Question 1

Which of the following statements best describes an absolute maximum of a function $$f on a $$set $$D$$?

Accepted Answer

$$f(c) is $$greater than or equal to $$f(x)$$ for every $$x in$$ $$D.$$

Question 2

Given the function $$f(x)$$ graphed on $$[a, b] as $$shown, which point is guaranteed to be an absolute minimum?

Accepted Answer

The point where $$f(x)$$ attains its smallest value on $$[a, b].$$

Question 3

Which of the following is NOT a necessary condition for the Mean Value Theorem to apply to a function $$f on$$ $$[a, b]$$?

Accepted Answer

$$f(a) = f(b).$$

Question 4

Suppose $$f'(x) = 0$$ for all $$x in an $$open interval $$I. $$What can you conclude about $$f on$$ $$I$$?

Accepted Answer

$$f is a $$constant function on $$I.$$

Question 5

If $$f''(c) \u003e 0$$ and $$f'(c) = 0$$, what does the Second Derivative Test tell you about $$f at$$ $$x = c$$?

Accepted Answer

$$f$$ has a local minimum at $$x = c.$$

Question 6

A function $$f is $$defined and differentiable on $$[a, b]. If$$ $$f'(x) \u003e 0$$ for all $$x in$$ $$(a, b)$$, what can you say about $$f$$?

Accepted Answer

$$f is $$increasing on $$[a, b].$$

Question 7

Which of the following is the correct linear approximation to $$f at$$ $$x = a$$?

Accepted Answer

$$L(x) = f(a) + f'(a)(x - a)$$

Question 8

Suppose $$f''(x)$$ changes sign at $$x = c. $$What does this indicate about the graph of $$f$$?

Accepted Answer

$$f$$ has an inflection point at $$x = c.$$

Question 9

Given $$f(x) = x^3$ - 3x^2$ + 2 on$$ $$[0, 3]$$, which steps would you use to find the absolute maximum and minimum values?

Accepted Answer

Find critical points in $$(0, 3)$$, evaluate $$f at $$critical points and endpoints, compare values.

Question 10

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

Accepted Answer

$$f''(x) \u003e 0$$ for all $$x in$$ $$I.$$

Question 11

A function $$f is $$defined 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 12

Which of the following best describes a local minimum of $$f at$$ $$x = c$$?

Accepted Answer

$$f(c) \leq f(x)$$ for all $$x$$ near $$c.$$

Question 13

If $$f'(x)$$ changes from positive to negative as $$x$$ increases through $$c$$, what does the First Derivative Test conclude?

Accepted Answer

$$f$$ has a local maximum at $$x = c.$$

Question 14

Suppose $$f(x) is $$differentiable and $$f'(x) = g'(x)$$ for all $$x in an $$interval $$I. $$What can you conclude?

Accepted Answer

$$f(x) - g(x) is a $$constant on $$I.$$

Question 15

Which of the following is the correct formula for the differential $$dy of a $$function $$f at$$ $$x$$?

Accepted Answer

$$dy = f'(x) dx$$

Question 16

If $$f(x) is $$increasing on an interval $$I$$, which of the following must be true for $$x_1$$ \u003c $$x_2 in$$ $$I$$?

Accepted Answer

$$f(x_1$) \u003c f(x_2$)$$

Question 17

Which of the following is the correct statement of the Mean Value Theorem?

Accepted Answer

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

Applications of the Derivative and Antiderivatives: Maxima, Minima, Mean Value Theorem, Concavity, Linear Approximation, and Differential Equations

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