# 1. Continuity (60 points)

For each of the following functions, state: the range, the interval(s) on which the function is continuous, and for each discontinuity whether or not the discontinuity is removable.
(a) $y = x^2 + 4$

(b) $y = \frac{x + 2}{x^2 - 4}$

(c) $y = 2 \lfloor x \rfloor$

(d) $y = 4\cos(\theta) + 1$

# 2. Slope and Derivative (50 points)

(a) Find an equation of the line tangent to $f(x) = x^2 + 1$ with slope $-2$. (20 points)

(b) Use the limit definition to find the derivative of the following functions (30 points; 15 points each)
i. $f(x) = x^2 - 1$

ii. $h(t) + 2\sqrt{t}$

# 3. Rates of Change (45 points)

(a) The profit $P$ from selling $x$ units of a product is given by $P = 15 + 12\sqrt{x} - \frac{81}{x}$ i. Find the average rate of change of $P$ on the interval $[1, 9]$. (10 points)

ii. Find a formula for the marginal profit. What is the marginal profit for $x = 9$? (15 points)

(b) A sphere of radius $3$ has its radius changing at a rate of $4$ inches per second. Howo quickly is the volume of the sphere changing? (20 points)

# 4. Derivative Tests (50 points)

(a) Apply the Second-Derivative Test to find the relative extrema of $f(x) = x^3 - 3x^2 + 1$. Show all of your work. (25 points)

(b) Apply the First-Derivative Test to find the relative extrema of $g(t) = t^4 - 4t^3 - 4$. Show all of your work. (25 points)

# 5. Optimization (50 points)

(a) Find two positive numbers such that their product is $75$, and so that the first plus three times the second is a minimum. (20 points)

(b) A rectangular page is to contain $18$ square inches of print. The margins at the top and bottom are $.5$ inches and on each side are $1$ inch. What dimensions minimize the amount of paper used? (30 points)

# 6. Sketching Graphs (60 points)

Sketch the graphs of the following functions, showing all work: state domain and range; label all symptotes, points of discontinuity, and intercepts; and show how you obtained relative extrema, points of inflection. (30 points each)
(a) $g(x) = \frac{x^2}{x^2 - 1}$

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

# 7. Differentials (45 points)

(a) Find the differential $dy$ if $y = \sqrt{x^2 + 1}$. (15 points)

(b) Find the differential $dy$ if $y = \sin \theta + \cos \theta$ (15 points)

(c) Use a differential to approximate the change in revenue $R$ corresponding to an increase in the number of sales $x$ of one unit, if $R = 50x - 2x^2$ and $x = 20$. (15 points)

# 8. True/False (40 points, 5 points each)

Mark each question as True or False.
a. The volume of a sphere of radius $r$ is $4\pi r^3/3$.

b. The absolute maximum of a function on a closed interval never occurs at the endpoints.

c. The derivative of $\sec(x)$ is $-\sec(x)\tan(x)$.

d. A function can only change from increasing to decreasing at a critical number.

e. A function cannot intersect its horizontal asymptotes.

f. If $f'(x) > 0$ for all $x$, then $f$ is concave upward for all $x$.

g. The Product Rule states: $\frac{d}{dx}[f(x)g(x)] = g(x)f'(x) + f(x)g'(x)$.

h. A point of inflection of $f$ can occur only at a critical number of $f'$.

Extra credit: State the definition of the derivative of a function $f(x)$. (4 points)
Extra credit: State the derivative of $|x|$. (4 points)