Q: Which of the following is the correct limit definition of the derivative of $$f(x) at$$ $$x = a$$?

$$\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

A car's position in meters after $$t$$ seconds is given by $$s(t) = 4t^2$ + 2t. $$What is the average rate of change of the car's position from $$t = 1 to$$ $$t = 4$$?

Accepted Answer

$$20$$

Question 2

Given the function $$f(x) = x^3$ - 2x + 1$$, what is the slope of the tangent line to the graph at $$x = 2$$?

Accepted Answer

$$6$$

Question 3

Which of the following best describes the instantaneous rate of change of a function $$f(x) at$$ $$x = a$$?

Accepted Answer

The slope of the tangent line to the graph of $$f(x) at$$ $$x = a$$

Question 4

If $$f(x) is $$differentiable at $$x = 3$$, which of the following must be true?

Accepted Answer

$$f(x) is $$continuous at $$x = 3$$

Question 5

Suppose the demand for a product is given by $$p = 100 - 2q$$, where $$p is $$the price and $$q is $$the quantity sold. What is the marginal revenue when $$q = 10$$?

Accepted Answer

$$-2$$

Question 6

Given the table of values for a function $$f(x)$$:

$$x$$: $$1$$ $$2$$ $$3$$ $$4$$

$$f(x)$$: $$2$$ $$5$$ $$10$$ $$17$$

Estimate $$f'(2)$$ using the symmetric difference quotient.

Accepted Answer

$$3.5$$

Question 7

Which of the following statements is true about the derivative $$f'(x) of a $$function $$f(x)$$?

Accepted Answer

$$f'(x)$$ gives the slope of the tangent line to $$f(x) at $$each point where it is defined$

Question 8

A population of bacteria grows according to $$P(t) = 500e$$^{0.03t}$$, where t$ is in hours. What is the instantaneous rate of growth at t = 10$ hours?

Accepted Answer

$$15e^{0.3}$$

Question 9

If $$f(x) is a $$function such that $$f'(x) = 0$$ for all $$x in an $$interval, what can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$constant$

Question 10

A company’s profit $$P (in $$thousands of dollars) as a function of advertising spending $$x (in $$thousands of dollars) is $$P(x) = -2x^2$ + 12x + 5. $$What amount of advertising maximizes the profit?

Accepted Answer

$$x = 3$$

Question 11

Which of the following is the correct limit definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\displaystyle \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 12

Given $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 13

Suppose $$f(x) is $$not differentiable at $$x = 2. $$Which of the following could be a reason?

Accepted Answer

Both A and B

Question 14

A ball is thrown upward with velocity $$v(t) = 20 - 9.8t. At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = 2.04$$

Question 15

If $$f(x) is $$increasing and $$f'(x) is $$negative for some interval, what can you conclude?

Accepted Answer

This is not possible; if $$f'(x) is $$negative, $$f(x)$$ must be decreasing

Question 16

Given $$f(x) = x^4$ - 4x^2$, find all x$ where the tangent line is horizontal.

Accepted Answer

$$x = 0, x = 2, x = -2$$

Question 17

A chemical reaction’s concentration $$C(t) in $$mol/L is modeled by $$C(t) = 5e$$^{-0.2t}$$. What is the rate of change of concentration at t = 5$ minutes?

Accepted Answer

$$-1e^{-1}$$

Question 18

Which of the following best describes the economic interpretation of the derivative of a cost function $$C(q)$$ with respect to quantity $$q$$?

Accepted Answer

The marginal cost, or the approximate cost of producing one more unit

Question 19

Suppose $$f(x) is a $$differentiable function and $$f'(a) \u003e 0. $$Which of the following is true?

Accepted Answer

$$f(x) is $$increasing at $$x = a$$

Calculus I: Derivatives, Rates of Change, and Applications – Study Guide

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