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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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Equation [04/03/1997]

Solve for x...

Equation for a Curved Line [01/19/1997]

Is there a formula into which you can enter a starting point, a starting slope, an ending point, and an ending slope that gives the equation of the curved line between these points?

Equation of a Circle [08/25/1997]

Write the equation of the specified circle in general form...

Equation of Parabola with Given Vertex, Point [8/4/1996]

How do I find the equations for all parabolas containing the origin whose vertex is at (2,1)?

Equations for the Paths of Four Lady Bugs [12/02/1998]

Four ladybugs travel from the corners of a square towards the center. Find the equations that describe the path.

Escaping L'Hopital's Loop [10/14/2003]

I'm trying to apply L'Hopital's rule, but I keep ending up with 0/0 after each application. What am I missing?

Euler Equation and DeMoivre's Theorem [05/18/1999]

Do you have a proof of the equation e^(i*Pi) + 1 = 0?

Evaluate the Limit [12/03/1997]

In Calculus today we were trying to evalulate the limit of (1-cos x)/ (x^2) as X->0.

Evaluating Two Definite Integrals [02/28/1998]

Evaluating a definite integral and determining an enclosed area.

Exact? Area under a Curve [03/13/1998]

Would you please explain why the area under a curve is exact and not an approximation?

Examples of Cooling, Growth and Decay [06/03/1999]

Can you give some real life problems involving Newton's cooling method, logistic growth, and exponential growth and decay?

Explaining the Intermediate Value Theorem [02/18/2004]

I want to know how to explain the Intermediate Value Theorem in its most simple form: If f is a continuous function on the closed interval [a,b] and N is any number between f(a) and f(b), there must be a number c in (a,b) such that f(c)= N.

Explanation of L'Hopital's Rule [02/06/2001]

I understand the method, and I somewhat understand the proof, but can you give me an explanation of why L'Hopital's Rule actually works?

Exploring the Distance from (0,0) to (1,1) with Limits [10/15/2006]

Any route traveling from (0,0) to (1,1) going only north and east will cover a total distance of 2 units. But the straight line distance from (0,0) to (1,1) is sqrt(2) units. It seems that if I think of a staircase connecting the two points and let the stairs become infinitely small, the limit of the north/east route distance should converge to sqrt(2). But it doesn't! What's going on here?

Exponential Growth & Compound Interest [6/2/1996]

(1) The cost of goods and services in an urban area increased 1.5% last month. If the rate continues, what will be the annual rate of increase? (2) Suppose a certain population increases by 30% every 10 years. By what percent does it increase each year?

Exponential Growth of a Population [05/03/1998]

A population P(t) is increasing at a rate directly proportional to 800 - P(t), where the constant of proportionality is k...

Exponential Growth of Bacteria [8/7/1995]

A colony of N bacteria increases with time according to the formula dN/dT = kN. If initially there are N' bacteria, show that the number at any time is given by N = N'e^kt.

Factoring Polynomials [02/12/1999]

What are some real-life applications of factoring polynomials?

The Figure of Maximum Area and Given Perimeter [06/02/1998]

Can you help me show, with and without calculus, that the geometric figure of a maximum area and given perimeter is a circle? What are the dimensions of a triangle with perimeter p that encloses the maximum area?

Find a Function, Integrals [12/17/1997]

Suppose the graph of f has the formula f(x)=-x+1 for 0<=x<=1; x-1 for 1<x<=2 ...

Find Curve Given Slope, Point [8/29/1996]

Find the curve whose slope at (x,y) is 3x^2 and which passes through (1,- 1).

Finding a Definite Integral with Natural Log and Sine [05/14/2008]

I'm trying to evaluate the definite integral of ln(sin(x)) dx from 0 to pi. I've tried integration by parts but am not getting anywhere.

Finding a Limit [02/28/2002]

My calculus teacher gave my class the answer to this limit; however, we cannot figure out how to get the answer that is given.

Finding a Vertical Asymptote [05/11/2000]

How can I find the vertical asymptote of the equation xy^2 - x^3y = 6?

Finding First Derivatives by Definition [10/30/1998]

Can you show me an easy example of how to find the first derivative of a function using the definition?

Finding Integrals Using Trigonometric Substitutions [10/2/1995]

What is the integral of 1/(x(1-x))^(1/2)? How can I integrate this function using only the derivative of arcsin or arctan?

Finding Limits Using Natural Logs and L'Hopital's Rule [05/03/1998]

I think I need to use l'Hopital's Rule and the natural logarithm function to find lim (x goes to 0) [(cos (2x))^(3/(x^2))].

Finding Maxima and Minima: Turning Points [05/10/1998]

Can you help me find all of the maxima and minima of the function: 5e^(- x/2)sin(2x)?

Finding Tangents to Curves Using Derivatives [09/29/1999]

Three problems that involve finding the tangents of a curve using calculus.

Finding the Area of Overlap between Two Circles [04/16/2008]

Suppose I know the centers and radii of two circles that overlap. How can I find the area of that overlapping region?

Finding the Derivative [07/27/1999]

Find dy/dN if y= A*(1 - e(-cN))* (1 - e(-cM)).

Finding the Domain of a Function [9/28/1995]

What is the domain of f(x)=((2-x)/(x-2))^.5

Finding the Equation of a Tangent Line [10/6/1995]

What is the equation of the tangent line to the graph of f(x)=x^(2/5) at the point (-32,4)?

Finding the Equation of a Tangent Line [11/24/1998]

I only have where the tangent intersects the y axis...

Finding the Extrema of a Function [10/25/1996]

Find the extrema of f(x)=(ln(1+exp(-a-bx-cx^2)))/((x+d)^2)...

Finding the Factorial of Non-Integers [01/06/2005]

I thought the factorial function was only defined for positive integers, but I notice that my calulator returns a value for non-integers like 3.5! What's going on?

Finding the Indefinite Integral [2/1/1996]

I am trying to find the indefinite integral of the following function: f(x) = x^x.

Finding the Limit [10/24/1995]

For a limit to exist, the right side and left side have to equal each other. Say you have a limit such as "lim as x->4 for function (3x-4). To see if this function exists, you have to check it on both sides. Would you plug in 3 for the left and for the right?

Finding the Maximum Area of a Rectangle [1/28/1996]

A rectangle is inscribed in a triangle (with certain dimensions). Find the maximum area of the rectangle.

Finding the Roots of a Function [11/4/1996]

How do you find the positive roots of x^9+3x^8-5x^3+4x+6=0? Show that x^n- 1=0 has exactly 2 roots if n is even, and only 1 real root if n is odd

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