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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?
- 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
- Finding the Shape of a Funnel Curve [04/08/2001]
Find the equation of the curve of revolution of a coin leaving a chute.
- Finding the Sum of Arithmetico-Geometric Series [09/13/2004]
Find the sum of the infinite series 1/7 + 4/(7^2) + 9/(7^3) + 16/(7^4)
+ ... I would also like to know if there is a general rule to find the sum
of (n^2/p^n) for n = 1 to infinity.
- Finding the Width of a Corridor [11/13/1995]
A steel girder 27 ft long is moved horizontally along a passageway 8 ft
wide and into a corridor at right angles to the passageway. How wide must
the corridor be for the girder to go around the corner? Neglect the
horizontal width of the girder.
- Finding Volumes: Washer/Disc Methods [01/05/1998]
I am having difficulties in the washer and disc methods of finding
volumes, as in the problem y=x^2, x=3 about the y axis.
- Find Polynomial G(x) (Impossibility) [7/1/1996]
For all real x, F(x) = x+3 and G(x) is a polynomial of degree two such
that G(F(x)) = x+2. Find G(x).
- Find the Density Function [01/24/2002]
Suppose X1, X2, X3 are independent and uniform on(0,1). Find the density
function of S = X1 + X2 + X3.
- Find the Path and the Time Taken [10/17/2001]
A man wants to cross a river 500m wide. His rowing speed relative to the
water is 3000 m/hr. The river flows at a speed of 1200 m/hr...
- Find x in Degrees, not Radians [01/14/1998]
I need to find the derivative of sin(x) when x is in degrees, not
- Fixed-Point Theorem [05/02/2002]
An intuitive reason why the theorem is true, and a proof.
- Floor and Mod Functions Discontinuous [10/10/2001]
What is f(x) = x mod b for some fixed b?
- Formula for the Curvative of a Curve [10/10/2002]
Is there a formula or theorem I can use to convince someone who
believes a wall has only side that there is another side to it?
- Formula for the Trapezoid Rule [07/31/1999]
I am writing a program to find the moment area of any shape...
- Formula to Sum a Series of Square Roots [07/03/2004]
Is there a way to calculate the sum of the series sqrt(x) as x goes
from 1 to n?
- Fresnel Integral of sin(x^2) [03/28/2002]
I can not see how this integral from 0 to infinity can have any limit at
all. If we used infinite series, we would not get a value for x =
- Frustum of a Pyramid with a Rectangular Base [02/20/2002]
I am an engineer with a water treatment agency and need to figure the
amount of water per foot of elevation in our reservoirs...
- Fuel Left in an Ellipsoidal Tank [10/19/2002]
How much fuel is left in a tank at any depth, given that the tank is
- Functions of a Complex Variable [10/05/2002]
Show that w=f(z*), where z*=x-iy and f is a differentiable function,
is not an analytic function of z.
- Function X is a variable... [1/9/1995]
Right now we are learning general exponential and logarithmic functions.
In one of my homework problems....
- The Gamma Function and Its Derivative [05/26/1998]
Does n! = the integral from 0 to infinity of (x^n)(e^-x)dx hold true for
all real numbers? If so, can we find the derivative of n!?
- Gamma Function Division [12/20/2005]
Is there a formula for Gamma(n)/Gamma(m) for n and m having integer
values >= 1 and m < n?
- Gas in a Cylindrical Tank with Hemispheric Caps [02/09/2010]
Doctor Jeremiah uses calculus to calculate how much gas a large pill-
shaped tank holds.
- General and Particular Solutions [01/08/1999]
What is the difference between a general solution and a particular
solution of a differential equation?
- Generalized Cubic Equation [05/22/2000]
How do we find a generalized cubic rule for a family of cubics that cut
through the point (1,1), have a gradient of 2 when x = 1, and have 1
- General Solution of dy/dt + y = t [10/25/1999]
How can I find the general solution to the equation dy/dt + y = t?
- Generating the Volume of an n-Sphere [02/07/2002]
Could you help me find an equation for the integration of a unit-ball in
- Geometric Proof of a Limit [12/06/2003]
Prove that lim[x->0](sinx)/x = 1 without using L'Hopital's rule.
- Goat and Silo [05/31/2001]
A goat is roped to a point A next to a silo 30 ft. in diameter. The rope
is 15pi feet long. How much area can goat graze?
- Goat tied to a barn [11/8/1994]
If there is a goat tied to a rectangular barn on a 50 foot lead and the
barn is 20 feet by 20 feet (floor), what is the maximum grazing area?
Assume the goat is tied to a corner.
- The Gradient [06/03/1997]
What is the geometrical interpretation of the gradient?
- Graphing Absolute Values [07/03/2002]
I have a function, f(x), that I can graph, but I don't understand how
to graph |f(x)|.
- Graphing Limits [09/12/2001]
I was hoping you could explain the concept of "limits" and how to read
them from graphs.
- Graphs and Gradients [05/13/2003]
Sketch graphs of f and f' for f(x) = sin x and guess the formula for
- Great Circle Parametric Equation [05/25/1998]
How can you calculate specific points of a great circle on a sphere? Can
you help me find the parametric equation?
- Half-Life of a Radioactive Substance [8/4/1995]
The half-life of a radioactive substance is d days. If you begin with a
sample weighing b g, how long will it be until c g remains?
- Halving and Halving Again - Zeno's Paradox [8/22/1996]
Since we can never really get to zero by reducing something by halves,
does that mean that we are floating on air?
- Harmonic Series [04/28/2003]
Prove that if in the sum 1 + 1/2 + 1/3 + ... + 1/n we throw out each
term that contains 9 as a digit in its denominator, then the sum of
the remaining terms is less than 80.
- Heat Supplied and Heat Loss of a Furnace [05/17/2000]
Heat is supplied by a furnace at a constant rate, but is lost at a rate
determined by the difference between its temperature and the temperature
of its surroundings...