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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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Conditions for Real-valued Roots [04/01/1998]

Using derivatives to find the conditions that ensure that an equation of degree 3 has 3 distinct real-valued roots.

Conical Drinking Cup [06/20/1997]

A conical drinking cup is made from a circular piece of paper of radius r by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup.

Conjugates [11/8/1994]

What's a conjugate? (I'm in calculus, I've heard the term before, I just had a mental malfunction...)

Constant Function Zero and Orthogonal Functions [02/08/2002]

The constant function f(x) = 0 will give zero no matter what function g(x) it is integrated with. Does this mean that the constant function zero is orthogonal to all functions? Also, what could be the geometrical interpretation of orthogonal functions?

Constant of Integration [03/04/2003]

What's wrong with this proof using integration by parts?

Continuity [9/1/1996]

Is the absolute function continuous?

Continuity and Differentiability of Piecewise Function [05/04/1998]

What values of "a" and "b" will make this piecewise function continuous and differentiable?

Continuity of f(x) = sin(x)/x at x = 0 [01/20/2004]

Given f(x) = (sinx)/(x) for x =/= 0 and 1 for x = 0, show that f(x) is continuous at x = 0.

Convergence of Cos(x^2) or Sin(x^2) [11/15/1997]

How would you prove that the integral from n=0 to infinite of sin(x^ 2)dx or cos(x^2)dx converges?

Creasing Paper [05/11/1997]

Find the shortest crease when folding a piece of paper 6 units wide by 25 units long.

The Critical Number [03/23/1998]

Find the critical number of y = 2 cos x + sin 2x where x is greater than or equal to 0 and less than or equal to 2 Pi.

Cube Root Algorithm [04/04/1997]

Is there an algorithm for working out the cube root of numbers without a calculator?

Cubic Equations [12/8/1994]

In calculus, we were looking for the solutions for a third degree polynomial equation. Using my TI-85 to find the solutions, I stumbled upon an interesting observation. Given a cubic of the form ax^3 + bx + c, the absolute value of the sum of any two zeros seems to be equal to the absolute value of the third. However, I have so far been unable to give a general proof.

Cubic Polynomial [6/27/1996]

Show that the graph of every cubic polynomial has a point of symmetry.

Curl of a Vector Field [05/22/1999]

What does the term curl mean in math, and how do we define it in terms of fluid mechanics?

Cycloid [01/30/1998]

What is a cycloid and what does it do?

Cylinder Problem [11/22/1997]

Is it possible to construct two cans with different volumes but the same surface area?

Cylinder Volume and Surface Area [04/14/1999]

How does one make a cylinder that holds the greatest volume if the entire surface area is 600 cm squared?

Definite Integrals [05/04/1997]

What is the difference between the area under the curve from a to b and the definite integral from a to b?

Definition of the Limit [11/05/2002]

I need some help with the definition of the limit, particularly choosing delta for a given epsilon.

Degree of a Rational Function [06/19/2003]

Given: f(x) = p(x)/q(x). Prove that if the highest degree of p(x) is smaller than the highest degree of q(x), the horizontal is y=0.

Degrees of Generality [03/13/1999]

Different types of integrals: Riemann, Riemann-Stieltjes, and Lebesque.

Delta-Epsilon Limit Proofs [09/05/1998]

Can you help me with some delta-epsilon proofs? How do you choose delta?

Derivation of Geometric Formulas [5/29/1996]

What are the formulas for the surface area, total surface area, and volume of a sphere, and volume of a pyramid and cone?

Derivative Definition and Division by Zero [03/31/2007]

When calculating derivatives by hand using the definition given in first principles, we seem to eventually just let h = 0. But doesn't that violate the rule about division by 0 since the definition starts with a denominator of h?

Derivative Interceptions [02/10/1998]

I'm trying to figure out if there is a way to determine the points at which a derivative intersects a function.

Derivative of a Function [04/01/1999]

What is the formula for calculating the Kth order derivative of the function: f(x) = exp(A(x))?

Derivative of a Function [10/05/2001]

I am to find the derivative of the function (x-6)(x+1)/(x-6) at x=6. I simplified the function to a linear function x+1 with a "hole" at x=6. Then I tried to take the derivative.... but my calculator says division by zero, infinite result...

Derivative of an Exponential Function [04/23/2007]

How can I find the derivative of an exponential function like y = 4^x?

Derivative of an Odd Function [02/27/2002]

How do you show that the derivative of an odd function is even?

Derivative of e^x [10/28/1999]

How can I prove that the derivative of e^x is e^x?

Derivative of e^x by Limit Definition [05/21/2002]

Show that the derivative of e^x is e^x, using the limit definition but not the Taylor series expansion.

The Derivative of f(x)^g(x) [09/26/1998]

Is there a general formula to find the derivative of f(x)^g(x)?

Derivative of Inverse Cosine [05/24/2005]

Why does the derivative of arccos[cos(x)] turn out to be 1 or -1? How can it be both?

Derivative of Sine Function and Use of Chain Rule [11/06/2003]

How do I differentiate a sine function like f(x) = 2sin(0.2(x - 5))? The sine part is the most confusing, but I'm also not sure whether to use the product or chain rule.

Derivative of Tan x [03/31/2001]

Can you show me how to get the derivative of tan x using this definition of derivative...?

Derivative of x! [05/27/2003]

My Pre-Calculus teacher gave my class several derivative practice problems to do in our spare time, one of which is: (d/dx)(x!).

Derivatives [12/07/1998]

Can you write x^2 = x+x+...+x (x times) if x is not a positive integer?

Derivatives and Antiderivatives [10/31/2001]

When I run the formula int(sqrt(cos(x^2)),x=0..1) in Maple, it just returns the formula. Why?

Deriving Arctan without Using Numbers [03/22/2002]

How do you do you derive arctan without using numbers, or how do you prove (arctan)' = 1/1+x^2?

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