Q: A function $$f is $$defined by $$f(x) = \begin{cases} x^2$ & x \leq 0 \\ x & x \u003e 0 \end{cases}. Is$$ $$f$$ continuous at $$x = 0$$?

Yes, because $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0).$$

Q: Which of the following is the correct definition of the derivative of a function $$f at a $$point $$a$$?

$$f'(a) = \lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$$

Q: A biologist models the population of bacteria by $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours. What is the instantaneous rate of change of the population at t = 10$?

$$P'(10) = 100 \times 0.05 \times e$$^{0.05 \times 10}$$

Q: A medical researcher models the concentration of a drug in the bloodstream by $$C(t) = 5e$$^{-0.2t}$$. What is the rate at which the concentration is changing at t = 3$ hours?

$$C'(3) = -0.2 \times 5e$$^{-0.6}$$

Question 1

Which of the following statements about the domain of the function $$f(x) = \sqrt{x-2} is $$correct?

Accepted Answer

The domain is all real numbers $$x$$ such that $$x \geq 2.$$

Question 2

Suppose you are measuring the velocity of a car at time $$t$$ using the function $$s(t)$$ for position. Which formula gives the average velocity over the interval $$[a, b]$$?

Accepted Answer

$$\dfrac{s(b) - s(a)}{b - a}$$

Question 3

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

Accepted Answer

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

Question 4

Given $$f(x) = x^3$ and g(x$$) = \sin(x)$$, what is the derivative of h(x) = f(x)g(x$$)$$ using the product rule?

Accepted Answer

$$h'(x) = 3x^2$ \sin(x) + x^3$ \cos(x)$$

Question 5

A function $$f is $$defined by $$f(x) = \begin{cases} x^2$ & x \leq 0 \ x & x \u003e 0 \end{cases}. Is$$ $$f$$ continuous at $$x = 0$$?

Accepted Answer

Yes, because $$\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^+} f(x) = f(0).$$

Question 6

Which of the following is the correct definition of the derivative of a function $$f at a $$point $$a$$?

Accepted Answer

$$f'(a) = \lim_{h 	o 0} \dfrac{f(a+h) - f(a)}{h}$$

Question 7

If $$f(x) = x^4$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 4x^3$

Question 8

A biologist models the population of bacteria by $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours. What is the instantaneous rate of change of the population at t = 10$?

Accepted Answer

$$P'(10) = 100 	imes 0.05 	imes e$$^{0.05 	imes 10}$$

Question 9

Which of the following is the correct application of the quotient rule for $$f(x) = \dfrac{x^2$}{x+1}$$?

Accepted Answer

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

Question 10

Which of the following statements about the limit $$\lim_{x 	o 0} \dfrac{\sin(x)}{x} is $$true?

Accepted Answer

The limit is $$1.$$

Question 11

A chemical reaction rate is modeled by $$R(t) = 3t^2$ - 2t + 1. $$What is the instantaneous rate of change of the reaction at $$t = 4$$?

Accepted Answer

$$R'(4) = 22$$

Question 12

Which of the following is the correct statement of the Sandwich (Squeeze) Theorem?

Accepted Answer

If $$f(x) \leq g(x) \leq h(x)$$ for all $$x$$ near $$a$$, and $$\lim_{x 	o a} f(x) = \lim_{x 	o a} h(x) = L$$, then $$\lim_{x 	o a} g(x) = L.$$

Question 13

Which of the following is true about the function $$f(x) = |x|$$?

Accepted Answer

$$f(x) is $$continuous everywhere but not differentiable at $$x = 0.$$

Question 14

A medical researcher models the concentration of a drug in the bloodstream by $$C(t) = 5e$$^{-0.2t}$$. What is the rate at which the concentration is changing at t = 3$ hours?

Accepted Answer

$$C'(3) = -0.2 	imes 5e$$^{-0.6}$$

Question 15

Which of the following is the correct application of the chain rule for $$f(x) = \sin(x^2$)$$?

Accepted Answer

$$f'(x) = \cos(x^2$) \cdot 2x$$

Question 16

Which of the following is the correct definition of a right-hand limit of a function $$f at$$ $$a$$?

Accepted Answer

$$\lim_{x 	o a^+} f(x) = L$$ means that for every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that if $$a \u003c x \u003c a + \delta$$, then $$|f(x) - L| \u003c \epsilon.$$

Question 17

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

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

Question 18

Which of the following is true about the function $$f(x) = \dfrac{1}{x}$$?

Accepted Answer

$$f(x) is $$continuous everywhere except at $$x = 0.$$

Calculus I: Numbers, Functions, Limits, and Derivatives

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