Question 1

What is the domain of the function $$y = \sin x$$?

Accepted Answer

$$x \in (-\infty, \infty)$$

Question 2

Which of the following best describes the range of $$y = \cos x$$?

Accepted Answer

$$y \in [-1, 1]$$

Question 3

What is the period $$T of $$the function $$y = \sin x$$?

Accepted Answer

$$T = 2\pi$$

Question 4

The function $$y = \cos(x)$$ can be written as a shifted version of $$y = \sin(x). $$Which equation correctly expresses this relationship?

Accepted Answer

$$y = \sin(x + \frac{\pi}{2})$$

Question 5

At which $$x-$$values does $$y = \sin x$$ intercept the $$x-$$axis?

Accepted Answer

$$x = k\pi$$, \ k \in \mathbb{Z}$

Question 6

Given the function $$y = A \sin(\omega(t-d)) + C$$, what does $$A$$ represent?

Accepted Answer

The amplitude

Question 7

A hormone level $$H(t)$$ fluctuates between 0 and 100 ng/mL in a periodic manner. What is the amplitude of the function modeling this hormone level?

Accepted Answer

$$50$$

Question 8

If $$y = \sin(5x)$$, what is the period of this function?

Accepted Answer

$$\frac{2\pi}{5}$$

Question 9

Suppose $$f(x) = \sqrt{x}$$ and you want to approximate $$f(2)$$ using linearization at $$x = 1. $$What is the linear approximation $$L(x)$$ near $$x = 1$$?

Accepted Answer

$$L(x) = 1 + \frac{1}{2}(x-1)$$

Question 10

Given $$f(x) = \sqrt{x}$$, use the linear approximation at $$x = 1 to $$estimate $$\sqrt{2}.$$

Accepted Answer

$$1.5$$

Question 11

If $$f(x) = \sqrt{x+4}$$, what is the linear approximation $$L(x)$$ near $$x = 5$$?

Accepted Answer

$$L(x) = 3 + \frac{1}{6}(x-5)$$

Question 12

A sphere's surface area is given by $$S = 4\pi r^2$. If the radius r$ is increasing at a rate of 2$ cm/s, what is the rate at which the surface area is increasing when r = 8$ cm?

Accepted Answer

$$128\pi cm$^2/s$$

Question 13

Which of the following is the correct formula for the vertical shift in $$y = A \sin(\omega(t-d)) + C$$?

Accepted Answer

$$C$$

Question 14

If $$y = \cos(x)$$, at which $$x-$$values does the function intercept the $$x-$$axis?

Accepted Answer

$$x = (2k+1)\frac{\pi}{2}$$, $$k$$ integer

Question 15

A periodic phenomenon is modeled by $$y = 2 \sin(3x) + 4. $$What is the amplitude, period, and vertical shift?

Accepted Answer

Amplitude: $$2$$, Period: $$\frac{2\pi}{3}$$, Vertical shift: $$4$$

Question 16

If $$y = \sin(x)$$, what is the maximum value of $$y$$?

Accepted Answer

$$1$$

Question 17

Given $$f(x) = \sqrt{x}$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{1}{2\sqrt{x}}$$

Question 18

A function $$f(x) is $$differentiable at $$x = a. $$The linear approximation $$L(x)$$ near $$x = a is $$given by:

Accepted Answer

$$L(x) = f(a) + f'(a)(x-a)$$

Question 19

If $$y = \sin(x)$$ models a physical process, what does the phase shift represent?

Accepted Answer

How much the graph is shifted horizontally

Study Notes: Sine and Cosine Functions, Periodic Modeling, and Linear Approximation

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