Q: Which of the following is the formal definition of the derivative $$f'(x)$$?

$$\displaystyle \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$$

Q: Suppose $$f(x) is $$defined as follows: $$f(x) = \begin{cases} x^2$, & x \u003c 2 \\ 4x - 4, & x \geq 2 \end{cases}. Is$$ $$f$$ continuous at $$x=2$$?

Yes, because $$\displaystyle \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2).$$

Q: Which of the following limits does NOT exist?

$$\displaystyle \lim_{x \to 0} \dfrac{1}{x}$$

Question 1

Suppose the population of a bacteria culture is modeled by $$P(t) = 500e$$^{0.2t}$$, where t$ is in hours. What is the average rate of change of the population from t=2$ to t=5$?

Accepted Answer

$$\dfrac{P(5) - P(2)}{5-2} = \dfrac{500e$$^{1}$$ - 500e$$^{0.4}$$}{3}$$

Question 2

Evaluate $$\displaystyle \lim_{x 	o 0} \dfrac{\sqrt{x+4} - 2}{x}.$$

Accepted Answer

$$\dfrac{1}{4}$$

Question 3

Which of the following statements about continuity is TRUE?

Accepted Answer

A function $$f is $$continuous at $$x=a if$$ $$f(a)$$ exists, $$\displaystyle \lim_{x 	o a} f(x)$$ exists, and $$\displaystyle \lim_{x 	o a} f(x) = f(a).$$

Question 4

Find the derivative of $$f(x) = x^3$ - 4x + 7.$$

Accepted Answer

$$3x^2 - 4$

Question 5

A medication's concentration in the bloodstream is given by $$C(t) = 10 - 2t$$ for $$0 \leq t \leq 5. $$What is the average rate of change of $$C$$ from $$t=1 to$$ $$t=4$$?

Accepted Answer

$$\dfrac{C(4) - C(1)}{4-1} = \dfrac{2 - 8}{3} = -2$$

Question 6

Evaluate $$\displaystyle \lim_{x 	o 3} \dfrac{x^2$ - 9}{x - 3}.$$

Accepted Answer

$$6$$

Question 7

Which of the following best describes a removable discontinuity?

Accepted Answer

A hole in the graph where the two-sided limit exists but $$f(a) is $$missing or not equal to the limit.

Question 8

Given $$f(x) = \sqrt{3x+7}$$, find the slope of the tangent line at $$x=3.$$

Accepted Answer

$$\dfrac{3}{8}$$

Question 9

Which of the following is the formal definition of the derivative $$f'(x)$$?

Accepted Answer

$$\displaystyle \lim_{h 	o 0} \dfrac{f(x+h) - f(x)}{h}$$

Question 10

Evaluate $$\displaystyle \lim_{x 	o \infty} \dfrac{3x^2$ + 1}{5x^2$ - 7}.$$

Accepted Answer

$$\dfrac{3}{5}$$

Question 11

Suppose $$f(x) is $$defined as follows: $$f(x) = \begin{cases} x^2$, & x \u003c 2 \ 4x - 4, & x \geq 2 \end{cases}. Is$$ $$f$$ continuous at $$x=2$$?

Accepted Answer

Yes, because $$\displaystyle \lim_{x 	o 2^-} f(x) = \lim_{x 	o 2^+} f(x) = f(2).$$

Question 12

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

Accepted Answer

$$2 \cdot 3 + 2 = 8$$

Question 13

Which of the following limits does NOT exist?

Accepted Answer

$$\displaystyle \lim_{x 	o 0} \dfrac{1}{x}$$

Question 14

If $$\displaystyle \lim_{x 	o 1^-} f(x) = 4$$ and $$\displaystyle \lim_{x 	o 1^+} f(x) = 6$$, what is $$\displaystyle \lim_{x 	o 1} f(x)$$?

Accepted Answer

Does not exist

Question 15

Find the derivative of $$f(x) = 2x$$^{-1/2}$$.

Accepted Answer

$$-x^{-3/2}$$

Question 16

Which of the following is the correct equation for the tangent line to $$f(x) = \sqrt{3x+7} at$$ $$x=3$$?

Accepted Answer

$$y - 4 = \dfrac{3}{8}(x - 3)$$

Question 17

Evaluate $$\displaystyle \lim_{x 	o 0} \dfrac{1 - \cos x}{x}.$$

Accepted Answer

$$0$$

Question 18

Which of the following is the correct application of the power rule for differentiation?

Accepted Answer

$$\dfrac{d}{dx} x^n = n x$$^{n-1}$$

Question 19

Suppose $$f(x) = \dfrac{2}{\sqrt{x}}. $$What is $$f'(x)$$?

Accepted Answer

$$-x^{-3/2}$$

Calculus I Unit 1 Study Guide: Limits, Continuity, Rates of Change, and Derivatives

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