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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?
- Absolute Values and e in an Integrand [05/10/1998]
Integrating INT(e^(-|x|)) dx by treating x > 0 and x < 0 as separate
- Adding Infinity [06/25/2003]
What is -infinity plus infinity?
- Algebraic Integration of Standard Normal Distribution Function [02/11/1998]
I've tried inspection and partial integration and don't seem to get the
right results - I wonder if this is a function whose integral cannot be
- Algebra, Limit and Derivative Problems [7/28/1996]
Let x+y=1 and x^3+y^3=19. What is the value of x^2 + y^2?...
f(x) = sqrt(2x+4), find f'(x)... d(8x^3+8x^2+6x+1)/d(2x) =
- Alternate Forms of the Derivative Formula [10/06/2001]
Why does f'(a) = limit as x approaches a of [f(x)-f(a)]/(x-a), and how do
you use this formula to find the derivative? I get confused about when to
plug in x for a or a for x.
- Alternating Harmonic Series [11/18/1997]
I am trying to find the proof for the sum of the alternating harmonic
series. I did find out that it is ln(2), but please tell me why?
- Analysis of Signs of Derivatives of an Inverse Function [06/22/2005]
Given a continuous function f with f'> 0 and f''< 0, can I conclude
anything about the signs of the derivatives of g, which is the inverse
function of f?
- Analytic Integration Problem [7/13/1995]
Is it possible to integrate [exp(-zcosh(t)] dt analytically?
- Analyzing a Function [05/19/1997]
Find the domain, intercepts, asymptotes, critical points, points of
inflection, and graph of the function: y = 3x^4-4x^3-12x^2.
- Another Grazing Cow [6/7/1995]
A man has a barn that is 20 ft by 10 ft. He tethers a cow to one corner
of the outside of the barn using a 50-ft rope. What is the total area
that the cow is capable of grazing?
- Antiderivative [11/17/1994]
One of my friends has asked me to solve a problem for her. I arrived at
the right answer using my method of doing it. Later, she showed me her
method. It seems correct, too. I was just wondering if you can find a
mistake for her. Question/answer: antiderivative of secant square 3X
times tangent 3X= 1/6 (cos3X) to the negative 2 plus C.
- AP Calculus - Minimizing Time Traveled [12/22/1998]
A person on a boat 9 km from the shore must go 12 km down the shoreline
in the shortest time possible. Find a function for where the person
should hit land...
- AP Calculus - Minimums and Maximums [12/23/1998]
Hints for finding specified x-coordinates...
- AP Calculus Test [8/24/1996]
Where can I find a site with information about the AP calculus test?
- Applications of Imaginary Numbers [10/14/1997]
Where are imaginary numbers used today in real life, as in the work force
or other areas that use math?
- Applied Problems in Maxima and Minima [08/30/1997]
At what first quadrant point on the parabola y = 4 - x^2 does the tangent
together with the coordinate axes determine a triangle of minimum area?
- Arc Length Formula [07/28/1998]
Find the arc length of a curve in symbolic form and decimal
- Arc Length of Sine Curve [7/18/1996]
What is the arc length of sin x from x = [0, 2pi]?
- Area Between a Line and a Curve [11/5/1994]
A high school senior writes: "The horizontal line y=c intersects the
curve y=2x-3x^3 in the first quadrant. Area A is bounded by y=c, x=0, and
the curve. Area B is bounded above by the curve and below by y=c. Find c
such that (area of A)=(area of B)." I can do this using integrals, but I
don't want to solve any cubics. There must be an easier way to find c.
- Area Between Curves [01/11/1997]
Find the area bounded on the right by x+y = 2, on the left by y = x^2,
and below by the x-axis.
- Area between two Curves [12/24/1995]
Find the area between the curve x = 1-y^2 and the y-axis?
- Area, Circumference of an Ellipse [7/29/1996]
How do I calculate the area and circumference of a given ellipse?
- Area of a Curved Figure [07/26/1997]
How can you find the area of a curved figure without using calculus?
- Area of an Ellipse [08/03/2001]
Show that the area enclosed by the ellipse x^2/a^2 + y^2/b^2 = 1 is
pi*a*b, where a is the longer horizontal part and b is the shorter
vertical part of an ellipse.
- Area of a Parabola [09/05/1997]
How do you find the area of a parabola? (I just finished Algebra 2.)
- Area of A Sector of An Ellipse [02/28/1998]
Finding the area of a sector of an ellipse, given the semiminor and major
axes and the angles of the 2 vectors bounding the sector.
- Area of a volume of revolution [11/22/1994]
I am a little confused about finding volumes when graphs are revolved
around an axis - specifically about finding the volume of a disk and the
volume of a washer.
- Area of Union of Two Circles [6/10/1996]
If the effective length of a rope tied to a goat is L, and the goat can
eat exactly half of the grass in a field, express L in terms of R.
- Area Under a Parabola [06/27/1997]
Why is the area under the parabola y = x^2 between the y-axis and the
line x = b equal to b^3/3?
- The Area Under a Parabola and the Length of a Curve [11/15/1995]
Find the area under the parabola f(x) = x^2 + 1 for x = 0 to x = 2, and y
> 0. Also find the length of the curve from x = 0 to x = 2.
- Area Under the Curve y = x^0 [02/14/2005]
Sketch the graph of y = x^0, where x e [0,1], and calculate the exact
- Arterial Branching Calculus [05/28/2003]
In the surgical connection of a small artery to an existing larger
artery, attention must be paid to minimising the viscous resistance
to the blood flow... Investigate the distance from B of the join and
hence the angle of attachment that minimises the resistance to flow.
- Art Gallery Problem [11/30/1997]
A picture is two meters high and is hanging so the bottom of the picture
is one meter above eye level. How far from the wall should you stand so
the angle of vision occupied by the picture is a maximum?
- Assigning a Piecewise Function in a Proof [01/11/2001]
Can you explain how we can arbitrarily assign e(u) = 0 when u = g(c) in
this chain rule proof of composite functions?
- Asymptote of a Function [06/02/2002]
Determine the value of A so that y = (Ax+5)/(3-6x) has a horizontal
asymptote at y = -2/3.
- Average Radial Distance of Points within a Circle [03/26/2003]
I'm trying to determine the average value of a circular/radial
gradient that is at full value (white, call it 100% brightness) in the
center, and drops in a linear fashion to zero (black, call it 0%
brightness) at the radius.
- Average Rate of Change [10/30/2002]
Find the average rate of change of y with respect to x on the interval
[2, 3], where y = x^3 + 2.
- Balance Point between Converging and Diverging Infinite Series [03/20/2002]
Is there a way to find the balance point between convergence and
divergence for any type of series - that point where the n(th)term gets
smaller JUST fast enough for the series to converge?
- Barycentric Calculus [01/06/1999]
How does barycentric calculus compare with trilinear or cartesian
- The Bay of Fundy [04/03/1997]
The Bay of Fundy in Canada is reputed to have the largest tides in the
world, with the difference between low and high water level being as much
as 15 meters...