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].