Math 2144 Final Exam NAME:

CRN:

INSTRUCTIONS: This exam is a closed book exam. You may not use your text,homework, or other aids except for a 3×5 inch notecard. You may use an allowable calculator,TI-83 or TI-84 to• perform operations on real numbers,

• evaluate functions at specific values, and

• look at graphs and/or tables.A TI-89, TI-Nspire, or a calculator with a computer algebra system, any technology withwireless or Internet capability (i.e. laptops, tablets, smart phones or watches), a QWERTYkeyboard, or a camera are not allowed. Unless otherwise stated, you must show all ofyour work including all steps needed to solve each problem and explain your reasoning inorder to earn full credit. This means that correct answers using incorrect reasoningmay not receive any credit. This comprehensive exam assesses your understanding ofmaterial covered throughout the entire course.

Turn off all noise-making devices and all devices with an internet connection and put themaway. Put away all headphones, earbuds, etc.

This exam consists of 10 problems on 13 pages. Make sure all problems and pages are present.

The exam is worth 134 points in total.

You have 1 hour and 50 minutes to work starting from the signal to begin. Good luck!

1

Math 2144 Final Exam

Final Exam Grade byProblem Number

No. Out of Pts.

1 12

2 12

3 16

4 11

5 16

6 10

7 12

8 16

9 8

10 12

Total 125

2

Math 2144 Final Exam

1. (2 points each) True (T) or False (F) statements. Circle your answer. You do not needto provide justification. True means always true.

(i) T F If limx→5

f(x) = 7, then f(5) = 7.

(ii) T F If g is a continuous function with g(4) = 9 and g(11) = −6, theng(x) = 0 must have a solution for some x in the interval (4, 11) bythe Intermediate Value Theorem.

(iii) T F Suppose the functions g, f , and k are integrable on [−2, 9]. Ifg(x) ≤ f(x) ≤ k(x) for all x in the interval [−2, 9], then∫ 9

−2g(x)dx ≤

∫ 9

−2f(x)dx ≤

∫ 9

−2k(x)dx.

(iv) T F If a function f is differentiable on [1, 8] and f ′(5) = 0, then f musthave a local maximum or minimum at x = 5.

(v) T F If g′(x) > 0 for all x in the interval (0, 7), then g(5) > g(2).

(vi) T F Suppose f is continuous on [−1, 7]. Let Rn represent the right-hand

Riemann sum approximation of

∫ 7

−1f(x)dx. If f is decreasing on

[−1, 7], then Rn <

∫ 7

−1f(x)dx.

(vii) T F If x = c is a critical point for f and f ′′(c) < 0, then f has a localminimum at x = c.

(viii) T F If g is continuous at x = a then g is differentiable at x = a.

3

Math 2144 Final Exam

2. (12 points) Answer the following questions based on the graph of f below. Assume thatall critical points, point of discontinuity, and points of inflection of f can be observedfrom the graph below. Asymptotes are indicated by dotted lines. (Note: Pay attentionto the numbers on the axes.)

(a) Give numeric values for each of the following. Write “DNE” if the value does notexist and “∞” or “−∞” as appropriate.

f(−3) = f(−6) = limx→−6

f(x) =

limx→−∞

f(x) = limx→∞

f(x) = limx→−4

f(x) =

limx→6−

f(x) = limx→6

f(x) = f ′(7) =

(b) List the x-values of all critical points in the interval (−4, 6).

(c) List the x-values of all inflection points in the interval (−4, 6).

4

Math 2144 Final Exam

3. (4 points each) Evaluate the following limits or state that they do not exist (“DNE”).Use ∞ or −∞ if either is appropriate. Numbers alone without justification (eitheralgebra and/or quoting theorems where applicable) will receive no credit. If you useL’Hopital’s Rule, clearly indicate when you do so.

(a) limx→−2

x2 + 8x+ 12

x+ 2=

(b) limx→π

2

cos(x) + 1

x=

(c) limt→∞

1 + 4t2

5t+ 3t2=

(d) limθ→0

θ

tan(θ)=

5

Math 2144 Final Exam

4(a) (6 points) Consider the graph of y = f(x) illustrated below.

Write each graphical quantity, A-F, in the blank next to corresponding expression onthe left. Each letter will be used exactly once.

Expression Graphical Quantity

h A

f(a) B

f(a+ h) C

f(a+ h)− f(a) D

f(a+ h)− f(a)

hE

limh→0

f(a+ h)− f(a)

hF

6

Math 2144 Final Exam

4(b) (5 points) The following image illustrates a Riemann sum using N terms:

Write each item on the right in the blank next to the corresponding expression onthe left. Items B, C, and A3 refer to the labeled graphical quantities above. Eachexpression will be used exactly once.

Expression Item

∆x B

f(x3)∆x C

f(x3) A3

N∑i=1

f(xi)∆x

∫ b

a

f(x)dx

limN→∞

N∑i=1

f(xi)∆x A1 + A2 + A3 + · · ·+ AN−1 + AN

7

Math 2144 Final Exam

5. (4 points each) Compute the following derivatives. Do not simplify.

(a) Let y =2x+ 5

3x− 2. Find

dy

dx.

(b) Let f(x) = x2 cos(x) + ln(x). Finddf

dx.

(c) Let g(θ) = tan(θ). Find g′′(θ).

(d) Finddy

dxfor the curve x+ sin(y) = xy. (You must solve for

dy

dxin terms of x and

y).

8

Math 2144 Final Exam

6. (10 points) A hot air balloon rising straight up from a level field is tracked by a rangefinder 500 feet from the lift-off point (see the image below). At the moment the rangefinder’s elevation angle is π/4 radians, the angle is increasing at a rate of 0.14 radiansper minute. How fast is the balloon rising at that moment?

9

Math 2144 Final Exam

7. (12 points) You create a box with no top (shown on the right) by cutting squares ofside length x out of a 9 × 24-inch piece of cardboard (shown on the left) and foldingup the sides.

(a) (4 points) Define a function V that gives the volume of the box in terms of sidelength x of the square cutout.

(b) (8 points) Use calculus to determine the value of x for which the box has maximalvolume. Justify your answer.

10

Math 2144 Final Exam

8. (4 points each) Evaluate each of the following. Show all of your work and give exactanswers. If the answer is a number, do not round.

(a)

∫ π

0

(1 + cos(x))dx =

(b)

∫(ex − 4 sin(x)) dx =

(c)d

dx

∫ 7

x

√2t4 + t+ 1dt =

(d)

∫x2√

5 + 2x3dx =

11

Math 2144 Final Exam

9. (8 points) The function A is defined as A(x) =

∫ x

−6f(t)dt. The graph of f is shown

below.

(a) (2 points) Shade the region corresponding to A(4) on the graph above.

(b) (4 points) Evaluate A(4) and A(−8).

(c) (2 points) Find A′(2).

12

Math 2144 Final Exam

10(a) (6 points) Determine the area of the region between y = x and y =√x over the interval

[0, 1].

10(b) (6 points) Suppose the region between y = x and y =√x over the interval [0, 1] is

rotated around the x-axis. Completely set up the integral that represents the volumeof the resulting solid but do not evaluate this integral.

13