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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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Raytrace of a Star Sapphire [12/11/1996]

How do I find the rotations necessary to create a raytrace of a star sapphire so that the star always faces the camera?

Rectification of a Quartic [12/02/1997]

I've been told that some 4th order equations can be rectified. Are there any Cassinian ovals whose perimeter can be calculated without looking up tables?

Related Rates [08/05/1997]

A tank contains 150 litres of brine solution. The concentration is 0.7 kg of salt per litre... Find the time when the salt content will be 90 kg.

Related Rates [03/11/1998]

The rate of change of the edges of a cube, given the rate of change of its volume.

Related Rates [03/01/1999]

A circle is inscribed in a square tangent to each side of the square. The circumference of the circle is increasing at a constant rate of 6 inches per second...

Related Rates [10/16/2003]

A related rate problem involving a 2 cm long hair lying on a spherical balloon as the balloon is inflated.

Related Rates and a Decreasing Radius [11/18/1998]

A nitroglycerin lump is decreasing at a rate of 2 cu.cm/sec. Use related rates to find how fast the lump's radius decreases at 9pi/4 cu.cm.

Related Rates - Clock Hands [04/05/1999]

How fast is the distance between the tips of the hands on a watch changing at one o'clock?

Related Rates Problems [01/15/1999]

A plane flying horizontally at an altitude of 1 mi and a speed of 500 miles/hour...

The Relation of the Speed of a Boat to the Water Line [2/27/1996]

What is the mathematical formula or equation for calculating the speed of a boat as it relates to the water line of the boat.?

The Relationship of the Tangent to a Curve and the Area under a Curve [10/25/1995]

The problem of drawing a tangent to a given curve at a given point is closely connected to the problem of finding areas ("areas under a curve"). What is the connection, and when and by whom was it discovered?

Relative and Absolute Extrema of a Function [01/07/2004]

What is the difference between the absolute extrema and the relative extrema in calculus?

Relativity [05/22/2002]

How do you demonstrate algebraically that [the following expressions] are proportional to each other even when they describe 'events' that are not on the expanding wavefront of light?

The Riemann Sum of Sin(x) [02/07/2001]

How do you evaluate the integral of sin(x) from 0 to pi by Riemann sum?

Riemann Sums and Antiderivatives [07/11/1999]

Are there any other ways of illustrating the link between Riemann sums and antiderivatives?

Riemann Sums and the Integral [01/18/1999]

Why does the limit of Riemann sums turn out to be the integral? How is that derived? What is behind it?

A Rumor Spreads [04/01/1999]

A rumor spreads through a community at the rate dy/dt = 2y(1-y), where y is the proportion of the population that has heard the rumor at time t...

Sea Navigation: Great Circles [11/4/1994]

In sea navigation, given two points A,B with LON and LAT coordinates, find the travel distance from A to B. The distance is actually the great circle passing A and B. The solution is needed in the form of a double integral where the first int covers the LONs and the second the LATs. I guess the difficult part is to form the vector.

Second Derivative of a Function With Absolute Values [02/27/1998]

Let f(x) = |x|^3. Find f"(0).

Second Order Differential Equation [11/04/2001]

By letting p = dy/dx, solve y(d2y/dx2) = 2 (dy/dx)2.

Sequences and Series [4/27/1996]

Determine the number of arithmetic means between two numbers whose sum is 5.5.

Setting Interest Rate to Maximize Profit [7/25/1996]

A loan company is limited to charging maximum 18 percent interest on a loan. The amount of money available for loans is proportional to the interest rate the company will pay its investors...

Shakespearean Differential Equations [10/27/1998]

Can you help me solve these differential equations: dG/dt=aJ-bG and dJ/ dt=cG?

Shortest Distance between a Point and a Circle [1/22/1996]

Find the shortest distance between the point (4,5) and the circumference of x^2 + y^2 = 9.

Simplifying and Solving a Natural Log Problem [11/15/1995]

Integrate ln(r/a)dr - the limits are a to b. Is this possible, or is there some table for it?

Simpson's Rule [04/12/1997]

Find the volume of the solid formed if the curve y = cos -1 x is rotated about the x-axis between x=0 and x=1.

Simpson's Rule - Alternating Pattern [09/20/1997]

I'm trying to apply Simpson's rule to a computer program, but I don't understand it well enough - can you tell me how it works?

Simpson's Rule for Cubics [10/23/2000]

Why is Simpson's rule exact for cubic polynomials?

Simpson's Rule/Trapezoid Rule [02/01/1998]

How is Simpson's rule derived and why it is a better approximation of the integral than the midpoint (rule?) or trapezoidal method?

Sketching a Function [07/06/1998]

Can you help me piece a function together so that the following hold? It is increasing and concave up on (-infinity, 1) ...

Sketching a Graph Given Information about Its Derivatives [07/30/2005]

We've been learning how to analyze a function by using the first and second derivatives to test if the graph is increasing/decreasing and concave up/down. But now we have to sketch the graph given some information about the derivatives and some specific points on the graph.

Slope of 3-Dimensional Equations [11/12/1997]

Is it possible to find the slope of three-dimensional equations?

Slope of root x. . . [12/14/1994]

How do I find the slope of: root x = -ln (xy) at (4,2) ?

Slope of the Tangent [1/9/1995]

Given y = A sinx + B tanx. Find A and B if the slope of the tangent to the curve at x1 = pi/4 is m1 = 4 + root 2, and at x = 0, m2 = 4.

Snail! [06/20/2002]

A snail is climbing a window-pane, beginning in the evening at a height of e minus 1 meter from the base. It loses 1 meter each night. On the second day, it doubles its altitude of the morning. On the third day, it triples the altitude of the morning, and so on. What will be its altitude on the 51st day at dawn?

Solve It Numerically [05/21/1997]

How do you solve the equation x^x = 100?

Solving a Differential Equation [6/5/1996]

The integral looking somewhat like this: S ((e^u)/u) du has resisted every attack on my part.

Solving a Logarithm [6/29/1996]

log a = -1.3

Solving an Equation using Power Series [05/18/1999]

How does one solve the differential equation (t^4)*x'' + x = 0 for x?

Solving an Equation with Infinite Exponents [03/15/2007]

If x^x^x^x^x^x^x^x^x^x^x...... = 2, solve for x. How can I solve that equation?

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