Q: The limit definition of the derivative at $$x = a is $$given by which formula?

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

Question 1

A car's position at time $$t (in $$hours) is given by $$s(t) = 4t^2$ + 2t. $$What is the average rate of change of the car's position from $$t = 1 to$$ $$t = 3$$?

Accepted Answer

$$18$$

Question 2

Given the function $$f(x) = x^3$ - 2x + 1$$, what is the instantaneous rate of change of $$f at$$ $$x = 2$$?

Accepted Answer

$$8$$

Question 3

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

Accepted Answer

The instantaneous rate of change of $$f at$$ $$x = a$$

Question 4

Suppose the demand for a product is modeled by $$D(p) = 100 - 5p$$, where $$p is $$the price. What does the derivative $$D'(p)$$ represent in this context?

Accepted Answer

The rate at which demand changes with respect to price

Question 5

Given $$f(x) = e$$^{2x}$$, which of the following is the correct derivative f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 6

A population of bacteria grows according to $$P(t) = 500e$$^{0.3t}$$, where t$ is in hours. What is the instantaneous rate of change of the population at t = 2$?

Accepted Answer

$$150e^{0.6}$$

Question 7

Which scenario would result in a function not being differentiable at a point?

Accepted Answer

The function has a sharp corner (cusp) at the point

Question 8

If $$f(x) is $$increasing wherever $$f'(x) \u003e 0$$, which of the following graphs best represents $$f'(x)$$ for a function that is increasing on $$(-, 2)$$ and decreasing on $$(2, 5)$$?

Accepted Answer

A graph that is positive on $$(-5, 2)$$ and negative on $$(2, 5)$$

Question 9

The limit definition of the derivative at $$x = a is $$given by which formula?

Accepted Answer

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

Question 10

A company models its revenue by $$R(x) = 200x - x^2$, where x$ is the number of units sold. What number of units maximizes revenue?

Accepted Answer

$$x = 100$$

Question 11

Given the graph of $$f(x)$$, how can you estimate $$f'(2)$$?

Accepted Answer

Find the slope of the tangent line to the graph at $$x = 2$$

Question 12

If $$f(x) =  5x^4$ - 3x^2$ + 2$$, what is $$f''(x)$$, the second derivative?

Accepted Answer

$$20x^2 - 6$

Question 13

Which of the following is NOT a correct interpretation of the derivative in a real-world context?

Accepted Answer

The total accumulated value of a function

Question 14

Suppose $$f(x) is $$differentiable and $$f'(x) = 0$$ only at $$x = 3. $$Which of the following statements is true?

Accepted Answer

$$x = 3 is a $$possible local maximum or minimum of $$f(x)$$

Question 15

A chemical reaction's concentration is modeled by $$C(t) = 10e$$^{-0.2t}$$. What is the rate of change of concentration at t = 5$?

Accepted Answer

$$-2e^{-0.2  5}$$

Question 16

Which of the following is a correct application of the closed interval method for optimization?

Accepted Answer

Evaluate the function at all critical points and endpoints of the interval

Question 17

If $$f(x) is a $$polynomial, which shortcut can be used to compute its derivative?

Accepted Answer

Power rule: $$\frac{d}{dx} x^n = nx$$^{n-1}$$

Question 18

A function $$f(x)$$ has a horizontal tangent at $$x = 4. $$What does this imply about $$f'(4)$$?

Accepted Answer

$$f'(4) = 0$$

Question 19

If $$f(x) is $$differentiable and $$f'(x) \u003c 0$$ for all $$x in an $$interval, what can be said about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$decreasing on that interval

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

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