Question 1

A body moves along a straight line with velocity $$v = F(t) as $$shown in the graph. At which time interval is the body moving in the negative direction?

Accepted Answer

$$t = 2 to$$ $$t = 5$$

Question 2

Given the velocity function $$v = F(t)$$, when is the body at its maximum speed?

Accepted Answer

When $$|F(t)| is at $$its maximum value

Question 3

If the length $$l of a $$rectangle is decreasing at a rate of 2 inches per second while the width $$w is $$increasing at a rate of 2 inches per second, what happens to the area $$A of $$the rectangle?

Accepted Answer

$$A$$ remains constant.

Question 4

Find $$\lim_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Question 5

Find $$\lim_{x 	o 0} \frac{\sin 3x}{x}.$$

Accepted Answer

$$3$$

Question 6

A cube's side is increasing at a constant rate of $$0.2 cm/s. $$What is the rate of change of the volume, in cubic centimeters per second, when the surface area is $$6 cm$^2$$?

Accepted Answer

$$0.15$$

Question 7

Brent is riding his bicycle north away from an intersection at a rate of $$14$$ miles per hour. Sally is driving her car toward the intersection from the west at a rate of $$36$$ miles per hour. If Brent is $$4$$ miles from the intersection and Sally is $$3$$ miles from the intersection, what is the rate at which the distance between them is changing?

Accepted Answer

$$22$$ miles per hour

Question 8

Given $$f(x) = (1 - \sin x)^3$, use the local linear approximation at x$$ = \frac{\pi}{2}$$ to estimate f(0.5$$)$$. Is the estimate an overestimate or underestimate?

Accepted Answer

The estimate is an overestimate.

Question 9

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

Accepted Answer

$$(2x + x^2$) e^x$$

Question 10

A medication is administered to a patient and the concentration $$C(t) in $$the bloodstream decreases according to $$C(t) = C_0$ e$$^{-kt}$$. What is the rate of change of concentration at time t$?

Accepted Answer

$$-k C_0$ e$$^{-kt}$$

Question 11

A car's position is given by $$s(t) = t^3$ - 6t^2$ + 9t + 2. At $$what time(s) is the car at rest?

Accepted Answer

$$t = 1$$ and $$t = 3$$

Question 12

A bacteria population grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of growth at t = 10$?

Accepted Answer

$$5e^{0.5}$$

Question 13

If $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 14

A ball is thrown upward with velocity $$v(t) = 20 - 9.8t. At $$what time does it reach its maximum height?

Accepted Answer

$$t = 2.04 s$$

Question 15

If $$f(x) is $$differentiable and $$f'(a) = 0$$ and $$f''(a) \u003e 0$$, what can be concluded about $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f(x)$$ has a local minimum at $$x = a$$

Question 16

A tank is being filled with water at a rate of $$5$$ liters per minute, but is leaking at a rate proportional to the amount of water present, $$L(t) = kW(t). $$Which differential equation models the amount of water $$W(t) in $$the tank?

Accepted Answer

$$\frac{dW}{dt} = 5 - kW$$

Unit 4 Review: Contextual Applications of Differentiation

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