Q: A function $$f(x) is $$increasing on an interval $$I if$$:

$$f'(x) \u003e 0$$ \text{ for all } x \in I$

Q: Which of the following is the correct derivative of $$\tan(x)$$?

$$\frac{d}{dx} \tan(x) = \sec^2$(x)$$

Q: A function $$f(x)$$ has a local maximum at $$x = c if$$:

$$f'(c) = 0$$ \text{ and } f''(c) \u003c 0$

Question 1

Suppose $$f(x) is a $$continuous function on $$[a, b]$$ and differentiable on $$(a, b). If$$ $$f'(c) = 0$$ for some $$c in$$ $$(a, b)$$, which of the following statements is always true?

Accepted Answer

The point $$x = c is a $$critical point of $$f(x).$$

Question 2

A pharmaceutical company models the concentration of a drug in the bloodstream by $$C(t) = 5e$$^{-0.2t}$$, where t$ is in hours. What is the rate of change of the concentration at t = 3$ hours?

Accepted Answer

$$C'(3) = -1e$$^{-0.6}$$

Question 3

Which of the following is the correct application of the Fundamental Theorem of Calculus, Part 1, for a continuous function $$f on$$ $$[a, b]$$?

Accepted Answer

$$\int_a^b f'(x) dx = f(b) - f(a)$$

Question 4

A function $$f(x) is $$increasing on an interval $$I if$$:

Accepted Answer

$$f'(x) \u003e 0$$ 	ext{ for all } x \in I$

Question 5

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

Accepted Answer

$$f'(x) = \frac{2x}{x^2$ + 1}$$

Question 6

A biologist models the population of a species by $$P(t) = 1000e$$^{0.05t}$$. What is the population growth rate at t = 10$?

Accepted Answer

$$P'(10) = 50e$$^{0.5}$$

Question 7

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

Accepted Answer

$$\frac{d}{dx} \arcsin(x) = \frac{1}{\sqrt{1 - x^2$}}$$

Question 8

A company’s profit function is $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold. How many units should be sold to maximize profit?

Accepted Answer

$$x = 4$$

Question 9

Which of the following is the correct anti-derivative of $$\cos(x)$$?

Accepted Answer

$$\int \cos(x) dx = \sin(x) + C$$

Question 10

If $$f(x) is $$concave up on $$[a, b]$$ and concave down on $$[b, c]$$, what is $$x = b$$ called?

Accepted Answer

An inflection point

Question 11

Which of the following is the correct logarithm rule for $$\ln(ab)$$?

Accepted Answer

$$\ln(ab) = \ln(a) + \ln(b)$$

Question 12

A patient’s temperature is modeled by $$T(t) = 98.6 + 2\sin(\frac{\pi}{12}t)$$, where $$t is $$hours after midnight. What is the maximum temperature during the day?

Accepted Answer

$$100.6$$

Question 13

Which of the following is the correct derivative of $$	an(x)$$?

Accepted Answer

$$\frac{d}{dx} 	an(x) = \sec^2$(x)$$

Question 14

Which of the following is the correct exponent rule for $$a^m \cdot a^n$$?

Accepted Answer

$$a^m \cdot a^n = a$$^{m+n}$$

Question 15

A function $$f(x)$$ has a local maximum at $$x = c if$$:

Accepted Answer

$$f'(c) = 0$$ 	ext{ and } f''(c) \u003c 0$

Question 16

Which of the following is the correct anti-derivative of $$\frac{1}{x}$$?

Accepted Answer

$$\int \frac{1}{x} dx = \ln|x| + C$$

Question 17

A chemical reaction rate is given by $$R(t) = 3t^2$ - 12t + 9. At $$what time $$t$$ does the reaction rate reach its minimum?

Accepted Answer

$$t = 2$$

Question 18

Which of the following is the correct derivative of $$e^{x^2}$$?

Accepted Answer

$$\frac{d}{dx} e$$^{x^2}$$ = 2x e$$^{x^2}$$

Question 19

Which of the following is the correct anti-derivative of $$\sec^2$(x)$$?

Accepted Answer

$$\int \sec^2$(x) dx = 	an(x) + C$$

Calculus I: Key Definitions, Theorems, and Reference Formulas

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