Question 1

A cylindrical coffee pot has a radius of 5 inches and a height of 12 inches. What is the volume of coffee it can hold? Use $$V = \pi r^2$ h$$ and round your answer to one decimal place.

Accepted Answer

$$942.5 in$^3$$

Question 2

A wind carries a kite horizontally away from a person at a rate of 10 ft/sec. If the kite is 100 ft above the ground, how fast is the string length increasing when the kite is 200 ft away from the person? Use $$x^2$$ + $$y^2$$ = $$s^2.$$

Accepted Answer

$$10.95 ft/sec$$

Question 3

Find the absolute maximum and minimum values for $$f(x) = \frac{1}{2}x^2$ - 5$$ over the interval $$-2 \leq x \leq 3.$$

Accepted Answer

Max: $$-2$$, Min: $$-5$$

Question 4

Find the absolute maximum and minimum values for $$f(x) = x^2$ + x$$ over the interval $$-1 \leq x \leq 4.$$

Accepted Answer

Max: $$20$$, Min: $$0$$

Question 5

Find the absolute maximum and minimum values for $$f(x) = \sin(2x)$$ over the interval $$0 \leq x \leq \pi.$$

Accepted Answer

Max: $$1$$, Min: $$-1$$

Question 6

Find the absolute maximum and minimum values for $$f(x) = x 	an(x)$$ over the interval $$-\frac{\pi}{5} \leq x \leq \frac{\pi}{5}.$$

Accepted Answer

Max: $$0.5$$, Min: $$-0.5$$

Question 7

Find the critical points and intervals of increasing and decreasing for $$f(x) = -2x^2$ - 2x^3$. Label your critical points as local max, local min, or neither.

Accepted Answer

Critical points: $$x = 0$$, $$x = -1$$; Increasing: $$(-\infty, -1)$$; Decreasing: $$(-1, 0)$$; Local max at $$x = -1$$, local min at $$x = 0$$

Question 8

Find the critical points and intervals of increasing and decreasing for $$f(x) = 15x^2$ - x^3$. Label your critical points as local max, local min, or neither.

Accepted Answer

Critical points: $$x = 0$$, $$x = 10$$; Increasing: $$(-\infty, 10)$$; Decreasing: $$(10, \infty)$$; Local max at $$x = 10$$, local min at $$x = 0$$

Question 9

Find the critical points and intervals of increasing and decreasing for $$f(x) = \frac{2x}{x^2$ - 1}. $$Label your critical points as local max, local min, or neither.

Accepted Answer

Critical point: $$x = 0$$; Increasing: $$(-\infty, 0)$$; Decreasing: $$(0, \infty)$$; Neither local max nor min at $$x = 0$$

Question 10

Find the points of inflection and intervals of concave up and down for $$f(x) = x^3$ - 2x^2$ + 3.$$

Accepted Answer

POI: $$x = \frac{2}{3}$$; Concave up: $$(\frac{2}{3}, \infty)$$; Concave down: $$(-\infty, \frac{2}{3})$$

Question 11

Find the points of inflection and intervals of concave up and down for $$f(x) = \sin(x) - \sqrt{3}\cos(x) on $$the interval $$[0, \pi].$$

Accepted Answer

POI: $$x = \frac{\pi}{6}$$; Concave up: $$(0, \frac{\pi}{6})$$; Concave down: $$(\frac{\pi}{6}, \pi)$$

Question 12

A rectangular garden is to be enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. What dimensions for the outer rectangle will require the smallest total length of fencing if the area is $$120 m$^2$$?

Accepted Answer

Dimensions: $$10 m by$$ $$15 m$$

Question 13

Evaluate the following limit: $$\lim_{x 	o 0} \frac{\sin(6x)}{4x}.$$

Accepted Answer

$$\frac{3}{2}$$

Question 14

Evaluate the following limit: $$\lim_{x 	o 0} \frac{\ln(e^x - 1)}{x}.$$

Accepted Answer

$$1$$

Question 15

Evaluate the following limit: $$\lim_{x 	o 0} \frac{1 - \cos(x)}{x^2$}.$$

Accepted Answer

$$\frac{1}{2}$$

Question 16

A company wants to design an aquarium with a square base and an open top. The aquarium must hold $$500$$ cubic feet of water. If you call each side of the base $$x$$, what dimensions will use the least amount of glass? Volume: $$V = x^2$ h.$$

Accepted Answer

Base: $$10 ft$$, Height: $$5 ft$$

Question 17

Find the product of two integers that is a minimum if one is $$12$$ more than twice the other. Use $$p = x(12 + 2x).$$

Accepted Answer

$$-16$$

Question 18

Evaluate the following limit: $$\lim_{x 	o 0} \frac{	an(2x)}{x}.$$

Accepted Answer

$$2$$

Calculus I: Critical Points, Extrema, Optimization, and Indeterminate Forms

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