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

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