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

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

Question 1

Which of the following is the correct domain for the function $$f(x) = \sqrt{4 - x^2$}$$?

Accepted Answer

$$[-2, 2]$$

Question 2

Suppose $$f(x) = 2x^3$ - 3x^2$ + 4. $$What is the equation of the tangent line to $$f(x) at$$ $$x = 1$$?

Accepted Answer

Question 3

Evaluate the limit: $$\displaystyle \lim_{x 	o 0} \frac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 4

Which of the following statements about the function $$f(x) = \frac{1}{x-2} is $$TRUE?

Accepted Answer

The function has an infinite discontinuity at $$x=2.$$

Question 5

Let $$f(x) = x^4$ - 4x^2$. Find all critical points of f(x$$)$$.

Accepted Answer

$$x = 0, 2, -2$$

Question 6

A bacteria culture grows according to the model $$N(t) = N_0$ e$$^{0.3t}$$, where t$ is in hours. If the initial population is 1000$, how many bacteria are present after 5$ hours (rounded to the nearest integer)?

Accepted Answer

$$1000 \cdot e$$^{1.5}$$ \approx 4482$$

Question 7

Which of the following is the correct application of the Mean Value Theorem for $$f(x) = x^2$ on $$[1, 4]$$?

Accepted Answer

There exists $$c in$$ $$(1,4)$$ such that $$f'(c) = \frac{f(4) - f(1)}{4-1}$$

Question 8

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

Accepted Answer

$$20$$

Question 9

Which of the following is the correct derivative of $$y = \ln(\sin x)$$?

Accepted Answer

$$\cot x$$

Question 10

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). If$$ $$f(a) = f(b)$$, what does Rolle’s Theorem guarantee?

Accepted Answer

There exists $$c in$$ $$(a, b)$$ such that $$f'(c) = 0$$

Question 11

Find $$\displaystyle \lim_{x 	o \infty} \frac{5x^2$ + 3}{2x^2$ - x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 12

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

Accepted Answer

$$-1.2 ft/sec$$

Question 13

Which of the following is the correct antiderivative of $$f(x) = 2x \cos(x^2$)$$?

Accepted Answer

$$\sin(x^2$) + C$$

Question 14

If $$f'(x) \u003e 0$$ for all $$x in an $$interval, which of the following is TRUE about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$increasing.

Question 15

Evaluate $$\displaystyle \$$int_0$$^\pi \sin x \, dx.$$

Accepted Answer

$$2$$

Question 16

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

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

Question 17

Suppose $$f(x) is $$even and continuous on $$[-a, a]. $$Which of the following is TRUE about $$\displaystyle \int_{-a}^a f(x) \, dx$$?

Accepted Answer

$$2 \$$int_0$$^a f(x) \, dx$$

Question 18

Find the average value of $$f(x) = x^2$ on $$[0, 2]$$.

Accepted Answer

$$\frac{4}{3}$$

Question 19

Which of the following is the correct derivative of $$y = \arctan(x)$$?

Accepted Answer

$$\frac{1}{1 + x^2$}$$

MA 161 Final Exam Study Guide: Core Calculus Concepts and Applications

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