Question 1

Which of the following intervals is the domain of the inverse sine function $\sin^{-1}(x)$?

Accepted Answer

$[-1, 1]$

Question 2

What is the derivative of $y = \sin^{-1}(x)$ with respect to $x$?

Accepted Answer

$\dfrac{1}{\sqrt{1-x^2}}$

Question 3

If $f(x) = x^7 \sin^{-1}(x)$, what is $f'(x)$?

Accepted Answer

$7x^6 \sin^{-1}(x) + \dfrac{x^7}{\sqrt{1-x^2}}$

Question 4

Which statement is true about the range of $\cos^{-1}(x)$?

Accepted Answer

$[0, \pi]$

Question 5

Find the derivative of $y = \cos^{-1}(x)$.

Accepted Answer

$-\dfrac{1}{\sqrt{1-x^2}}$

Question 6

Evaluate $\sin^{-1}(\sin(\frac{\pi}{6}))$.

Accepted Answer

$\frac{\pi}{6}$

Question 7

Which of the following is the correct formula for the derivative of $y = \sin^{-1}(g(x))$?

Accepted Answer

$\dfrac{1}{\sqrt{1-[g(x)]^2}} \cdot g'(x)$

Question 8

If $y = \cos^{-1}(\sqrt{x})$, what is $\dfrac{dy}{dx}$?

Accepted Answer

$-\dfrac{1}{2\sqrt{1-x} \sqrt{x}}$

Question 9

Which of the following is the correct domain for $	an^{-1}(x)$?

Accepted Answer

$(-\infty, \infty)$

Question 10

What is the derivative of $y = 	an^{-1}(x)$?

Accepted Answer

$\dfrac{1}{1+x^2}$

Question 11

A medical imaging device uses the function $f(x) = \sin^{-1}(x)$ to process signals. If the input signal is $x = 0.8$, what is the rate of change of the output signal?

Accepted Answer

$\dfrac{1}{\sqrt{1-0.8^2}}$

Question 12

Which of the following is the correct range for $\sin^{-1}(x)$?

Accepted Answer

$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

Question 13

If $y = \sin^{-1}\left(\frac{1}{2}\right)$, what is $y$?

Accepted Answer

$\frac{\pi}{6}$

Question 14

A chemical reaction rate is modeled by $f(x) = \cos^{-1}(x)$. If $x = 0.5$, what is the instantaneous rate of change of the reaction rate?

Accepted Answer

$-\dfrac{1}{\sqrt{1-0.25}}$

Question 15

Which of the following is true about the inverse cosine function $\cos^{-1}(x)$?

Accepted Answer

It is strictly monotonic on $[0, \pi]$

Question 16

If $y = \sin^{-1}(g(x))$, which of the following is the correct expression for $\dfrac{dy}{dx}$?

Accepted Answer

$\dfrac{g'(x)}{\sqrt{1-[g(x)]^2}}$

Question 17

Which of the following is the correct formula for the derivative of $y = \cos^{-1}(g(x))$?

Accepted Answer

$-\dfrac{g'(x)}{\sqrt{1-[g(x)]^2}}$

Inverse Trigonometric Functions and Their Derivatives: Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards