Question 1

Given the function $$y = x^7$ \ln(x)$$, what is the second derivative $$y''$$?

Accepted Answer

$$y'' = 42x^5$ \ln(x) + 84x^5$

Question 2

Find the second derivative of $$y = e$$^{3x}$$.

Accepted Answer

$$y'' = 9e$$^{3x}$$

Question 3

What is the fifth derivative of $$f(x) = x^7$ + 6x^5$ + 17x^3$ + 2x + 1190$$?

Accepted Answer

$$f^{(5)}(x$$) = $$5040x^2 + 720$

Question 4

Evaluate $$\lim_{x 	o 3} \frac{x^4$ - 81}{x - 3}.$$

Accepted Answer

$$108$$

Question 5

For the function $$f(x) = x^3$ - 9x^2$ + 24x - 10$$, on which interval is the function increasing?

Accepted Answer

$$x  4$$

Question 6

For $$f(x) = x^3$ - 9x^2$ + 24x - 10$$, where is the function concave upward?

Accepted Answer

$$x \u003e 3$$

Question 7

Given $$f(x) = \frac{1}{3}x^3$ + \frac{7}{2}x^2$ + 12x + 90 on$$ $$[-3, 4]$$, what is the absolute maximum value?

Accepted Answer

$$f(4)$$

Question 8

A backpack manufacturer models production by $$T(x) = -0.25x^4$ + 3x^3$ for 0$$ \leq x \leq 15$$. At what value of x$ does the maximum rate of change of backpack production occur?

Accepted Answer

$$x = 9$$

Question 9

An open box is made from a $$24 	imes 24$$ inch cardboard by cutting squares from each corner and folding up the sides. If the depth must be between 3 and 6 inches, what size square should be cut to maximize the volume?

Accepted Answer

$$x = 5$$

Question 10

Susan wants to build a 10,000-square foot pen along a river, using $$13/	ext{ft}$$ maple fencing on the side opposite the river and $$5/	ext{ft}$$ birch fencing on the other two sides. What is the minimum cost for the fencing?

Accepted Answer

$$\2$$,828.43$

Question 11

Given the graph below (first image), at which $$x-$$values does $$f(x)$$ have a local maximum?

Accepted Answer

$$x = b, d$$

Question 12

Given the graph below (first image), on which interval is $$f(x)$$ decreasing?

Accepted Answer

$$[a, b]$$ and $$[e, h]$$

Question 13

Which of the following graphs (seventh to tenth images) shows a function that is concave upward on $$(a, b)$$?

Accepted Answer

Second image

Question 14

Which of the following graphs (eleventh to fourteenth images) shows a function that is concave downward on $$(a, b)$$?

Accepted Answer

First image

Question 15

Given $$s = 7t^3$ + 5t^2$ + 9t + 2$$, what is the acceleration at $$t = 1$$?

Accepted Answer

$$a = 42$$

Question 16

A function $$f is $$continuous and twice differentiable for all real numbers. The solutions to $$f'(x) = 0$$ are $$x = 2$$ and $$x = 4. If$$ $$f''(2)  0$$, what can you conclude?

Accepted Answer

$$x = 2 is a $$local maximum, $$x = 4 is a $$local minimum

Question 17

A function $$f is $$continuous and twice differentiable for all real numbers. The solutions to $$f''(x) = 0$$ are $$x = -1$$ and $$x = 2. If$$ $$f''(x) \u003e 0$$ for $$x \u003c -1$$ and $$f''(x) \u003c 0$$ for $$-1 \u003c x \u003c 2$$, what is the concavity of $$f on$$ $$(-\infty, -1)$$?

Accepted Answer

Concave upward

Question 18

If $$f''(x)$$ changes sign at $$x = 2$$, what does this indicate about the function $$f$$?

Accepted Answer

$$x = 2 is a $$point of inflection

Question 19

Given the function $$f(x) = x^3$ - 9x + 1 on $$the interval $$(-3, 0)$$, which theorem guarantees the existence of an absolute maximum?

Accepted Answer

Extreme Value Theorem

Analyzing Local Extrema from a Graph

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