Q: An architect needs to design a rectangular room with an area of $$72 ft$^2. $$What dimensions should she use in order to minimize the perimeter? Round to the nearest tenth if necessary.

$$8.5 ft$$ $$\times$$ $$8.5 ft$$

Question 1

Given the function $$y = 2x^4$ + 3x^2$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = 8x^3$ + 6x$$

Question 2

Suppose the population of a certain type of insect in a region near the equator is given by $$P(t) = 7t^2$ + 11$$, where $$t$$ represents the time in days. Find the rate of change of the population when $$t = 5.$$

Accepted Answer

$$81$$ insects/day

Question 3

Find the location and value of all local extrema for the function $$f(x)$$ shown in the graph (parabola opening downward with vertex at $$(2, -1)).$$

Accepted Answer

A local maximum at $$(2, -1)$$

Question 4

Identify the intervals where the function $$f(x) is $$increasing, given the graph of a parabola opening upward with vertex at $$(1, 0).$$

Accepted Answer

$$(1, \infty)$$

Question 5

The graph of the derivative function $$f'(x) is $$given. Find the critical numbers of the function $$f(x) if$$ $$f'(x)$$ crosses the $$x-$$axis at $$x = 0$$ and $$x = 1.$$

Accepted Answer

$$0$$ and $$1$$

Question 6

Determine the location of all local extrema of the function $$f(x) = x^3$ - 4.5x^2$ + 12x - 3.$$

Accepted Answer

Local maximum at $$x = 1$$, local minimum at $$x = 4$$

Question 7

Use the first derivative test to determine the location of each local extremum and the value of the function at that extremum for $$f(x) = 5x^4$.

Accepted Answer

No local extrema

Question 8

Find the location of the absolute minimum for the function $$f(x)$$ shown in the graph (minimum at $$x = -3).$$

Accepted Answer

$$x = -3$$

Question 9

Find the absolute maximum of $$f(x) = -2x^2$ + 4x + 2 on$$ $$[1, 2].$$

Accepted Answer

$$4$$

Question 10

Find the coordinates of the points of inflection for the function $$f(x) = x^3$ - 12x^2$ + 63x + 51.$$

Accepted Answer

$$(4, -13)$$

Question 11

Find the largest open intervals where the function $$f(x) = \frac{4}{x} is $$concave upward.

Accepted Answer

$$(-\infty, 0)$$

Question 12

If the price charged for a bolt is $$p$$ cents, then $$x$$ thousand bolts will be sold in a certain hardware store, where $$p = 40 - 2x. $$How many bolts must be sold to maximize revenue?

Accepted Answer

$$10$$ thousand bolts

Question 13

The cost of a computer system increases with increased processor speeds. The cost $$C of a $$system as a function of processor speed $$S is $$estimated as $$C = 16S^2$ - 7S + 2,000. $$What is the processor speed for which cost is at a minimum?

Accepted Answer

$$0.22$$ MHz

Question 14

An architect needs to design a rectangular room with an area of $$72 ft$^2. $$What dimensions should she use in order to minimize the perimeter? Round to the nearest tenth if necessary.

Accepted Answer

$$8.5 ft$$ $$	imes$$ $$8.5 ft$$

Question 15

Given $$y = \ln 8x^2$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{2}{x}$$

Question 16

Given $$y = 5e$$^{2x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$10e^{2x}$$

Question 17

Suppose a company’s revenue $$R (in $$thousands of dollars) is given by $$R(x) = -2x^2$ + 12x$$, where $$x is $$the number of units sold (in thousands). How many units should be sold to maximize revenue?

Accepted Answer

$$3$$ thousand units

Question 18

A medication is administered to a patient and the concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the rate of change of the concentration at t = 2$ hours?

Accepted Answer

$$-3.68$$ units/hour

Question 19

A ball is thrown upward and its height (in meters) after $$t$$ seconds is given by $$h(t) = -4.9t^2$ + 20t + 1.5. At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = 2.04$$	ext{ s}$

Question 20

A company’s profit $$P(x) (in $$dollars) is given by $$P(x) = -x^2$ + 6x - 5$$, where $$x is $$the number of items sold. What is the maximum profit?

Accepted Answer

$$13$$

Precalculus Study Guide: Derivatives, Extrema, and Applications

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