Question 1

Evaluate the following limit, which is commonly used in calculus:

$$
\lim_{x 	o 0} \frac{\sin x}{x}
$$
What is the value of this limit?

Accepted Answer

$$1$$

Question 2

Which of the following is the correct derivative of $$\sin x$$ with respect to $$x$$?

Accepted Answer

$$\cos x$$

Question 3

A pendulum swings such that its angle from the vertical at time $$t is $$given by $$	heta(t) = \sin(2t). $$What is the instantaneous rate of change of the angle at $$t = \frac{\pi}{4}$$ seconds?

Accepted Answer

$$2\cos\left(2 \cdot \frac{\pi}{4}\right) = 2\cos\left(\frac{\pi}{2}\right) = 0$$

Question 4

Find the derivative of $$f(x) = 	an x.$$

Accepted Answer

$$\sec^2$ x$$

Question 5

Evaluate the following limit:

$$
\lim_{x 	o 0} \frac{\sin(3x)}{x}
$$

Accepted Answer

$$3$$

Question 6

Which of the following is the correct derivative of $$\sec x$$?

Accepted Answer

$$\sec x 	an x$$

Question 7

A biologist is modeling the population of a species using $$P(t) = 100 \cos(\frac{\pi}{6} t)$$, where $$t is in $$months. What is the instantaneous rate of change of the population at $$t = 0$$?

Accepted Answer

$$-100 \cdot \frac{\pi}{6} \sin(0) = 0$$

Question 8

Which of the following is the correct derivative of $$\cot x$$?

Accepted Answer

$$-\csc^2$ x$$

Question 9

Evaluate the following limit:

$$
\lim_{x 	o 0} \frac{1 - \cos x}{x}
$$

Accepted Answer

$$0$$

Question 10

Suppose $$f(x) = \sin x \cos x. $$What is $$f'(x)$$?

Accepted Answer

$$\cos^2$ x - \sin^2$ x$$

Question 11

A sound wave is modeled by $$s(t) = \cos(440 \pi t). $$What is the instantaneous rate of change of the wave at $$t = 0$$?

Accepted Answer

$$-440\pi \sin(0) = 0$$

Question 12

Which of the following is the correct derivative of $$\cos x$$?

Accepted Answer

$$-\sin x$$

Question 13

Evaluate the following limit:

$$
\lim_{x 	o 0} \frac{\sin(2x)}{3x}
$$

Accepted Answer

$$\frac{2}{3}$$

Question 14

If $$y = \sec x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\sec x 	an x$$

Question 15

A chemical reaction's rate is modeled by $$R(t) = \sin(5t). $$What is the instantaneous rate of change of the reaction at $$t = 0$$?

Accepted Answer

$$5\cos(0) = 5$$

Question 16

Which of the following is the correct derivative of $$\csc x$$?

Accepted Answer

$$-\csc x \cot x$$

Question 17

Evaluate the following limit:

$$
\lim_{x 	o 0} \frac{\sin^2$ x}{x^2$}
$$

Accepted Answer

$$1$$

Question 18

Suppose $$f(x) = 	an x. $$Using the quotient rule, which of the following is the correct derivative?

Accepted Answer

$$\sec^2$ x$$

Question 19

A medical device measures heart rate as $$H(t) = 70 + 10\sin(\frac{\pi}{30} t)$$, where $$t is in $$seconds. What is the rate of change of heart rate at $$t = 0$$?

Accepted Answer

$$10 \cdot \frac{\pi}{30} \cos(0) = \frac{\pi}{3}$$

Question 20

Which of the following is the correct derivative of $$\sin(2x)$$?

Accepted Answer

$$2\cos(2x)$$

Limits and Derivatives Involving Trigonometric Functions

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