Question 1

A car travels along a straight road such that its position at time $$t (in $$hours) is given by $$s(t) = 4t^3$ - 3t^2$ + 2t (in $$miles). What is the average rate of change of the car's position from $$t = 1 to$$ $$t = 3$$?

Accepted Answer

$$34$$ miles per hour

Question 2

Evaluate the following limit: $$\displaystyle \lim_{x 	o 0} \frac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 3

Which of the following functions is NOT continuous at $$x = 2$$?

Accepted Answer

$$f(x) = \frac{x^2$ - 4}{x - 2}$$

Question 4

Given $$f(x) = x^2$ + 3x$$, use the limit definition of the derivative to find $$f'(2).$$

Accepted Answer

$$7$$

Question 5

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

Accepted Answer

$$2x \cos(x^2$)$$

Question 6

A medication is administered such that its concentration in the bloodstream at time $$t (in $$hours) is $$C(t) = 10e$$^{-0.5t}$$. What is the rate of change of the concentration at t = 2$ hours?

Accepted Answer

$$-3.68$$ units/hour

Question 7

Find $$\frac{dy}{dx} if$$ $$x^2$$ + $$y^2 = 25$.

Accepted Answer

$$-\frac{x}{y}$$

Question 8

Let $$f(x) = \ln(x^2$ + 1). $$What is $$f'(x)$$?

Accepted Answer

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

Question 9

A spherical balloon is being inflated so that its radius increases at a rate of $$2 cm/$$min. At the moment when the radius is $$5 cm$$, how fast is the volume of the balloon increasing? (Recall $$V = \frac{4}{3}\pi r^3$.)

Accepted Answer

$$200\pi cm$^3/$$min

Question 10

Use linear approximation to estimate $$\sqrt{25.2}.$$

Accepted Answer

$$5.02$$

Question 11

Given $$f(x) = x^3$ - 3x^2$ + 2$$, find all critical points.

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 12

For the function $$f(x) = x^4$ - 4x^3$, on which intervals is f(x$$)$$ concave up?

Accepted Answer

$$(-\infty, 1) \cup (3, \infty)$$

Question 13

A company wants to design a cylindrical can that will hold $$500 cm$^3 of $$soup. What dimensions will minimize the amount of material used (surface area)?

Accepted Answer

Height equals diameter

Question 14

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{e^x - 1}{x}$$ using L'Hospital's Rule.

Accepted Answer

$$1$$

Question 15

Use Newton's Method to approximate a root of $$f(x) = x^2$ - 2$$ starting at $$x_0 = 1$. What is x_1$?

Accepted Answer

$$1.5$$

Question 16

Which of the following statements about the Mean Value Theorem is TRUE?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, then there exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Question 17

Find $$\int \sec^2$(x) dx.$$

Accepted Answer

$$	an(x) + C$$

Question 18

A factory's power consumption (in kW) at time $$t (in $$hours) is given by $$P(t) = 2t + 3. $$What is the average power consumption from $$t = 0 to$$ $$t = 4$$?

Accepted Answer

$$7 kW$$

Question 19

State the Fundamental Theorem of Calculus, Part 1.

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$F(x) = \int_a^x f(t) dt$$, then $$F'(x) = f(x).$$

Question 20

Solve the differential equation $$\frac{dy}{dx} = 3y$$ with initial condition $$y(0) = 2.$$

Accepted Answer

$$y(x) = 2e$$^{3x}$$

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