Question 1

Which of the following statements best describes a global maximum of a function $$f(x) on a $$closed interval $$[a, b]$$?

Accepted Answer

$$f(c) is $$greater than or equal to $$f(x)$$ for all $$x in$$ $$[a, b].$$

Question 2

Given $$f(x) = x^3$ - 3x^2$ + 4$$, find all critical points.

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 3

For the function $$f(x) = -x^2$ + 4x + 5$$, on which interval is $$f(x)$$ increasing?

Accepted Answer

$$(-\infty, 2)$$

Question 4

Suppose $$f'(x)$$ changes from positive to negative at $$x = c. $$What does this indicate about $$f(x) at$$ $$x = c$$?

Accepted Answer

$$f(x)$$ has a local maximum at $$x = c$$

Question 5

Given $$f(x) = 2x^3$ + 12x^2$ + 9x - 4$$, for which intervals is $$f(x)$$ concave up?

Accepted Answer

$$(-\infty, -2)$$ and $$(0, \infty)$$

Question 6

A company’s revenue is given by $$R(x) = (100 - \frac{x}{4})x. $$What value of $$x$$ maximizes revenue?

Accepted Answer

$$x = 200$$

Question 7

If $$f(x) = x^2$ + 4x + 17$$, what is the local minimum value?

Accepted Answer

$$f(-2) = 13$$

Question 8

A box with a square base and no lid must have a volume of $$125\ 	ext{in}^3. $$What should the dimensions be to minimize the surface area?

Accepted Answer

Base side $$= 5 in$$, height $$= 2.5 in$$

Question 9

Given $$f(x) = x^3$ - 8x + 6$$, find the differential $$dy$$ when $$x = 2$$ and $$dx = 0.2.$$

Accepted Answer

$$dy = -2.8$$

Question 10

A 10-meter-tall tree stands 5 meters from a vertical cliff. What is the length of the shortest rope that can be tied from the ground over the top of the tree to the top of the cliff?

Accepted Answer

$$\sqrt{5^2$ + 10^2$} + 10 = 21.18$$ meters

Question 11

If $$y = x^3$ and $$\frac{dx}{dt} = 2$$ when x = 4$, what is $$\frac{dy}{dt}$$?

Accepted Answer

$$\frac{dy}{dt} = 48$$

Question 12

Given $$f(x) = x^3$ + x$$, use the linearization at $$x = 2 to $$approximate $$f(2.1). $$What is the approximate value?

Accepted Answer

$$f(2.1) \approx 10.7$$

Question 13

Find $$\frac{dy}{dx}$$ for $$x^2$$ + $$y^2 = 49$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = -\frac{x}{y}$$

Question 14

A sphere is growing at a rate of $$50\ 	ext{cm}^3/	ext{s}. At $$what rate is the radius increasing when the radius is $$5\ 	ext{cm}$$?

Accepted Answer

$$\frac{dr}{dt} = \frac{50}{4\pi (5^2$)}$$

Question 15

If $$f''(c) \u003e 0 at a $$critical point $$c$$, what does the Second Derivative Test tell us?

Accepted Answer

$$f(x)$$ has a local minimum at $$x = c$$

Question 16

A right triangle has a base of $$10\ 	ext{cm}$$ and a height of $$12\ 	ext{cm}. If $$the height is decreasing at $$0.4\ 	ext{cm/s}$$, at what rate is the area decreasing?

Accepted Answer

$$2\ 	ext{cm}^2/	ext{s}$$

Question 17

Given $$f(x) = x^2$/3 (4-x)$$, find the open interval where $$f(x) is $$increasing.

Accepted Answer

$$(0, 2)$$

Question 18

A company sells 400 calculators per year. It costs $$2 to store one calculator for a year, and 25$ per order. What is the cost function C(x$$)$$ for x$ items per order?

Accepted Answer

$$C(x) = 2\frac{x}{2} + 25\frac{400}{x}$$

Applications of Differentiation: Extrema, Concavity, Optimization, and Related Rates

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