Thread: BC Calculus Questions

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.

Originally Posted by defjammer91

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.

1. Yes, it can be expressed as . Does that make finding the domain and range easier?

2. b) Find the first derivative and set it equal to 0 to find critical points. Any of the critical points that satisfy are relative maxima.

c) For the graph to be increasing, its derivative is positive. So solve

3. Notice that the enclosed region occurs where .

So evaluate the points of intersection (which we'll call a and b, where b>a) then evaluate

.

As for b) I need to refresh my memory. I'll get back to you.

3. b) where and .