Solutions - University of Georgia · Math 2250 Final Exam December 2019 8. [10 pts] Use calculus...

University of GeorgiaDepartment of Mathematics

Math 2250Final Exam December 2019

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ProblemNumber

PointsPossible

PointsEarned

1 22

2 15

3 18

4 8

5 15

6 18

7 14

8 10

9 25

10 14

11 10

12 10

13 15

14 16

15 20

Total: 230

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Solutions

Math 2250 Final Exam December 2019

1. Determine the following limits; briefly explain your thinking on each one. If you applyL’Hopital’s rule, indicate where you have applied it and why you can apply it. If your finalanswer is “does not exist,” 1, or �1, briefly explain your answer. (You will not receivefull credit for a “does not exist” answer if the answer is 1 or �1.)

(a) [4 pts] limx!1

x2 � 9

x2 + x� 6

(b) [6 pts] limx!2+

x2 � 9

x2 + x� 6

(c) [6 pts] limp!0

ln(1 + 5p)� p

sin(3p)

(d) [6 pts] limx!1

xe�2x

A of 17

I 9 T1 or 2

I useddirectsub pluggedin

D

429.7 8 solimitDNEmaybe IN29 5 asX 217 I so Wx'tX 6 0andispositiveas x ZtJ

in Ftpla I O

P topf indeterminateformcanuseL'Hopital'sRule LH

ox'in finds D

A o isanindeterminateformcan'tuseLHyetisanindeterminateformcanuseLH


2. (a) [5 pts] State the limit definition of the derivative of f(x).

(b) [10 pts] Use the limit definition of the derivative to determine the derivative of

f(x) =1

3� 2x. No points will be awarded for the application of di↵erentiation rules

(and L’Hopital’s rule is not allowed).

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fhittingthh or flatling

flashing OR fiakyy.tkfala1higo3 xt hitIx figa a

This.tl n t a a

ya 229432113.251

dingo's D x'ima.EEaTEashimoth43IhxThiT3IxD zap213 272


3. Determine the first derivative of each of the following functions. Remember to use correctnotation to write your final answer.

(a) [6 pts] f(x) = x2 � sin(x) cos(x)

(b) [6 pts] g(t) = arctan(ln(t))

(c) [6 pts] h(x) =e3x

x+ 1

4. [8 pts] Determine the second derivative of the function

f(t) = 3t2 �pt

A of 17

OkfG 2x sincxksinlxdtcoslxkoslxDEf.mil2xtSin4x cos4x answer

g'Ctl 1_ I tffkalthrifty L answer

LtttlenHF

e3 xtiokfinal

h'CxktiK3e3xLe3q

danswers

OR h11841 152 txti5l3e3Xt'T e3xfcxt.ptxtDl3D3Xe3xt3e3e3x xti2631413111Http3 3 232

1 1526316 2

a1 2

f t _letEE'tf ft 6t4t32


5. Consider the curve defined by the equation 3x2 + 3xy + y2 = 5.

(a) [10 pts] Determinedy

dx.

(b) [5 pts] Determine all values of x for which the point (x, y) on the curve3x2 + 3xy + y2 = 5 has a horizontal tangent line. The graph of the curve is providedbelow for your reference.

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6 3171y3 tLy1 0

3 12y1 tox3gDYE 6323 2yothercorrectformsfor1Ff 6537 6533 t g

6 13506x 3g2x y32131271425 53 2 62 4 2 5

2 5X i


6. Determine the following indefinite integrals.

(a) [6 pts]

Z �x3 � 5x+ 7

�dx

(b) [6 pts]

Z(ln(x) + 4)10

xdx

(c) [6 pts]

Zsin(t)

1� 2 cos(t)dt

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4x4_5zX2t

fudu f u tC 4 lnlxH4u built4du f dx

I futdu luluItc 1zlnl2coshu I 2costdu 2sinCtldt


7. Evaluate the following definite integrals.

(a) [6 pts]

Z 1

0

✓ex � 3

1 + x2

◆dx

(b) [8 pts]

Z 3

0

f(x) dx, where f(x) is the function given by f(x) =

(sin (x) 0 x < ⇡

2

1 ⇡2 x 3

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e 3arctanxDo

e 3arctank Ceo3arctanCoD

e 3 I

x fcosxD Ccosth fcos6 O lD I

x Cx E3 th

ffundx It3TYz 4


8. [10 pts] Use calculus techniques to determine the x-coordinates of the local (relative) ex-trema for the function below. Be sure to label each extremum as a local (relative) maximumor as a local (relative) minimum.

f(x) = x6 � 4x3

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domainCupf41 6 5122

643 2x 0,352

1stderivativetest

to o o tz 352 is

ftpf'N6lD f 24644signftbehfdef deny ingrelmin

NoteThesecondderivativetestisinconclusiveatx o


9. The graph of y = f(x) is given below and consists of line segments and circle arcs. Thedomain of f is (0, 6).

x

y

1 2 3 4 5 6

1

2

(a) [5 pts] Determine all values of x in (0, 6) for which f 0(x) > 0. Write your answer usinginterval notation.

(b) [5 pts] Determine all values of x in (0, 6) for which f 0(x) is undefined.

(c) [5 pts] Estimate f 0(1.5).

(d) [5 pts] Determine whether f 00(3) is positive, negative, or zero. (Write “positive,” “neg-ative,” or “zero.” No explanation is needed.)

