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Section 3.8

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1. (a) f has a maximum value of 12 at x = 4 and a minimum value of -4 at x = 0 .

(c) g has a maximum value of \sqrt{2} at t = -\dfrac{\pi}{4} and a minimum value of -\sqrt{2} at t = \dfrac{3\pi}{4} .

(e) g has a maximum value of 0.5498 at x = -1.0678 and at x = 1.0678 , and a minimum value of -1.6646 at x = -2 and x = 2 .

(g) f has a maximum value of 3.9453 at t = 2.2889 and a minimum value of 0 at t = 0 and t = \pi .

3. The side parallel to the river should be 250 yards long and the other two sides should be 125 yards long.

5. \$30.94

6. The total area is maximized when the entire length of the wire is used for the circle. The total area is minimized when a piece of length \dfrac{\pi}{4 + \pi} is used for the circle and the remaining piece for the square.

7. (a) f has local minimum of 5 at x = 0 . f does not have a local maximum.

(c) g has a local maximum of 4 at t = -2 and a local minimum of 0 at t = 0 .

(e) f has a local maximum of 1 at x = 0 . f does not have a local minimum.

(g) g has a local maximum of \displaystyle{\frac{6}{25}\sqrt{\frac{3}{5}}} at \displaystyle{x = -\sqrt{\frac{3}{5}}} and a local minimum of \displaystyle{-\frac{6}{25}\sqrt{\frac{3}{5}}} at \displaystyle{x = \sqrt{\frac{3}{5}}}.

(i) g has a local maximum of 0 at t = 0 and a local minimum of -\dfrac{256}{81} at t = \dfrac{4}{3} .

8. (a) f''(x) = 6

(c) \displaystyle{\frac{d^2s}{dt^2} = \frac{8}{(2t - 1)^3}}

(e) \displaystyle{\frac{d^2x}{dt^2} = -20\sin(2t)\cos(4t) - 16\cos(2t)\sin(4t)}

9. f has a maximum value on (0, \infty) of \dfrac{1}{4} at x = 2 . f does not have a minimum value on (0, \infty) .

11. 10 \text{ feet} \times 10 \text{ feet} \times 10 \text{ feet}

13. The base of the bin should be 11.45 \text{ feet} \times 11.45 \text{ feet} and the height should be 7.63 \text{ feet} .

15.The radius of the can should be \displaystyle{\sqrt[3]{\frac{250}{\pi}}} and the height should be \displaystyle{4\sqrt[3]{\frac{250}{\pi}}}.

16. The radius of the can should be \sqrt[3]{62.5} and the height should be \displaystyle{\frac{8}{\pi}\sqrt[3]{62.5}}.

20. (b) L is maximized when \displaystyle{\theta = \frac{11}{20}}. The probabilty of AA is 0.3025 , the probability of Aa is 0.4950 , and the probabilty of aa is 0.2025 .

22. (a) v(t) = 3\pi\cos(\pi t)

(b) a(t) = -3\pi^2\sin(\pi t)

(c) From time t = 0 until time t = \dfrac{1}{2} , the object moves to the right from x = 0 to x = 3 and is slowing down. From time t = \dfrac{1}{2} until time t = 1 , the object moves to the left from x = 3 to x = 0 and is speeding up. From time t = 1 until time t = \dfrac{3}{2} , the object moves to the left from x = 0 to x = -3 and is slowing down. From time t = \dfrac{3}{2} until time t = 2 , the object moves to the right from x = -3 to x = 0 and is speeding up.

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