I was stuck on a couple of free response problems.
1) Given f(x) = |sin x|, -pi < x < pi, and g(x) = (x^2)
for all real x.
a. On the axes provided, sketch the graph of f.
b. Let H(x) = g(f(x)). Write an expression for H(x).
c. Find the domain and range of H.
d. Find an equation of the line tangent to the graph of H at the point where x =pi/4.
--For this one, I figured out the picture of the graph, but I could not figure out what to do after that. Is H(x) expressed as (|sin x|)^2?
2) Given the function f defined by f(x) = cos(x) - (cos^2)x
x for -pi < x < pi.
a. Find the x-intercepts of the graph of f.
b. Find the x- and y-coordinates of all relative maximum points of f. Justify your answer.
c. Find the intervals on which the graph of f is increasing.
d. Using the information found in parts a, b, and c, sketch the graph of f on the axes provided.
--For this one, I got pi/2, 3pi/2, and 0 as the x-intercepts, but I can't figure out what to do after that.
3) Let R be the region enclosed by the graphs of y = (x^3) and y= sqrt(x).
a. Find the area of R.
b. Find the volume of the solid generated by revolving R about the x-axis.
--I actually don't understand how to do this one at all.