advertisement

Math 165 – Quiz 5B, inverse trig functions – solutions Problem 1 Consider the function f (x) = cos−1 (0.5x). a) Find f 0 (x). b) Which of the graphs below is the graph of f (x)? Circle A, B, or C. A B C Solution a) Use the Constant Multiple Rule and the Chain Rule with u = 0.5x and du/dx = 0.5 to get 0.5 1 . f 0 (x) = − p = −√ 2 4 − x2 1 − (0.5x) b) The correct answer is None of the Above. You can graph f (x) to see this, here is what you can see without a calculator. • cos−1 (u) is defined for −1 ≤ u ≤ 1, so f (x) is defined for −2 ≤ x ≤ 2 (rules out B, C). • f 0 (x) < 0 for all x, so the graph of f (x) must be going down (f (x) is decreasing, rules out A, C). • the range of cos−1 (x), same as that of f (x), is [0, π]. This rules out all of A, B, C. You could say the graph looks like A but flipped about the x-axis, or like B but stretched and shifted up by π/2.