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.