Question 1

Given the graph below (first image), at which x-value does the function have a local minimum?

Accepted Answer

$$x = 0$$

Question 2

Using the table provided (second image), calculate $$p'(2)$$ where $$p(x) = f(x) \cdot g(x).$$

Accepted Answer

$$p'(2) = 10$$

Question 3

Given the function $$f(x) = x^3$ - 3x^2$ - 36x$$, as shown in the third image, how many critical points does the function have?

Accepted Answer

2

Question 4

Given the table (second image), find $$q'(1)$$ where $$q(x) = \frac{f(x)}{g(x)}.$$

Accepted Answer

$$q'(1) = \frac{1 \cdot 2 - 3 \cdot 4}{2^2$} = \frac{2 - 12}{4} = -2.5$$

Question 5

For the function $$f(x) = x^3$ - 3x^2$ - 36x$$, as shown in the third image, on which interval is the function concave up?

Accepted Answer

$$x \u003e 2$$

Question 6

Using the table (second image), find $$h'(4)$$ where $$h(x) = g(f(x)).$$

Accepted Answer

$$h'(4) = g'(f(4)) \cdot f'(4) = g'(4) \cdot f'(4) = 1 \cdot 3 = 3$$

Question 7

Given the graph in the first image, at which x-value does the function change concavity (point of inflection)?

Accepted Answer

$$x = 2$$

Question 8

Given the table (second image), estimate the instantaneous rate of change of $$f(x) at$$ $$x = 2$$ using the average rate of change between $$x = 1$$ and $$x = 3.$$

Accepted Answer

$$\frac{f(3) - f(1)}{3 - 1} = \frac{1 - 3}{2} = -1$$

Question 9

For the function $$f(x) = x^3$ - 3x^2$ - 36x$$, as shown in the third image, which of the following statements is true about the local extrema?

Accepted Answer

Question 10

Given the table (second image), use the product rule to find the derivative of $$p(x) = f(x)g(x) at$$ $$x = 3.$$

Accepted Answer

$$p'(3) = f'(3)g(3) + f(3)g'(3) = 2 \cdot 4 + 1 \cdot 3 = 8 + 3 = 11$$

Question 11

Given the graph in the first image, which of the following best describes the behavior of the function at the leftmost green point?

Accepted Answer

The function has a local maximum and is concave down.

Question 12

Using the table (second image), find the value of $$g''(2).$$

Accepted Answer

$$g''(2) = 2$$

Question 13

Given the graph in the third image, which interval is the function decreasing?

Accepted Answer

$$-2 \u003c x \u003c 2$$

Question 14

Given the table (second image), use the quotient rule to find the derivative of $$q(x) = \frac{f(x)}{g(x)} at$$ $$x = 4.$$

Accepted Answer

$$q'(4) = \frac{f'(4)g(4) - f(4)g'(4)}{[g(4)]^2} = \frac{3 \cdot 3 - 4 \cdot 1}{9} = \frac{9 - 4}{9} = \frac{5}{9}$$

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