersection [08/31/1997]

Estimate all points of intersection of the graphs f(x) = x + sin x and g(x) = x^3 by using a calculator or computer. How do you know you have found them all?

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