Expert: Paul Klarreich - 5/16/2007

Question
1) let f and g be odd functions.  if p, r, and s are nonzero functoins defined as follows, which must be odd?
I. p(x) = f(g(x))
II. r(x) = f(x) + g(x)
III. s(x) = f(x)g(x)

2) what is the average value of y for the part of the curve y=3x-x^2 which is in the first quadrant?

3) the volume of the solid obtained by revolving the region enclosed by the ellipse x^2 + 9 y^2 = 9 about the xaxis is?

4)the volume of a cylindrical tin can with a top and bottom is to be 16pie cubic inches.  if a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can?

5) integral from pie/2 to 0 of x cosx = ?

6) the lim h--> 0
tan3(x+h) - tan(3x)/
h is?

7) a region in the first quadrant is enclosed by the graphs y = e^2x, x=1 and the coordinate axes.  If the region is rotated about the y axix, the volume of the solid that is generated is represented by what?

Answer
Questioner:   Cobb
Category:  Calculus
 
Question:  1) let f and g be odd functions.  if p, r, and s are nonzero functoins defined as follows, which must be odd?
I. p(x) = f(g(x))
II. r(x) = f(x) + g(x)
III. s(x) = f(x)g(x)

2) what is the average value of y for the part of the curve y=3x-x^2 which is in the first quadrant?

3) the volume of the solid obtained by revolving the region enclosed by the ellipse x^2 + 9 y^2 = 9 about the xaxis is?

4)the volume of a cylindrical tin can with a top and bottom is to be 16pie cubic inches.  if a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can?

5) integral from pie/2 to 0 of x cosx = ?

6) the lim h--> 0
tan3(x+h) - tan(3x)/
h is?

7) a region in the first quadrant is enclosed by the graphs y = e^2x, x=1 and the coordinate axes.  If the region is rotated about the y axix, the volume of the solid that is generated is represented by what?
........................................
Hi, Cobb,

This is a lot of questions.  Is this a take-home test or something?  These questions come from all over the standard calculus curriculum.

I'll answer a couple of them now.  When I get some explanation from you, I'll consider the others.

Question:  1) let f and g be odd functions.  if p, r, and s are nonzero functoins defined as follows, which must be odd?

I. p(x) = f(g(x))

p(-x) = f(g(-x)) = f(-g(x)) = - f(g(x)) = - p(x).  p is an odd function.

II. r(x) = f(x) + g(x)

r(-x) = f(-x) + g(-x) = -f(x) - g(x) = -(f(x) + g(x)) = - r(x)

r is an odd function.

III. s(x) = f(x)g(x)

s(-x) = f(-x)g(-x) = (-f(x)) (-g(x)) = f(x)g(x) = s(x)

s is an even function.

..........................  
6) the lim h--> 0
tan3(x+h) - tan(3x)/
h is?
    tan(3(x+h)) - tan(3x)
lim  --------------------- = the derivative of tan(3x)
h->0           h

= 3 sec^2(3x)