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Math Help - BC Calculus Questions

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by defjammer91 View Post
    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 |\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 x_{critical} that satisfy f''(x_{critical}) < 0 are relative maxima.

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

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

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

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