Sample final exam
Math 30, Fall 2016
As usual, you should treat this sample exam not as a guide to what will be covered in
our exam, but as a guide to what the questions will be like. As always, your best guide to
what will be on the exam is the homework.
You will be allowed to use the usual calculators and ONE 3 × 5 notecard. Unless
otherwise stated, you must show all your work in a problem to receive full credit.
1. Compute the following derivatives. Show all your work, and do not simplify your final
answers.
(a) (6 points) Let w = (x3 − 5x − 7) ln(3x2 + 4). Find
dw
.
dx
2 sin2 x + 1
. Find f 0 (x).
3x2 − ex
√
(c) (6 points) Let y = cos
x4 + 4 . Find y 0 .
(b) (6 points) Let f (x) =
2. (10 points) Find the most general antiderivative of f (x) = 3x5 − 7x2 + cos x. Show all
your work. You do not need to simplify your final answer.
3. Consider the function g(x) graphed below. All parts of this question deal with this same
function g(x), which is defined for −5 ≤ x ≤ 5.
y
2
1
x
−5
5
(a) (10 points) If lim g(x) exists, compute the limit and briefly explain why lim g(x)
x→−3
x→−3
is what you say it is; if lim g(x) does not exist, briefly explain why lim g(x) does
x→−3
x→−3
not exist.
(b) (10 points) If lim g(x) exists, compute the limit and briefly explain why lim g(x) is
x→2
x→2
what you say it is; if lim g(x) does not exist, briefly explain why lim g(x) does not
x→2
x→2
exist.
(c) (8 points) Find a value of x, −5 < x < 5, such that g(x) is not differentiable at x, and
briefly explain why g(x) is not differentiable at that value of x.
4. (10 points) Let f (x) = x2 +2x . To calculate f 0 (−5) from the definition of the derivative,
you would have to evaluate a certain limit. Write down the limit you would have to calculate
to calculate f 0 (−5) from its definition. (In other words, write down the definition of f 0 (−5),
as specifically applied to this function f (x) = x2 + 2x .) DO NOT CALCULATE this
limit; just write down what you would need to calculate.
5. (14 points) Find the absolute minimum and maximum (i.e., the global minimum and
maximum) of the function h(x) = 7 − 4x − x2 on the closed interval [−4, −1] (i.e., in the
domain −4 ≤ x ≤ −1). Show all your work and give x and y values for all absolute
min/max.
6. (12 points) Find an equation of the tangent line to the graph y = h(x) = (x3 − 1)−2 at
the point where x = 2. No explanation necessary, but show all your work. You do not need
to simplify your final answer; all numbers in your final answer should be given as exact
values or rounded off to 4 decimal places.
7. Let f (x) be a function such that f (2) = −3, f 0 (2) = 7, f (−3) = 5, and f 0 (−3) = −11.
(a) (8 points) Let h(x) = tan(f (x)). Compute h0 (2). You do not need to simplify your
answer, but if you use decimals, please round off your final answer to 4 decimal places.
(b) (8 points) Let k(x) = f (x)(x2 + 3)−2/3 . Compute k 0 (−3). You do not need to simplify
your answer, but if you use decimals, please round off your final answer to 4 decimal
places.
8. (15 points) Suppose f (x) is a function that is continuous on the closed interval [2, 5] and
differentiable on the open interval (2, 5). Suppose also that f (2) = 7 and f (5) = −6. What
does the Mean Value Theorem tell you about this function f (x)? Briefly but precisely
EXPLAIN your answer. (Note that you do not need to explain why the Mean Value
Theorem is true; you just have to explain what the Mean Value Theorem says.)
9. (15 points) Andi is walking her dog Oscar on a 10 foot leash. Andi and Oscar are both
creatures of habit, so they always keep the leash stretched out to its full 10 feet, and they
always stay on the L-shaped sidewalk shown below (un-shaded area). If Oscar is 8 feet
north of the corner of the L, and is moving north at 5 ft/sec, at what speed is Andi moving,
and in which direction (west or east)?
Oscar
10 ft
8 ft
Andi
Show all your work, and put your final answer in the form of a complete sentence.
10. Suppose f (x) is a function whose DERIVATIVE f 0 (x) is graphed below on the domain −2 ≤ x ≤ 4.
y
x
−2
4
(a) (7 points) Find the values of x (−2 ≤ x ≤ 4) for which f (x) is decreasing, and name
the one feature of the graph of f 0 (x) that justifies your answer. (I.e., your justification
should begin, “f (x) is decreasing here because. . . ”.)
(b) (7 points) Find the values of x (−2 ≤ x ≤ 4) for which f 00 (x) is positive, and name
the one feature of the graph of f 0 (x) that justifies your answer.
(c) (6 points) Find the x values of the inflection points of f (x) (−2 < x < 4), and name
the one feature of the graph of f 0 (x) that justifies your answer.
11. (20 points) David and Eric are building a rectangular fenced-in play area for their cat
Nikita. They have 4 feet of fence to build with, but to get the most use out of their fence,
they decide to build the play area against a wall, with three sides of the rectangle built
from fence, and the fourth side coming from the wall, as shown in the picture below. What
are the dimensions of the largest play area they can build?
play area
wall
Show all your work, and put your final answer in the form of a complete sentence.
12. Let f (x) be a function such that
f 0 (x) = (x2 − 1)e−x .
Note that this is the formula of the DERIVATIVE of f (x), not f (x) itself. All parts of
this question refer to this same function f (x), in the domain −2 ≤ x ≤ 2.
(a) (9 points) Find the exact values of all critical numbers of f (x) for −2 < x < 2, and
classify each critical number as a local minimum, local maximum, or neither. Show all
your work.
(b) (5 points) Find the exact values of all inflection points of f (x) for −2 < x < 2. Show
all your work.
(c) (8 points) Sketch one possible graph of f (x) for −2 ≤ x ≤ 2. Clearly label all critical
numbers, local maxima and minima, and inflection points, and make sure that the
concavity of f (x) is clear throughout your graph.