Ask Dr. Math

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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Area of an Ellipse using Integral Calculus [11/4/1996]

How do you find the area of an ellipse?

Calculus Chain Rule [11/23/1997]

Could you give me the chain rule in easy terms, not a formula?

Coordinates of Relative Maxima or Minima [12/31/1994]

Given f(x) = 15 x^(2/3) + 5x, find coordinates of relative max. or min. For what x values is the function concave down?

Counting Bacteria with Calculus [2/15/1996]

The number of bacteria in a culture increases from 3000 to 9000 in 8 hours. If the rate of increase is proportional to the number present, estimate the number of bacteria at the end of 24 hours.

Critical points [12/19/1994]

Find all critical points (if any) in the following problem:
k(t)=1/the square root of (t-squared +1).

Definite Integral Across Addition, Subtraction [8/10/1996]

What is the integral of 3(x^2)-5x+9 from 0 to 7?

Definition of the Signum Function [05/31/2000]

Can you give me a simple definition of the signum function, and any practical examples of its usage?

Derivatives [10/28/1996]

What is a derivative?

Differentiation [05/17/1998]

I'm trying to find a GOOD definition for "differentiation."

Finding the Limit [1/25/1996]

I have a question about the limit of a sequence ...

Finding the Minimum Average Cost [2/12/1996]

The cost of producing x units of a certain product is given by C = 10,000 + 5x + (1/9)x^2. Find the value of x that gives the minimum average cost.

Find the Value of a... [1/9/1995]

Find the value of "a" so that the circles with equations (x-a)^2 + y^2 = 2 and (x+a)^2 + y^2 = 2 intersect at points where their tangents are perpendicular.

Formal Definition of a Limit [10/17/2001]

Could you please explain the formal definition of a limit? I need help specifically with finding a delta for a given epsilon and using the epsilon-delta definiton of a limit.

Functions Without a Second Derivative [6/28/1996]

What are some examples of functions of a real variable whose derivatives don't have derivatives?

Head start on Calculus: Differentiation, Integration [07/08/1997]

I'm soon to be a Calculus student in high school. Could you please teach me some Calculus over the internet?

An Introduction to Derivatives [05/17/1998]

I need to find out about third derivatives for a project. Can you give me some information, including uses, of derivatives?

Lorenz Equations [12/25/2002]

How do Lorenz equations work?

Mean Value Theorem [11/18/1994]

This Theorem states that "If f(X) is continuous in [a,b], then there is a 'z' in (a,b) such that the integral from a to b f(x)dx = f(z)(b-a). How does the Mean Value Theorem apply to life? Why do we need this theorem?

Murdered Professor [06/26/1997]

Your professor has been murdered. Using exponents and differential equations, prove that it did not happen while you were in his office.

Open Box Problem [06/22/2003]

Find the formula for the greatest volume box you can make from a sheet of cardboard with different-sized corners cut out of it.

Product and Rules, How to Remember Them [10/19/1995]

I have forgotten the formula for the integral of f(x)*g(x) and f(x)/g(x). Can you help me?

Proof of L'Hopital's Rule [12/23/1998]

Can you show me a proof of L'Hopital's Rule and say how it relates to the different versions of the Rule?

Proof of Product and Quotient Rules [02/07/2002]

I would like to know how the product rule and the quotient rule came about so I can better understand calculus.

Proving the Fundamental Theorems of Calculus [02/02/2001]

Can you provide a proof of the first and second Fundamental Theorems of Calculus?

Related Rates [11/26/1998]

A ladder 4m long is leaning against the vertical wall of a house... how fast is the top of the ladder sliding down that wall...?

Techniques of Integration - Change of Variables [02/17/1999]

Solve: (integral sign) sin 2x/ sq rt (9-cos^4 x) dx .

Tips for Integrating Functions [05/27/1998]

Can you give me some tips to help me integrate functions?

Two Ways to Differentiate [6/30/1996]

How can it be that if y=x^2, then y'=2x, but if y = x+x+...+x (x*x) then y'=1+1+...+1 (1*x) = x?

Volume equations for a sphere and pyramid [6/10/1996]

I am now thoroughly confused: we just learned the formulas for volume of a sphere and volume of a pyramid, but, he wouldn't tell me how to do it.

Volume of Liquid in the Bottom of a Tank [08/11/2003]

I need to be able to calculate the volume of stored liquid in the parabolic bowl of a cylindrical vertical tank based on the liquid level.

What is a Derivative? [06/07/1998]

What is the concise definition of a derivative? How do we use derivatives?

What Is An Integral? [10/14/2003]

I'm in eighth grade, and curious about what an integral does.

What is Calculus? [05/06/1997]

What is calculus and how does it work?

What is the Purpose of Determining a Derivative? [1/12/1995]

What is the purpose of the derivative?

Why is "e" so important? [1/8/1995]

Why is "e" so important? How significant is "e" compared with "pi?" How did it come about? How is it defined? Why is it taught only at higher level mathematics? Are there other numbers like "e"?

0 Raised to a Power [04/25/2002]

What is the limit of 0^n as n approaches infinity? Can you explain why it equals zero or why it can't equal zero, or if it's undefined?

1 = 0 Fallacy [06/22/2001]

Here is a 'proof' we ran up against during high school in the Netherlands that makes use of integral calculus.

1/4 Tank Dipstick Problem (from Car Talk) [12/04/2002]

The gauge on Rick's 18-wheeler is broken, so he uses a dowel to measure the diesel in his tank, which is cylinder-shaped, 20 inches in diameter, and sits on its side. How can he mark the dipstick to show 1/4 of a tank of fuel?

2nd Derivative [11/21/1994]

I am a grade twelve student taking calculus and was wondering if you could help me with this problem: y=2x(6x+5)exp4 - solve to the second derivative.

Absolute Values and e in an Integrand [05/10/1998]

Integrating INT(e^(-|x|)) dx by treating x > 0 and x < 0 as separate cases.

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