Q: Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$According to the Mean Value Theorem, which of the following must be true?

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

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

$$\lim_{x \to 0} \frac{\cos x}{1} = 1$$

Question 1

Which of the following best describes the derivative of a function $$f(x) at a $$point $$x = a$$?

Accepted Answer

The slope of the tangent line to the graph of $$f(x) at$$ $$x = a.$$

Question 2

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

Accepted Answer

$$4$$

Question 3

A medication is administered to a patient, and the amount in the bloodstream at time $$t$$ hours is modeled by $$A(t) = 100e$$^{-0.5t}$$. What is the rate at which the medication is leaving the bloodstream at t = 2$ hours?

Accepted Answer

$$-25e^{-1}$$

Question 4

Which of the following statements about the Fundamental Theorem of Calculus is correct?

Accepted Answer

It relates the process of differentiation and integration.

Question 5

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$According to the Mean Value Theorem, which of the following must be true?

Accepted Answer

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

Question 6

Find the derivative of $$f(x) = \ln(x^2$ + 1).$$

Accepted Answer

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

Question 7

A company’s profit $$P(x) (in $$dollars) from selling $$x$$ units is given by $$P(x) = -2x^2$ + 40x - 100. $$How many units should the company sell to maximize profit?

Accepted Answer

$$x = 10$$

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 application of L’Hôpital’s Rule to $$\displaystyle \lim_{x 	o 0} \frac{\sin x}{x}$$?

Accepted Answer

$$\lim_{x 	o 0} \frac{\cos x}{1} = 1$$

Question 10

If $$y = x^2$ \sin(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 11

A ladder 10 ft long is leaning against a wall. If the bottom of the ladder slides away from the wall at a rate of $$1 ft/sec$$, how fast is the top of the ladder sliding down the wall when the bottom is $$6 ft $$from the wall?

Accepted Answer

$$-0.75 ft/sec$$

Question 12

Which of the following functions is NOT continuous at $$x = 0$$?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 13

Find the antiderivative of $$f(x) = 6x^2$ - 4x + 3.$$

Accepted Answer

$$2x^3$$ - $$2x^2 + 3x + C$

Question 14

Suppose $$f(x) is $$differentiable and $$f'(x) \u003e 0$$ for all $$x in $$its domain. Which of the following must be true?

Accepted Answer

$$f(x) is $$increasing everywhere.

Question 15

Evaluate $$\displaystyle \int \cos(3x)\,dx.$$

Accepted Answer

$$\frac{1}{3}\sin(3x) + C$$

Question 16

Which of the following best describes the meaning of a definite integral $$\int_a^b f(x)\,dx$$?

Accepted Answer

The net area under the curve $$y = f(x)$$ from $$x = a to$$ $$x = b.$$

Question 17

If $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 18

A tank is being filled with water so that the volume at time $$t is$$ $$V(t) = 5t^2$ liters. What is the instantaneous rate at which water is entering the tank at t = 3$ minutes?

Accepted Answer

$$30$$ liters/minute

Question 19

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

Accepted Answer

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

Calculus I: Syllabus and Core Concepts Overview

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