Question 1

Let $$f(x) = 15x^4$ - 4x^3$ on the interval $$[-2, 2]$$. What is the absolute maximum value of f(x$$)$$ on this interval, and at which x$-value does it occur?

Accepted Answer

$$f(-2) = 208 at$$ $$x = -2$$

Question 2

Given $$h(x) = x e$$^{5x}$$, find all critical values.

Accepted Answer

There are no critical values.

Question 3

Suppose the cost function for producing $$x$$ chairs is $$C(x) = 0.006x^3$ + 0.07x^2$ + 15x + 600. $$What is the marginal cost of producing the 40th chair?

Accepted Answer

$$C'(40) = 0.018(40)^2$ + 0.14(40) + 15$$

Question 4

A function $$f is $$concave down on $$(1,4)$$ and concave up on $$(5,6). $$Which of the following intervals could contain a point of inflection?

Accepted Answer

$$(4,5)$$

Question 5

Let $$M(t) = -3t^2$ + 150t + 120$$ for $$0 \leq t \leq 40. At $$what value of $$t is $$the maximum productivity achieved?

Accepted Answer

$$t = 25$$

Question 6

Given $$g(x) = \ln(x^2$ + 2x + 4)$$, find any critical values.

Accepted Answer

The critical value is $$x = -1$$

Question 7

For the function $$f(x) = x^3$ + \frac{17}{2}x^2$ + 10x + 1$$, what are the $$x-$$values where relative extrema may occur?

Accepted Answer

$$x = -4$$ and $$x = 0$$

Question 8

A business has revenue $$R(x) = 3x$$ and cost $$C(x) = 0.05x^2$ + 0.4x + 1. $$What is the production level for maximum profit?

Accepted Answer

$$x = 29$$

Question 9

Given the graph of $$f'(x)$$, how do you determine where $$f(x) is $$increasing?

Accepted Answer

$$f(x) is $$increasing where $$f'(x) \u003e 0$$

Question 10

Suppose $$M(v)$$ models fuel economy (mpg) as a function of speed $$v ($$mph), and $$M(v)$$ has a maximum at $$v = 35. $$What does this mean in practical terms?

Accepted Answer

The best fuel economy is achieved at 35 mph.

Question 11

For $$h(x) = x e$$^{5x}$$, on which interval is h(x$$)$$ increasing?

Accepted Answer

$$\left(-\frac{1}{5}, \infty\right)$$

Question 12

Which of the following is true about points of inflection?

Accepted Answer

A point of inflection is where the concavity of a function changes.

Question 13

Given $$f(x) = 15x^4$ - 4x^3$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 60x^3$ - 12x^2$

Question 14

If $$f''(x) \u003e 0 on an $$interval, what can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$concave up.

Question 15

For $$g(x) = \ln(x^2$ + 2x + 4)$$, what is the $$x-$$value of the relative minimum?

Accepted Answer

$$x = -1$$

Question 16

Suppose $$C(x) = 0.006x^3$ + 0.07x^2$ + 15x + 600$$ and $$x = 40. $$What is the actual cost of producing 40 chairs?

Accepted Answer

$$C(40) = 10192$$

Calculus Chapter 3 Review: Critical Values, Extrema, Concavity, and Applications

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