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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?
- Derivatives [12/07/1998]
Can you write x^2 = x+x+...+x (x times) if x is not a positive integer?
- Derivatives and Antiderivatives [10/31/2001]
When I run the formula int(sqrt(cos(x^2)),x=0..1) in Maple, it just
returns the formula. Why?
- Derivatives, and the Big Deal with Little Intervals [04/07/2015]
A teen struggles with the concept of instantaneous rates of change. Throughout
several rounds of clarification and analogies, Doctor Ian underscores how remarkably well calculus
models natural phenomena.
- Deriving Arctan without Using Numbers [03/22/2002]
How do you do you derive arctan without using numbers, or how do you
prove (arctan)' = 1/1+x^2?
- Deriving Simpson's Rule [06/07/1998]
Can you show me a derivation of Simpson's Rule?
- Deriving the Area of a Sphere [10/21/2003]
I know the area of a sphere is 4phi(r^2), but I'm wondering how to
derive that formula. I know it should be done in cylindrical
coordinates, and I'm thinking that the arc of a circle is defined as
rd(theta) and it's multiplied with rd(phi) to get (r^2)d(theta)d(phi).
Could you please help explain this?
- Deriving the Gamma Function [12/15/2000]
How can you prove that sqrt(pi)/2 = (1/2)!, and what is a fractional
factorial like that equal to?
- Deriving the Integral for the Surface Area of a Sphere [06/28/2001]
I need to find the cost per square foot of steel that makes up a tank, so
I need a way to derive the surface area of part of a sphere with a given
- Deriving the Quotient Rule [9/5/1995]
How do I prove this derivation: (f/g)' = fg'-fg'/g^2 ?
- Deriving Trig Functions; Taylor Series [05/01/2001]
How would I find, from first principles - no tables, no calculator - for
example, 32 degrees? If I use a formula, how is it derived?
- Descartes' Method for Tangents [09/02/1998]
Can you help me find the equation of the tangent of y^2 = 2x using
- Design a More Efficient Soda Can [4/23/1995]
The problem is to design a more efficient soda can that holds the regular
12 oz. of liquid. The can needs to have the least possible surface area.
- Determining Tangent to an Ellipse, Minimizing Area [02/20/1998]
Find the equation of the tangent to the ellipse that forms, with the
coordinate axes, the triangle of smallest possible area.
- Determining the Length of a Coil of Ribbon [08/31/2001]
How will calculus give me the same answer I found with algebra and
- Determining Whether a Function is Continuous [06/10/2000]
How can you tell whether or not a function is continuous?
- Differences in Differentiation Notation? [06/06/2013]
A calculus textbook variously represents derivatives as dy/dx, d/dy, d/dx(f(x))....
What does it mean to differentiate "with respect to x" anyway? Doctor Peterson
makes sense of the many notations, starting with the basics.
- Differentiability, Intervals, Inflection Points of Piecewise Function [05/13/1998]
For what values of k and p will the function be continuous and
differentiable? On what interval will it be increasing? Find all points
- Differential Equations and Flow Rate [02/14/1999]
A solution containing 3 lb/gal of dye flows into a 100 gal tank at 5
- Differential Notation [04/16/2009]
Why is the second derivative written as d^2y/dx^2?
- Differentials [09/18/1998]
What is the difference between finding the derivatives of a function
(dy/dx), and finding its differentials (dy,dx)?
- Differentiate Twice [11/14/1997]
If y = xsin(3x) prove that y''+9y = 6cos(3x), where y'' is its second
- Differentiating [01/14/1998]
Please help me differentiate (y^2) * cos (1/y) = 2x + 2y.
- Differentiating and Integrating the Formula for Area of Circle [05/11/1998]
The formula for a circle's circumference is the derivative of the formula
for its area. What is the significance of this?
- Differentiating a Polynomial [04/25/1999]
Differentiate with respect to x: f(x) = x^8 + 3x^5 + (3-5x)^4.
- Differentiating the Ceiling Function [10/16/2000]
How do I find dD/dp for an equation involving the ceiling function? How
do I minimize D as a function of P?
- Differentiating the Rate of Decay [3/14/1996]
A lump of radioactive subtance is disintegrating at time 't' days after
it was first observed to have mass 10 grams and: (dm/dt) = -km (where k
is a positive constant). Find the time, in days, for the substance to
reduce to 1 gram in mass, given that its half life is 8 days.
- Differentiating Under the Integral Sign [01/11/2001]
Can you give me an example of the integration method called
"differentiating under the integral sign"?
- Differentiating y with Respect to ... y? [11/05/2010]
A student familiar with differentiation struggles to take the derivative of a function with
respect to the self-same variable. Doctor Ali puts the student back on track with a
- Differentiating y = x^x [11/3/1994]
Please could you differentiate y=X^X (that's X to the power of X)?
- Differentiation Problem [11/15/1997]
A light shines from the top of a pole 50 ft. high. A ball is dropped from
the same height at a point 30 ft. away from the light...
- Distance From a Point to a Plane [03/31/1998]
Can you show me the proof of the formula for the distance between a point
and a plane?
- Distance to the Sun [04/16/1999]
Find the distance from Earth to the sun when t = 90 days...
- Does f Have a Local Extrema at x = 0? [07/10/2003]
f(x) = (x^3)/6 + (x^2)/2 + cosx - 1. Does f have a local extremum at x
- Domain, Asymptotes, Intercepts of a Function [04/01/2003]
What is the domain of this function? What asymptotes does it have?
What are the x and y intercepts? Etc...
- Domain/Range of a Function [01/22/1997]
How do you find the domain and range of the function f(x) = 2x^2-3x+1?
(Both with and without calculus.)
- Dominant Terms [03/25/1998]
What are dominant terms, and how do you obtain their values?
- Donkey Grazing Half a Field [08/08/1997]
A donkey is attached by a rope to a point on the perimeter of a circular
field. How long should the rope be so that the donkey can graze exactly
half the field?
- Double Integration in Polar Coordinates [03/27/2003]
Evaluate double integral x-y/x*2+y*2 over x*2+y*2 equal to or less
- e as a Series and a Limit [03/30/1998]
Why does e = 1 + 1/2! + 1/3! + 1/4! + ... and lim (1 + 1/n) ^ n, as n --
- An Easy Definition of Calculus [9/4/1995]
What is calculus?