Q: Water is being drained from a container shaped as an inverted right circular cone with a radius of $$6.00$$ inches at the top and a height of $$8.00$$ inches. When the water is $$6.00$$ inches deep and the surface level is falling at $$0.9 in./sec$$, what is the rate at which water is being drained from the container?

$$54.7$$ \text{ in}^3/\text{s}$

Q: A company is constructing an open-top, square-based, rectangular metal tank with a volume of $$64 ft$^3. $$What dimensions yield the minimum surface area?

$$4 ft$$ $$\times$$ $$4 ft$$ $$\times$$ $$4 ft$$

Question 1

Find the derivative of $$s = t^7$ \cos t - 7t \sin t - 7 \cos t$$ with respect to $$t.$$

Accepted Answer

$$\frac{ds}{dt} = -t^7$ \sin t + 7t^6$ \cos t - 7t \cos t - 14 \sin t$$

Question 2

Find the second derivative of $$y = 8x \sin x.$$

Accepted Answer

$$y'' = 16 \cos x - 8x \sin x$$

Question 3

Use implicit differentiation to find $$\frac{dy}{dx}$$ for $$xy + x + y = x^2$y^2$.

Accepted Answer

$$\frac{dy}{dx} = \frac{2xy^2$ - y - 1}{-2x^2$y + x + 1}$$

Question 4

A wheel with radius $$3 m $$rolls at $$18$$ rad/s. How fast is a point on the rim of the wheel rising when the point is $$\frac{\pi}{3}$$ radians above the horizontal (and rising)? (Round your answer to one decimal place.)

Accepted Answer

$$54.0 m/s$$

Question 5

Water is being drained from a container shaped as an inverted right circular cone with a radius of $$6.00$$ inches at the top and a height of $$8.00$$ inches. When the water is $$6.00$$ inches deep and the surface level is falling at $$0.9 in./sec$$, what is the rate at which water is being drained from the container?

Accepted Answer

$$54.7$$ 	ext{ in}^3/	ext{s}$

Question 6

Find the absolute maximum and minimum values of $$g(x) = -x^2$ + 13x - 40 on $$the interval $$5 \leq x \leq 8.$$

Accepted Answer

Absolute maximum is $$9 at$$ $$x = 6.5$$; absolute minimum is $$-15 at$$ $$x = 5.$$

Question 7

Given $$f(x) = x^3$ - 9x^2$ + 27x + 3$$, identify the function's local and absolute extreme values, if any, and where they occur.

Accepted Answer

Local maximum at $$x = 3.$$

Question 8

Use Newton's method to estimate the real solution of $$x^3 + 5x + 2 = 0$ starting at x_0$ = -1. $$What is $$x_2$$?

Accepted Answer

$$-0.44$$

Question 9

Find an antiderivative of $$7\sqrt{x} + 5.$$

Accepted Answer

$$\frac{14}{3}x$$^{3/2}$$ + 5x$$

Question 10

A company is constructing an open-top, square-based, rectangular metal tank with a volume of $$64 ft$^3. $$What dimensions yield the minimum surface area?

Accepted Answer

$$4 ft$$ $$	imes$$ $$4 ft$$ $$	imes$$ $$4 ft$$

Question 11

Given $$y = (x + 1)^2$(x^2$$ + $$1)^{-3}$$, find $$\frac{dy}{dx}$.

Accepted Answer

$$\frac{2(x + 1)(2x^2$ + 3x - 1)}{(x^2$ + 1)^4$}$$

Question 12

If $$xy^2 = 4$ and $$\frac{dx}{dt} = -5$$, what is $$\frac{dy}{dt}$$ when x = 4$ and y = 1$?

Accepted Answer

$$-\frac{8}{5}$$

Question 13

Find the value(s) of $$c$$ that satisfy the Mean Value Theorem for $$f(x) = x + \frac{75}{x} on$$ $$[3, 25].$$

Accepted Answer

$$c = \frac{5}{3}$$

Question 14

Given $$f'(x) = (x - 1)^2$(x + 6)$$, what are the critical points of $$f(x)$$?

Accepted Answer

$$x = -6, 1$$

Question 15

Given $$f'(x) = x$$^{1/3}$$(x - 1)$$, on which intervals is $$f(x)$$ increasing?

Accepted Answer

Increasing on $$(-\infty, 0) \cup (1, \infty)$$

Question 16

Find the most general antiderivative of $$\sqrt{y} + 8/y$$ with respect to $$y.$$

Accepted Answer

$$\frac{2}{3}y$$^{3/2}$$ + 8\ln|y| + C$$

Question 17

Use differentiation to determine whether the following integral formula is correct: $$\int (2x - 3)^4$ dx = \frac{(2x - 3)^5$}{10} + C.$$

Accepted Answer

Yes

Question 18

Given the table $$x: -1, 1, 3$$ and $$f(x): 0, 	ext{does not exist}, 0$$, which graph best matches this data?

Accepted Answer

Graph 3

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