Q: Which of the following limits represents the definition of the derivative of $$f(x) at$$ $$x = a$$?

$$\displaystyle \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Q: Which of the following is an application of L'Hôpital's Rule?

Evaluating $$\displaystyle \lim_{x \to 0} \frac{e^x - 1}{x}$$

Question 1

Which of the following statements best describes the meaning of the derivative $$f'(a) of a $$function $$f(x) at a $$point $$x = a$$?

Accepted Answer

It gives the instantaneous rate of change of $$f(x) at$$ $$x = a.$$

Question 2

Evaluate the following limit:

$$\displaystyle \lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 3

A car's position at time $$t (in $$seconds) is given by $$s(t) = 3t^2$ + 2t + 1 (in $$meters). What is the car's instantaneous velocity at $$t = 4$$ seconds?

Accepted Answer

$$26 m/s$

Question 4

Which of the following functions is NOT continuous at $$x = 0$$?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 5

Suppose $$f(x) is $$differentiable everywhere and $$f'(x) = 0$$ for all $$x. $$Which of the following must be true?

Accepted Answer

$$f(x) is a $$constant function.

Question 6

A pharmaceutical company models the concentration $$C (in mg/L) of a $$drug in the bloodstream $$t$$ hours after injection by $$C(t) = 10e$$^{-0.5t}$$. What is the rate of change of concentration at t = 2$ hours?

Accepted Answer

$$-5e^{-1} mg/L/hr$

Question 7

Which of the following is the correct application of the Chain Rule to compute $$\frac{d}{dx} \left[ \sin(x^2$) \right]$$?

Accepted Answer

$$2x \cos(x^2$)$$

Question 8

A company finds that the cost $$C (in $$dollars) to produce $$x$$ units is $$C(x) = 500 + 20x + 0.1x^2$. What is the marginal cost when x = 100$ units?

Accepted Answer

$$40$$

Question 9

Evaluate the following definite integral using the Fundamental Theorem of Calculus:

$$\displaystyle \$$int_0$$^2 (3x^2$ + 2x)\,dx$$

Accepted Answer

$$14$$

Question 10

Which of the following statements about the Mean Value Theorem is TRUE?

Accepted Answer

If $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, then there exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Question 11

A tank is being filled with water so that the volume $$V (in $$liters) at time $$t (in $$minutes) is $$V(t) = 5t^2$. How fast is the water level rising at t = 3$ minutes?

Accepted Answer

$$30 L/$$min

Question 12

Which of the following limits represents the definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\displaystyle \lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$$

Question 13

Find the derivative of $$f(x) = x^3$ \ln(x).$$

Accepted Answer

$$3x^2$$ \ln(x) + $$x^2$$

Question 14

A ball is thrown upward with an initial velocity of $$20 m/s $$from a height of $$50 m. $$Its height after $$t$$ seconds is $$h(t) = -4.9t^2$ + 20t + 50. $$What is the maximum height reached by the ball?

Accepted Answer

$$70.41 m$$

Question 15

Evaluate the following limit:

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

Accepted Answer

$$3$$

Question 16

Which of the following is the correct antiderivative of $$f(x) = 2x + 3$$?

Accepted Answer

$$x^2 + 3x + C$

Question 17

A population of bacteria grows according to $$P(t) = 100e$$^{0.2t}$$, where t$ is in hours. What is the rate of growth of the population at t = 5$ hours?

Accepted Answer

$$20e^{0.2 	imes 5}$$

Question 18

Which of the following is an application of L'Hôpital's Rule?

Accepted Answer

Evaluating $$\displaystyle \lim_{x 	o 0} \frac{e^x - 1}{x}$$

Question 19

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). If$$ $$f'(x) \u003e 0$$ for all $$x in$$ $$(a, b)$$, which of the following is true?

Accepted Answer

$$f(x) is $$increasing on $$[a, b].$$

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