Q: Which of the following is the correct application of the Mean Value Theorem (MVT) for a function $$f$$ continuous on $$[a, b]$$ and differentiable on $$(a, b)$$?

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$ \, \text{for some} \, c \in (a, b)$

Question 1

Which of the following statements about vertical asymptotes is correct?

Accepted Answer

A vertical asymptote occurs where the denominator of a rational function is zero and the numerator is not zero.

Question 2

Evaluate the following limit: $$\lim_{x 	o 0} \frac{\sin x}{x}$$

Accepted Answer

$$1$$

Question 3

Which of the following is the correct derivative of $$f(x) = e$$^{2x}$$?

Accepted Answer

$$2e^{2x}$$

Question 4

A medication's concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$, where t$ is in hours. What is the rate of change of concentration at t = 3$ hours?

Accepted Answer

$$-1e^{-0.6}$$

Question 5

Which of the following is the correct application of the Mean Value Theorem (MVT) for a function $$f$$ continuous on $$[a, b]$$ and differentiable on $$(a, b)$$?

Accepted Answer

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$ \, 	ext{for some} \, c \in (a, b)$

Question 6

Given $$f(x) = x^3$ - 3x^2$ + 2x$$, find all critical points.

Accepted Answer

$$x = 1, 2$$

Question 7

Which of the following is the correct formula for the average value of a continuous function $$f on$$ $$[a, b]$$?

Accepted Answer

$$\frac{1}{b-a} \int_a^b f(x) \, dx$$

Question 8

A company wants to minimize the cost $$C(x) = x^2$ - 8x + 20$$ for producing $$x$$ units. What is the optimal number of units to produce?

Accepted Answer

$$x = 4$$

Question 9

Which of the following is the correct indefinite integral of $$\int x^2$ \, dx$$?

Accepted Answer

$$\frac{1}{3}x^3$ + C$$

Question 10

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

Accepted Answer

$$12$$

Question 11

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

Accepted Answer

$$u = x^2$

Question 12

A right Riemann sum for $$f(x) = x^2$ on $$[0,2]$$ with n=2$ subintervals is:

Accepted Answer

$$f(1) \Delta x + f(2) \Delta x$$

Question 13

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

Accepted Answer

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

Question 14

Which of the following is the correct summation notation for the sum $$1 + 2 + 3 + \ldots + 10$$?

Accepted Answer

$$\sum_{k=1}^{10} k$$

Question 15

Which of the following is the correct application of the Fundamental Theorem of Calculus?

Accepted Answer

$$\int_a^b f'(x) \, dx = f(b) - f(a)$$

Question 16

A tank is being filled with water at a rate given by $$r(t) = 4t$$ liters per minute. How much water is added between $$t=0$$ and $$t=5$$ minutes?

Accepted Answer

$$\$$int_0$$^5 4t \, dt = 50$$ liters

Question 17

Which of the following is the correct antiderivative of $$\int \frac{1}{x} \, dx$$?

Accepted Answer

$$\ln|x| + C$$

Question 18

If $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, which statement is true according to the Mean Value Theorem?

Accepted Answer

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

Question 19

Which of the following is the correct definite integral for the area under $$f(x) = x$$ from $$x=1 to$$ $$x=3$$?

Accepted Answer

$$\$$int_1$$^3 x \, dx$$

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