Question 1

Which of the following best describes the relationship between the graphs of $$y = \cos x$$ and its derivative $$y' = -\sin x as $$shown in the image?

Accepted Answer

The derivative $$y' = -\sin x is $$always positive where $$y = \cos x is $$increasing.

Question 2

At which $$x-$$values does the graph of $$y = \cos x$$ have a horizontal tangent, based on the image?

Accepted Answer

$$x = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \ldots$$

Question 3

If $$y = \cos x$$, what is the value of the derivative $$y' at$$ $$x = 0$$?

Accepted Answer

$$y' = 0$$

Question 4

Which statement is true about the maximum and minimum points of $$y = \cos x as $$shown in the image?

Accepted Answer

At the maximum and minimum points, the derivative $$y' = -\sin x is $$zero.

Question 5

Based on the image, what is the period of both $$y = \cos x$$ and $$y' = -\sin x$$?

Accepted Answer

$$2\pi$$

Question 6

If $$y = \cos x$$, at which $$x-$$values does $$y' = -\sin x$$ reach its maximum value?

Accepted Answer

$$x = \frac{3\pi}{2}, -\frac{\pi}{2}, \ldots$$

Question 7

Suppose $$y = \cos x$$ models the position of a particle. What does the graph of $$y' = -\sin x$$ represent in this context?

Accepted Answer

The velocity of the particle.

Question 8

According to the image, what is the value of $$y' = -\sin x at$$ $$x = \pi$$?

Accepted Answer

$$0$$

Question 9

Which of the following best describes the phase relationship between $$y = \cos x$$ and $$y' = -\sin x as $$shown in the image?

Accepted Answer

$$y' = -\sin x$$ lags $$y = \cos x by$$ $$\frac{\pi}{2}.$$

Question 10

If $$y = \cos x$$, what is the slope of the tangent line to the curve at $$x = \frac{\pi}{2}$$?

Accepted Answer

$$-1$$

Derivatives of Trigonometric Functions

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