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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?
- 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
- Solving a Window Problem with Calculus [11/11/1995]
My question is just like the Norman window question, but I have an
isosceles triangle on top. Where do I start? I seem to have 3 unknowns.
b, h and x.
- Solving Differential Equations [5/8/1996]
(2x^2 + 2y^2 - y)dx + (x^2y + y^3 + x)dy = 0
- Solving Equations Numerically: Newton's Method, Bisection Method [11/05/2002]
Can you tell me methods to work out equations that can only be solved
numerically and not algebraically, and suggest some good equations
with which I could try out these methods on using graphs?
- Solving for Arc Length [9/10/1996]
How do I use the formula for arc length solve for the length of the
- Solving for Rate [8/8/1996]
In the equation P=V*R/(1-(1+R)^(-n)), how do you solve for "R"?
- Solving Functional Equations [08/11/2005]
How can I solve an equation like f(f(x)) + 5f(x) + 7 = x^2 + 6x?
- The Speed of Change of a Shadow [12/12/1995]
A street lamp whose light is 5 meters from the ground is 6 meters from a
vertical brick wall. A person, 2 meters tall...
- A Sphere's Surface Area and Volume [12/17/1998]
What is the relation between a sphere's surface area and its volume? How
does their ratio change?
- Sphere Surface Area Precision [04/22/2003]
How can the formula 4*pi*r^2 for the surface area of a sphere be
- Spread of a Virus Through a City [05/09/2002]
A Flu-like virus is spreading through a city of population 260000 at
a rate proportional to the product of the number of people already
infected and the number of people stilll uninfected.
- Square Roots and Limits [03/12/2003]
Given a value of x, find the value of the expression sqrt(x sqrt(x
- The Squeeze Theorem [10/12/1998]
How do you use the Squeeze Theorem to find the limit of f(x) = x *
sin(1/x) as x approaches 0?
- Standard vs. Symmetric Derivative of Sin(x) [02/07/2006]
Can I find the derivative of f(x) = sin(x) by using the limit
(sin(x+h) - sin(x-h))/2h as h->0 and then using a trig identity to
simplify the difference of sines?
- Stationary Point [10/27/1999]
How can I show that the curve y = x*ln(2x-3) has no stationary point in
the region x greater than 2?
- Stokes-Greens-Gauss Theorems [03/25/1997]
Are the Stokes-Greens-Gauss) theorems related? What is their
- Strongest Possible Beam From a Log [07/11/2002]
Find the dimensions of the strongest beam that can be cut from a
cylindrical log of radius 'a'.
- Subdividing a Field; Falling Water Level [7/28/1996]
What's the maximum area I can fence with a thousand yards? A water tank
has a shape of a right circular cone... how fast is the water level
- Substituting to Simplify the Integral [08/04/2003]
What is the integral of tan^3 x * secx dx?
- Substitution [06/18/1997]
Integrate sqrt(x^3-1)/x dx.
- Summation Formulas for Trigonometric Functions [04/24/2001]
How do you find the area under the curve of a trigonometric function
using the definition of a limit and not an approximation? Are there
summation formulas for trigonometric functions?
- Surface Area and Volume Derivative [10/30/2000]
For what 3D figures is the derivative of the volume formula equal to the
formula for surface area? With respect to which variable would you need
- Surface Area and Volume: Related Rates [02/22/1999]
The surface area of a cube is changing at a rate of 8 in./s^2. How fast
is the volume changing when the surface area is 60 square inches?
- Surface Area of a Football [02/12/1999]
Calculate the surface area of a football with a circumference of 22
inches and an arc length of 14 inches.
- Surface Area of an n-dimensional Sphere [07/28/1997]
I was wondering how to calculate the surface area of a sphere in n
- The Surface Area of a Rotational Curve [04/17/1998]
I'm having difficulty understanding how my math professor proved the
formula for the surface area of this curve...
- Surface Area of a Sphere [10/03/1997]
How is the surface area of a sphere calculated, and why?
- Surface Area of a Sphere [04/10/1998]
Can you derive the formula for the surface area of a sphere?
- Taking the Partial Derivative of a Function [09/06/2002]
Are the left and right sides of an equation always symmetric?
- Tangents to a Curve That Pass through a Fixed Point [08/08/2008]
How many tangent lines to the curve y = x/(x+1) pass through the point
(1,2)? At which points do those tangent lines touch the curve?