Q: An ice cube is melting so its volume decreases at $$0.1\ \text{cm}^3/\text{sec}. $$What is the rate at which its side length is changing when the side is $$2\ \text{cm}$$?

$$\frac{ds}{dt} = -\frac{0.1}{12} = -0.0083\ \text{cm/sec}$$

Q: A spherical balloon is being inflated so its volume increases at $$5\ \text{cm}^3/\text{sec}. $$What is the rate of change of its radius when the radius is $$10\ \text{cm}$$?

$$\frac{dr}{dt} = \frac{5}{400\pi} = 0.00398\ \text{cm/sec}$$

Q: A 13-ft ladder is leaning against a wall. The base slides away from the wall at $$2\ \text{ft/sec}. $$When the base is $$12\ \text{ft}$$ from the wall, how fast is the top sliding down?

$$\frac{dy}{dt} = -\frac{24}{5} = -4.8\ \text{ft/sec}$$

Q: A cylindrical water heater tank with radius $$1\ \text{ft}$$ and height $$4\ \text{ft} is $$being drained. If the water level drops at $$0.5\ \text{ft/min}$$, how fast is the water draining out?

$$\frac{dV}{dt} = 2\pi \cdot 1^2$ \cdot 0.5 = \pi\ \text{ft}^3/\text{min}$$

Question 1

Given the function $$y = 2x^4$ + 7x^3$ + \frac{3}{5}x - \frac{2}{x} + x$$^{\frac{3}$${2}} - \frac{9}{\sqrt{x}}$$, what is its derivative $$y'$$?

Accepted Answer

$$y' = 8x^3$ + 21x^2$ + \frac{3}{5} - 2x$$^{-2}$$ + \frac{3}{2}x$$^{\frac{1}$${2}} + \frac{9}{2}x$$^{-\frac{3}$${2}}$$

Question 2

For the function $$y = e^x(x^2$ + x - 2)$$, at what $$x-$$values is the rate of change zero?

Accepted Answer

$$x = 1$$ and $$x = -2$$

Question 3

An ice cube is melting so its volume decreases at $$0.1\ 	ext{cm}^3/	ext{sec}. $$What is the rate at which its side length is changing when the side is $$2\ 	ext{cm}$$?

Accepted Answer

$$\frac{ds}{dt} = -\frac{0.1}{12} = -0.0083\ 	ext{cm/sec}$$

Question 4

A spherical balloon is being inflated so its volume increases at $$5\ 	ext{cm}^3/	ext{sec}. $$What is the rate of change of its radius when the radius is $$10\ 	ext{cm}$$?

Accepted Answer

$$\frac{dr}{dt} = \frac{5}{400\pi} = 0.00398\ 	ext{cm/sec}$$

Question 5

A 13-ft ladder is leaning against a wall. The base slides away from the wall at $$2\ 	ext{ft/sec}. $$When the base is $$12\ 	ext{ft}$$ from the wall, how fast is the top sliding down?

Accepted Answer

$$\frac{dy}{dt} = -\frac{24}{5} = -4.8\ 	ext{ft/sec}$$

Question 6

For the function $$y = \cos^2$ x - \sin x on$$ $$[0, 2\pi]$$, at which $$x-$$values is the rate of change zero?

Accepted Answer

$$x = \frac{\pi}{2},\ x = \frac{7\pi}{6},\ x = \frac{11\pi}{6}$$

Question 7

The position function of a particle is $$y = 6 - \sin x. $$What is the velocity function?

Accepted Answer

$$v(x) = -\cos x$$

Question 8

For the function $$y = \ln(\cos x + 1)$$, between $$(-2\pi, 2\pi)$$, at what $$x-$$values is the rate of change zero?

Accepted Answer

$$x = 0$$ and $$x = \pm 2\pi$$

Question 9

A cylindrical water heater tank with radius $$1\ 	ext{ft}$$ and height $$4\ 	ext{ft} is $$being drained. If the water level drops at $$0.5\ 	ext{ft/min}$$, how fast is the water draining out?

Accepted Answer

$$\frac{dV}{dt} = 2\pi \cdot 1^2$ \cdot 0.5 = \pi\ 	ext{ft}^3/	ext{min}$$

Question 10

For the function $$y = \sec(x^2$ + 2x - 4)$$, give one $$x-$$value where the rate of change is zero.

Accepted Answer

$$x = 1$$

Question 11

Find the derivative of $$y = (2x - 1$$)^{34}$$(7x - 10$$)^{12}$$.

Accepted Answer

$$y' = (2x - 1$$)^{33}$$(7x - 10$$)^{11}$$ \left[34 (2x - 1$$)^{1}$$ \cdot 2 (7x - 10$$)^{12}$$ + 12 (7x - 10$$)^{1}$$ \cdot 7 (2x - 1$$)^{34}$$ \right]$$

Question 12

Given $$y = \ln(\cos x + 1)$$, what is the equation of the tangent at $$x = \frac{\pi}{2}$$?

Accepted Answer

$$y = 0 + 0(x - \frac{\pi}{2})$$

Question 13

For the function $$y = x^2$y^2$$ - $$x^3 + xy = 5$, when x = 1$, what are the two possible y$-values?

Accepted Answer

$$y = 2$$ and $$y = -2$$

Question 14

For the function $$y = e^x(x^2$ - x - 11)$$, what is its derivative?

Accepted Answer

$$y' = e^x(x^2$ - x - 11) + e^x(2x - 1)$$

Question 15

If $$y = \sin$$^{-1}$$(e^x)$$, what is $$y'$$?

Accepted Answer

$$y' = \frac{e^x}{\sqrt{1 - e$$^{2x}$$}}$$

Question 16

A particle moves according to $$y = 6 - \sin x. $$What is its velocity at $$x = \frac{\pi}{6}$$?

Accepted Answer

$$v(\frac{\pi}{6}) = -\cos(\frac{\pi}{6}) = -\frac{\sqrt{3}}{2}$$

Question 17

For the function $$y = \ln(\cos x + 1)$$, what is its derivative?

Accepted Answer

$$y' = \frac{-\sin x}{\cos x + 1}$$

Question 18

If the surface area $$A of a $$cube with side $$s is$$ $$A = 6s^2$, and the side is decreasing at -0.0083$$\ 	ext{cm/sec}$$ when s = 2$$\ 	ext{cm}$$, what is the rate of change of surface area?

Accepted Answer

$$\frac{dA}{dt} = 24s \cdot \frac{ds}{dt} = 24 \cdot 2 \cdot (-0.0083) = -0.399\ 	ext{cm}^2/	ext{sec}$$

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