(e) [5 pts] Determine

Z 6

0

f(x)dx.

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2,3 14,5

1,24,5

2

negative

2TIt 2 2 5tIz


10. [14 pts] On the axes provided below, sketch a graph of a function y = f(x) which meetsthe following criteria:

• Its domain is (�1, 2) [ (2,1).

• The line x = 2 is a vertical asymptote to the graph of y = f(x).

• The function f is continuous at every point in its domain.

• The sign chart for the derivative of f is the following:

interval (�1,�2) (�2, 2) (2, 4) (4, 6) (6,1)sign of f 0 � + + � +

• The function satisfies f(4) = 4 and f(6) = 1.

• limx!1

f(x) = 1

• The line y = 4 is a horizontal asymptote to the graph of y = f(x).

x

y

-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

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daga

ThdMyT.io ii.t

outs

Furse'Erfewiff g and I and

µ


11. [10 pts] Determine all values of c for which the function below is continuous on (�1,1).Use the limit definition of continuity to explain your answer.

f(x) =

8<

:

9� x2

3� x, x < 3

c, x � 3

A of 17

Note fisalreadycontinuouseverywhereexceptx 3

Tomakefcontinuousatx 3 weneed yHx f13

f3 c

ngfix c

3tk figmz 9zx lijg3tf kxliggf3txl6

Tomakef continuousatx 3 weneedc 6


12. [10 pts] Find the total area of the region enclosed by the curves y = sin(x) (solid) andy = � sin(x) (dashed) from x = 0 to x = 2⇡. The relevant graphs are provided below foryour reference.

⇡2

⇡ 32⇡ 2⇡

�1

1

x

y

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fFinkLsinkDdx sindxEnosxD LnostrilfuoscoD 2 2 4

Togettotalareaenclosedcanuseanyofthemethodsbelow

It'sink fsinkDdx4usingabovesteps

fffsinksinkDdx1,72sinkdxEwsHD 2 l214

Total414187

Sosink fsinkDdx 4usingabovesteps

Totalarea2141181

fo'sinMdx CcosxD I I 2

Totalarea4480

Ijtihadx foodxDoh o lD 1Totalarea8i D


13. [15 pts] Karina and Juan both leave an intersection at the same time, both driving on long,straight roads. Karina drives north at a speed of 50 miles per hour, and Juan drives east at60 miles per hour. When 1.5 hours have elapsed since they left the intersection, how fast isthe distance between Karina and Juan changing? Use calculus to justify your answer.

A of 17

KK d 5omph goal 1 whent 1.5ZY n d 60mPh

x j

E x y t1.5hours11604.5190miles

2zd 2x t2yddY g504.5175miles

ftp.XI tydYdE It

2

d 9g If m

Note d isapproximately78mphAlsoprovidetheexactanswershownintheboxabove ifyougiveanapproximateanswerlike78mph


14. You plan to make a rectangular box having an open top. Each face of the box is a rectangle,and the top face of the box is missing so that the box is open. For the base of the box, youwant the length to be twice the width; you also want the total surface area of the resultingbox to be 200 square inches.

(a) [12 pts] Let w represent the width of the box in inches. Determine a formula V (w) forthe total volume of the box as a function of w alone. Your final answer should includethe variable w and can not include any other variables.

(b) [4 pts] Suppose you want to determine the dimensions of the box that will result inthe maximum possible volume. Determine an appropriate domain for V (w). Brieflyexplain your answer. (Note: We will not actually maximize V (w) in this problem.)

A of 17

I1h v ewhb w zoo_ewt2wht2lhl 2w Constraintequation

20042Wlwt2wht22wh

HM 200 2w2t2wht4wh

DT 200 2026Whzoo2w2 refinedconstrainth e Tw h equationUwww.eooii7gEaeIiIetwhegEsaIIaev

2wko602w7 lGoow2w3 answer

0,10 or 0,10Foviolatesconstraintequations wcan'tbenegative

largerwcorrespondstosmallerhjsethO

0 200zurTw0200202WE200w lo


15. Suppose you run a factory that processes raw materials in two possible ways.

• Using method A, you can process x tons of material at a cost of x2 dollars.

• Using method B, you can process y tons of material at a cost of 10y dollars.

You want to process a total of 200 tons of material, so your process is constrained by theequation x+ y = 200. The overall cost (in dollars) of processing the two materials is

C = x2 + 10y.

You can process a partial ton of material using each method, so x and y do not have to bewhole numbers. You plan to minimize the overall cost.

(a) [4 pts] Determine the overall cost C as a function of x. (The variable y should notbe used in your answer.)

(b) [4 pts] Determine an appropriate domain for your cost function C from part (a), andexplain briefly.

[This problem continues on the next page.]

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Xty 200y200X

C X'tloyx'tlozoox X't200010x Theseareallcorrectwaystof toxt2000 writeyourfinalanswer

XzoousingmethodAexclusivelyI

0200px o usingmethodBexclusively


[Problem 15, continued]

(c) [12 pts] Use calculus techniques to determine the value of x which results in the mini-mum overall cost. Write a sentence, using appropriate units, to summarize your answer.

A of 17

C x 10 2000

0,200

c 2 10x 5

Extremevaluetheoremclosedintervalmethod

oo min5 255020001975 cost20040000 20001200040000

WhenX 5tonstheoverallcostisminimized


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Solutions - University of Georgia · Math 2250 Final Exam December 2019 8. [10 pts] Use calculus...

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