Question 1

Given the graph above, which of the following statements best describes the relationship between $$y = \cos x$$ and its derivative $$y' = -\sin x$$?

Accepted Answer

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

Question 2

What is the derivative of $$y = \cos x$$ with respect to $$x$$?

Accepted Answer

$$\frac{d}{dx} \cos x = -\sin x$$

Question 3

If $$y = \cos x$$, at which $$x-$$values does the derivative $$y' = -\sin x$$ equal zero?

Accepted Answer

$$x = 0, \pi, 2\pi, \ldots$$

Question 4

Which of the following best describes the amplitude and period of $$y = \cos x$$ and its derivative $$y' = -\sin x$$?

Accepted Answer

Both have amplitude 1 and period $$2\pi.$$

Question 5

At $$x = 0$$, what are the values of $$y = \cos x$$ and $$y' = -\sin x$$?

Accepted Answer

$$y = 1$$, $$y' = 0$$

Question 6

Which of the following real-world scenarios could be modeled by $$y = \cos x$$ and its derivative $$y' = -\sin x$$?

Accepted Answer

The position and velocity of a point on a rotating wheel.

Question 7

If $$y = \cos x$$, at which points does $$y$$ reach a maximum, and what is the value of $$y' at $$those points?

Accepted Answer

$$x = 2n\pi$$, $$y' = 0$$

Question 8

Which of the following statements is true about the phase relationship between $$y = \cos x$$ and $$y' = -\sin x$$?

Accepted Answer

$$y' = -\sin x is a $$quarter period behind $$y = \cos x.$$

Question 9

Suppose $$y = \cos x$$ models the vertical position of a mass on a spring. What does $$y' = -\sin x$$ represent in this context?

Accepted Answer

The velocity of the mass.

Question 10

If $$y = \cos x$$, what is the second derivative $$y''$$?

Accepted Answer

$$y'' = -\cos x$$

Derivatives of Trigonometric Functions: Calculus Study Notes

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