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Math 165 – Quiz 5A, inverse trig functions – solutions Problem 1 Consider the function f (x) = − sin−1 (2x). 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 = 2x and du/dx = 2 to get f 0 (x) = − √ 2 . 1 − 4x2 b) The correct answer is C. You can graph f (x) to see this, but here are a few clues to see it without a calculator. • sin−1 (u) is defined for −1 ≤ u ≤ 1, so f (x) is defined for −0.5 ≤ x ≤ 0.5. • f 0 (x) < 0 for all x, so the graph of f (x) must be going down (f (x) is decreasing). • the range of sin−1 (x), same as that of f (x), is [−π/2, π/2].