Q: Which of the following limits does NOT exist?

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

Q: Suppose $$f(x) is $$continuous at $$x = 3$$ and $$f(3) = 7. $$Which of the following must be true?

$$\displaystyle \lim_{x \to 3} f(x) = 7$$

Q: Which of the following is the correct application of L'Hôpital's Rule to $$\displaystyle \lim_{x \to 0} \frac{e^x - 1}{x}$$?

$$\displaystyle \lim_{x \to 0} \frac{e^x}{1} = 1$$

Question 1

Which of the following statements best describes the relationship between a function $$f(x)$$ and its inverse $$f^{-1}(x$$)$$?

Accepted Answer

The graph of $$f^{-1}(x$$)$$ is a reflection of the graph of f(x$$)$$ across the line y = x$.

Question 2

Evaluate the following limit using limit laws: $$\displaystyle \lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 3

Which of the following is the correct derivative of $$f(x) = \sin(x) \cdot e^x$$?

Accepted Answer

$$\cos(x) \cdot e^x + \sin(x) \cdot e^x$$

Question 4

A medication is administered to a patient and the amount in the bloodstream at time $$t (in $$hours) is modeled by $$A(t) = 100e$$^{-0.2t}$$. What is the rate at which the medication is leaving the bloodstream at t = 5$ hours?

Accepted Answer

$$-20e^{-1}$$

Question 5

Which of the following limits does NOT exist?

Accepted Answer

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

Question 6

Suppose $$f(x) is $$continuous at $$x = 3$$ and $$f(3) = 7. $$Which of the following must be true?

Accepted Answer

$$\displaystyle \lim_{x 	o 3} f(x) = 7$$

Question 7

A ladder 10 ft long is leaning against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2 ft/s, how fast is the top of the ladder sliding down the wall when the bottom is 6 ft from the wall?

Accepted Answer

$$-1.5 ft/s$$

Question 8

Find the $$x-$$coordinate of the critical point(s) of $$f(x) = x^3$ - 3x^2$ + 2.$$

Accepted Answer

$$x = 1$$ and $$x = 2$$

Question 9

Which of the following is the correct application of L'Hôpital's Rule to $$\displaystyle \lim_{x 	o 0} \frac{e^x - 1}{x}$$?

Accepted Answer

$$\displaystyle \lim_{x 	o 0} \frac{e^x}{1} = 1$$

Question 10

A company wants to design a cylindrical can that will hold 500 mL of liquid. What is the radius $$r (in cm) $$that minimizes the surface area of the can? (Recall: $$V = \pi r^2$ h$$ and $$A = 2\pi r h + 2\pi r^2$)

Accepted Answer

$$r = \sqrt[3]{\frac{250}{\pi}}$$

Question 11

Find the indefinite integral: $$\displaystyle \int (2x^3$ - 5x + 4)\,dx$$

Accepted Answer

$$\frac{1}{2}x^4$ - \frac{5}{2}x^2$ + 4x + C$$

Question 12

Evaluate the definite integral: $$\displaystyle \$$int_0$$^2 (3x^2$ + 2x)\,dx$$

Accepted Answer

$$20$$

Question 13

Which of the following is the correct statement of the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$F(x) = \int_a^x f(t)\,dt$$, then $$F'(x) = f(x).$$

Question 14

A car's velocity (in m/s) at time $$t (in $$seconds) is given by $$v(t) = 3t^2$. How far does the car travel between t = 0$ and t = 4$ seconds?

Accepted Answer

$$64 m$$

Question 15

Find the area between the curves $$y = x^2$ and y = x + 2$ from x = 0$ to x = 2$.

Accepted Answer

$$\frac{10}{3}$$

Question 16

Which substitution is appropriate for evaluating $$\displaystyle \int x \cos(x^2$)\,dx$$?

Accepted Answer

Let $$u = x^2$, so du = 2x$$\,dx$$

Question 17

The volume of the solid obtained by rotating the region bounded by $$y = x^2$ and y = 0$ about the x$-axis from x = 0$ to x = 1$ is:

Accepted Answer

$$\frac{\pi}{5}$$

Question 18

If $$f(x) = \ln(x)$$, what is $$f'(e^2$)$$?

Accepted Answer

$$\frac{1}{e^2$}$$

Question 19

Which of the following is the correct antiderivative of $$f(x) = \frac{1}{x}$$ for $$x \u003e 0$$?

Accepted Answer

$$\ln|x| + C$$

Syllabus Overview and Core Topics for Calculus I (MATH 1210)

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