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.