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Math 165 – Quiz 5C, 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, or write ’none’. 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 =√ . 4 − x2 1 − (0.5x)2 b) The correct answer is B. 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 A, C). • f 0 (x) > 0 for all x, so the graph of f (x) must be going up (f (x) is increasing, rules out A). • the range of cos−1 (x) is [0, π], that of f (x) is the opposite, so [−π, 0]. This rules out A, C.