Question 1

Suppose a function $$f(x)$$ has a critical point at $$x = 2. $$The second derivative $$f''(2) = -5. $$What can you conclude about $$f(x) at$$ $$x = 2$$?

Accepted Answer

There is a local maximum at $$x = 2.$$

Question 2

A company models its profit $$P(x) (in $$thousands of dollars) as a function of the number of units $$x$$ produced: $$P(x) = -2x^2$ + 8x + 10. $$What is the maximum profit, and how many units should be produced to achieve it?

Accepted Answer

Maximum profit is $$18 at$$ $$x = 2$$ units.

Question 3

Given $$f(x) = x^3$ - 3x^2$ + 4$$, find all intervals where $$f(x) is $$concave up.

Accepted Answer

$$(2, \infty)$$

Question 4

Which of the following statements about inflection points is TRUE?

Accepted Answer

An inflection point occurs where $$f''(x)$$ changes sign.

Question 5

A function $$f(x)$$ has a vertical asymptote at $$x = 3$$ and a horizontal asymptote at $$y = 2. $$Which of the following could be the equation for $$f(x)$$?

Accepted Answer

$$f(x) = \dfrac{2x}{x-3}$$

Question 6

Evaluate $$\displaystyle \$$int_0$$^2 (3x^2$ - 4x + 1)\,dx.$$

Accepted Answer

$$2/3$$

Question 7

Which of the following best describes the end behavior of $$f(x) = \dfrac{2x^2$ + 1}{x^2$ - 4} as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o 2$$

Question 8

Suppose $$f(x) is $$continuous on $$[1, 4]$$ and differentiable on $$(1, 4). If$$ $$f(1) = 2$$ and $$f(4) = 8$$, what does the Mean Value Theorem guarantee?

Accepted Answer

There exists $$c \in (1, 4)$$ such that $$f'(c) = 2$$

Question 9

Find the $$x-$$coordinate of the inflection point for $$f(x) = x^4$ - 4x^2$.

Accepted Answer

$$x = 0$$

Question 10

A medication's concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the rate of change of concentration at t = 2$ hours?

Accepted Answer

$$-5e^{-1}$$

Question 11

Evaluate $$\displaystyle \$$int_1$$^3 x^2$\,dx.$$

Accepted Answer

$$26/3$$

Question 12

Which of the following is NOT a requirement for Rolle's Theorem to apply to a function $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$f'(a) = f'(b)$$

Question 13

Let $$f(x) = \dfrac{1}{x-1}. $$What is the domain of $$f(x)$$?

Accepted Answer

$$(-\infty, 1) \cup (1, \infty)$$

Question 14

If $$f(x) is $$increasing and concave down on an interval, which of the following must be true?

Accepted Answer

$$f'(x) \u003e 0$$ and $$f''(x) \u003c 0$$

Question 15

A tank is being filled with water at a rate given by $$R(t) = 5t^2$ liters per minute. How much water is added to the tank from t = 0$ to t = 3$ minutes?

Accepted Answer

$$75$$ liters

Question 16

Find the equation of the tangent line to $$y = x^2$ - 4x + 1 at$$ $$x = 1.$$

Accepted Answer

$$y = -2x + 4$$

Question 17

Which of the following definite integrals gives the area between $$y = x^2$ and y = x$ from x = 0$ to x = 1$?

Accepted Answer

$$\$$int_0$$^1 (x - x^2$)\,dx$$

Question 18

If $$f(x) is $$continuous on $$[a, b]$$ and $$f'(x) \u003e 0$$ for all $$x in$$ $$(a, b)$$, which of the following is true?

Accepted Answer

$$f(x) is $$increasing on $$[a, b]$$

Question 19

Evaluate $$\displaystyle \$$int_0$$^1 e$$^{2x}$$\,dx.$$

Accepted Answer

$$\dfrac{e^2$ - 1}{2}$$

Question 20

A function $$f(x)$$ has a local minimum at $$x = 0$$ and a local maximum at $$x = 2. $$Which of the following could be the sign chart for $$f'(x)$$?

Accepted Answer

Negative for $$x  2$$

Calculus I Final Exam Review: Limits, Derivatives, and Integrals

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