1. ## 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.

2. 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 $\displaystyle |\sin{x}|^2 = \sin^2{x}$. 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 $\displaystyle x_{critical}$ that satisfy $\displaystyle f''(x_{critical}) < 0$ are relative maxima.

c) For the graph to be increasing, its derivative is positive. So solve $\displaystyle f'(x) > 0$

3. Notice that the enclosed region occurs where $\displaystyle \sqrt{x} > x^3$.

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

$\displaystyle \int_a^b{\sqrt{x}} - \int_a^b{x^3}$.

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

3. 3. b) $\displaystyle V = \pi \int_a^b {|f(x)^2 - g(x)^2|\,dx}$ where $\displaystyle f(x) = \sqrt{x}$ and $\displaystyle g(x) = x^3$.