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Selected answers to common questions:
Maximizing the volume of a box.
Maximizing the volume of a cylinder.
Volume of a tank.
What is a derivative?
- Rate of Change Constant? [03/20/2002]
The rate of change of e^x is e^x. Does this mean that the rate of change
is constant? Why are sinx,cosx,... and sinhx,coshx,.... similar?
- Rate of Change of a Function [6/10/1996]
Show that the expression for the rate of change can also be given by -cos
x cot^2 x.
- Rate of Change of Angle [8/12/1996]
Find the rate at which angle BXA is changing in radians per second...
- Rate of Change of Radius of a Cone [03/07/1999]
Using the measured rate of change in volume of water in a cone, calculate
the rate at which the circular area of water reduces when the radius is r
- Ratioing Errors [01/07/2002]
How do you calculate the error bars for ratios?
- Rational Function [04/03/1997]
Given the rational function y=r(x)=(4x^6-x^4)/(x^5+5) describe its end
- 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
- Recurrence Relation Resolution [11/25/2013]
A student struggles to determine the limit points and explicit formula for a recurrence
relation complicated by powers and other operations. Exploiting derivatives and
ratios, Doctor Vogler shows the way.
- 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
- 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
- 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
- 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
- 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
- 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/
- 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) ?