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 car's average velocity between $$t = 1$$ and $$t = 3$$ hours?

Accepted Answer

$$30$$ miles/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

$$g(x) = \frac{1}{x-2}$$

Question 4

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

Accepted Answer

$$6$$

Question 5

If $$y = x^3$ + 2x$$, what is the second derivative $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

$$6x$$

Question 6

Find the equation of the tangent line to $$f(x) = \ln(x) at$$ $$x = 1.$$

Accepted Answer

$$y = x - 1$$

Question 7

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

Accepted Answer

$$5e^{0.5}$$ \approx 8.24$$

Question 8

If $$y^2$$ + $$x^2 = 25$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

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

Question 9

Find the derivative of $$f(x) = \arctan(x).$$

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

Use linear approximation to estimate $$\sqrt{4.1}.$$

Accepted Answer

$$2.05$$

Question 12

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

Accepted Answer

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

Question 13

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 14

A farmer wants to build a rectangular pen with one side against a barn (so no fence needed on that side). If the farmer has $$100$$ meters of fencing, what is the maximum area that can be enclosed?

Accepted Answer

$$1250 m$^2$$

Question 15

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{3x^2$ + 2}{x^2$ - 5}.$$

Accepted Answer

$$3$$

Question 16

Use L'Hôpital's Rule to evaluate $$\displaystyle \lim_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Question 17

Use Newton's Method with initial guess $$x_0 = 1$ to find the next approximation x_1$ for a root of f(x$$) = $$x^2 - 2$.

Accepted Answer

$$1.5$$

Question 18

Which of the following statements is the Mean Value Theorem for derivatives?

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 19

Find $$\$$int_0$$^2 (3x^2$ + 2) dx.$$

Accepted Answer

$$14$$

Question 20

A particle moves with acceleration $$a(t) = 6t$$, and its initial velocity is $$v(0) = 2. $$What is the velocity function $$v(t)$$?

Accepted Answer

$$v(t) = 3t^2$ + 2$$

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