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High School Archive

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?

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Big O: 2 Functions Map from N to N+ [6/25/1996]

If f, g: N -> N+, then either f(x) is O(g(x)) or g(x) is O(f(x)). Why?

Brackets or Parentheses? [01/07/1997]

When using interval notation to describe when a function is increasing and decreasing, how do I know whether to use brackets or parentheses?

A Brief Overview of Calculus [01/16/2001]

I am required to take a calculus course, but I have no experience with it. What should I concentrate on?

Calculate the Derivative for Sqrt(x^5) [03/09/2003]

Why does the calculator fail when I calculate the derivative for sqrt(x^5)?

Calculating Any Root [10/13/1997]

I need to find an algorithm to determine any root of a number. I was told I could determine the estimated value by using Newton's Method...

Calculating the Shortest Flight Path [10/30/1997]

Is this spherical trigonometry or can it be solved using simpler geometric equations?

Calculus (Area of a Plane Region) [5/20/1996]

Problem: y = 4-x2 ; x axis - a) Draw a figure showing the region and a rectangular element of area; b) express the area of the region as the limit of a Riemann sum; c) find the limit in part b by evaluating a definite integral by the second fundamental theorem of the calculus.

Calculus Constants [11/16/1994]

If a "u substitution" is made for different parts of the function, I get different answers. However the back of my book says that only one of the answers is correct. Here is the problem: Integral of secant squared of 3x times tangent of 3x.

Calculus - limits [12/10/1995]

Find all the discontinuities of the following function and label those that are removable...

Calculus: Limits [11/24/1997]

Why does my textbook define the limit of (1/x) as x approaches zero as infinity?

Calculus (Minimizing Volume) [5/13/1996]

The arch y=sin x,0 less than or = to x less than or = to pi, is revolved around the line y=c to generate a solid. Find the value of c that minimizes the volume.

Calculus of Piecewise Functions [06/07/2003]

Can I take the integral or derivative of a piecewise function like the floor function [u] or the absolute value function |u| and still notate it in concise form, |U| or [U]?

Calculus Problem Using Taylor Series [2/16/1996]

Looking at the equation: 1 + x*2^(x-1)

A Calculus Question [02/09/1998]

I figured this worked out to -1/2: lim (x - (x^2 + x)^(1/2)) (x going to infinity).

Calculus Questions [1/27/1995]

A student from Riverdale has interesting integration questions.

Calculus: Rate of Change in Volume [07/27/1997]

The radius of a right circular cylinder is decreasing at the rate of 4 feet per minute, while the height is increasing at the rate of 2 feet per minute. Find the rate of change in the volume when the radius is 2 feet and the height is 6 feet.

Calculus - Shell or Washer Method? [08/23/1997]

I have a question regarding how to set up integrals for the volume of revolution.

Calculus Starting Points [10/23/2001]

I want to learn calculus because I have heard that it is important for applied math in engineering and I want to be an engineer. What should I learn, and in what order?

Calculus Word Problems [04/03/1997]

Minimizing functions and costs.

Can dy/dx Be Treated as a Fraction? [08/26/2004]

I learned that dy/dx was a notation that implied 'derivative of y with respect to x.' But in some calculus work, dy and dx seem to have individual meanings and dy/dx is treated as a fraction. How is this possible?

Can f'(-1) Equal Zero and f''(-1) Not Equal Zero? [03/23/2004]

Is it possible to have a derivative of zero and then have a double derivative that is not zero at that same x value? How?

Carbon Dating the Shroud of Turin [05/16/2000]

In 1988, scientists found that the Shroud of Turin contained 91% of the amount of carbon-14 contained in freshly made cloth of the same material. How old is the Shroud according to the data?

Cardan's Cubic Method [7/19/1996]

Cardan's well known formula solution of cubic equation x^3 - 3x^2 - x - Sqrt[2] = 0 gives us a single "monstrous" algebraic real root which appears to defy further simplification....

Cases Where the Newton-Raphson Method Fails [06/30/2005]

Why does the Newton-Raphson method work for some functions but not for others?

Catenary Curve [03/30/1999]

Find the vertex of a catenary curve.

Cauchy Principal Value Integral [09/17/2002]

Why isn't an integral from negative infinity to positive infinity defined as the limit as t (or any other variable) approaches infinity of the integral from -t to t?

Center of Mass [02/22/1999]

Find the center of mass of a thin plate of constant density covering the region bounded by the parabolas y = 2x^2-4x and y = 2x-x^2.

Center of Mass of a Semicircle [06/14/1999]

Is there a standard formula I can use to know where the center of mass of a semicircle is?

Centroid of an Arc [03/20/2002]

How do I find the centroid (or centre of gravity) of an arc of a circle?

Chain Rule [10/23/1996]

I have a question on the "chain rule" when finding the derivatives of polynomials.

Chain Rule and Product Rule: Differentiating [02/19/1998]

Why isn't the result what I expect when I use the Chain Rule and the Product Rule to differentiate?

Chain Rule Applied to Exponential Functions [10/11/2004]

How do I use the Chain Rule to find the derivative of a function such as f(t) = 27 e^(-.14t)?

Chain Rule: Function Notation [9/5/1996]

How can I apply the chain rule to find du/dx for dy/dx=(nu^n-1)*(du/dx)?

Chain Rule Notation [04/21/1999]

Taking the derivative of a function of a function.

Chain Rule: Prove the Quotient [05/30/2001]

Prove the quotient where u/v is uv^-1 using the product rule and the chain rule for (dv^-1/dx...

Changing Angle of a Tank [06/11/2003]

Points A and B represent pressure sensors in fixed positions on the base of a round tank. The chord through CD represents the water level in the tank. Lines a and b are the heights of water registered by each sensor...

Choosing Factors When Integrating by Parts [10/08/2003]

When integrating by parts, how do I decide what factors to use?

Classifying Cubics into Three Basic Types [08/11/2002]

How can I classify each of y=x^3+x^2, y=x^3-2x^2, y=x^3-2x^2+x, y=x^3- 2x^2+2x, and y=-x^3+x^2-x into one of three basic types without actually drawing their graphs?

Cola Consumption [05/03/1999]

The rate of consumption of Cola in the U.S. is S(t)= Ce^kt where S is measured in billions of gallons per year and t is measured in years from the beginning of 1980...

Computing e^(-2) Using a Power Series [04/16/2005]

Use a power series to compute e^(-2) to four decimal place accuracy.

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