Q: Let $$f(x) = e^x$$ and $$g(x) = \ln(x). $$Which of the following statements is true about their limits as $$x \to 1$$?

$$\displaystyle \lim_{x \to 1} e^x = 1$$ and $$\displaystyle \lim_{x \to 1} \ln(x) = 0$$

Q: Which of the following is NOT a valid limit law?

$$\displaystyle \lim_{x \to a} f(g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$$

Question 1

Suppose the position of a car moving along a straight road is given by $$s(t) = 3t^2$ + 2t + 1$$, where $$s is in $$meters and $$t is in $$seconds. What is the average rate of change of the car's position from $$t = 2 to$$ $$t = 5$$?

Accepted Answer

$$23$$ meters per second

Question 2

Given the function $$f(x) = x^2$ - 4x + 7$$, what is the slope of the curve at the point $$x = 3$$?

Accepted Answer

$$1$$

Question 3

Consider the graph of $$f(x)$$ which passes through the points $$(1, 2)$$, $$(2, 4)$$, and $$(3, 6). $$Based on the graph, what is $$\displaystyle \lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 4

A table of values for $$g(x)$$ near $$x = 1 is $$given: $$g(0.9) = 2.1$$, $$g(0.99) = 2.01$$, $$g(1.01) = 1.99$$, $$g(1.1) = 1.9. $$What is the best estimate for $$\displaystyle \lim_{x 	o 1} g(x)$$?

Accepted Answer

$$2$$

Question 5

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{\sin x}{x}$$ using limit laws.

Accepted Answer

$$1$$

Question 6

Which of the following is the correct application of the limit laws to $$\displaystyle \lim_{x 	o 2} (3x^2$ + 5x)$$?

Accepted Answer

$$3(2^2) + 5(2) = 12 + 10 = 22$

Question 7

Let $$f(x) = e^x$$ and $$g(x) = \ln(x). $$Which of the following statements is true about their limits as $$x 	o 1$$?

Accepted Answer

$$\displaystyle \lim_{x 	o 1} e^x = 1$$ and $$\displaystyle \lim_{x 	o 1} \ln(x) = 0$$

Question 8

The average rate of change of the temperature $$T (in$$ $$^\circC) in a $$chemical reaction is given by $$T(t) = 5t + 20. $$What is the average rate of change from $$t = 2$$ minutes to $$t = 6$$ minutes?

Accepted Answer

$$5$$ $$^\circC/$$min

Question 9

Evaluate $$\displaystyle \lim_{h 	o 0} \frac{(2 + h)^2$ - 4}{h}.$$

Accepted Answer

$$4$$

Question 10

Suppose $$f(x) is $$squeezed between $$x^2$$ and $$2x^2$$ for all $$x$$ near $$0$$, i.e., $$x^2$$ \leq f(x) \leq $$2x^2. $$What is $$\displaystyle \lim_{x 	o 0} f(x) by $$the squeeze theorem?

Accepted Answer

$$0$$

Question 11

A population of bacteria grows according to $$P(t) = 100e$$^{0.5t}$$. What is the average rate of change of the population from t = 0$ to t = 2$ hours?

Accepted Answer

$$100(e^{1} - 1$$)$$

Question 12

Given $$f(x) = \frac{x^2$ - 9}{x - 3}$$, evaluate $$\displaystyle \lim_{x 	o 3} f(x).$$

Accepted Answer

$$6$$

Question 13

If $$f(x) = \ln(x)$$, use the table values $$f(0.9) = -0.105$$, $$f(1.0) = 0$$, $$f(1.1) = 0.095 to $$estimate $$\displaystyle \lim_{x 	o 1} f(x).$$

Accepted Answer

$$0$$

Question 14

Which of the following is NOT a valid limit law?

Accepted Answer

$$\displaystyle \lim_{x 	o a} f(g(x)) = \lim_{x 	o a} f(x) \cdot \lim_{x 	o a} g(x)$$

Question 15

A medication's concentration in the bloodstream is modeled by $$C(t) = 50e$$^{-0.2t}$$. What is the average rate of change of C$ from t = 0$ to t = 5$ hours?

Accepted Answer

$$\frac{50(e$$^{-1}$$ - 1)}{5}$$

Question 16

Evaluate $$\displaystyle \lim_{x 	o 0} x \cos\left(\frac{1}{x}ight)$$ using the squeeze theorem.

Accepted Answer

$$0$$

Question 17

If $$f(x) = |x|$$, what is $$\displaystyle \lim_{x 	o 0} \frac{|x|}{x}$$?

Accepted Answer

The limit does not exist

Question 18

Suppose $$f(x) is $$continuous and $$\displaystyle \lim_{x 	o 2} f(x) = 5. $$Which of the following must be true?

Accepted Answer

$$f(2) = 5$$

Question 19

A ball is thrown upward and its height is given by $$h(t) = -4.9t^2$ + 20t + 1.5. $$What is the average rate of change of height from $$t = 1 to$$ $$t = 3$$ seconds?

Accepted Answer

$$-4.3$$

Calculus I: Limits, Rates of Change, and the Slope of a Curve – Study Notes

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