SOLUTIONS Name: Section: F01 (Rhodes) F02 (Bueler) Rules

Math F251 Midterm 2 Spring 2019

Name: Section: ⇤ F01 (Rhodes)⇤ F02 (Bueler)

Rules:You have 60 minutes to complete the exam.Partial credit will be awarded, but you must show your work.No calculators, books, notes, or other aids are permitted.

Circle your final answer to each question where appropriate.If you need extra space, you can use the back sides of the pages. (Clearly label any work you want graded.)Turn o↵ anything that might go beep during the exam.Good luck!

Problem Possible Score

1 102 103 104 105 156 157 108 10

Extra Credit 3Total 90

1

SOLUTIONS

Math 251: Midterm 2 10 April 2019

1. (10 points)

Consider the functionf (x) = x � 2 sin(x).

a. Find the critical number(s) of f (x) on the interval [0, ⇡]

b. Find the absolute maximum and absolute minimum values of f (x) on the interval [0, ⇡].

2. (10 points)

a. Find the linearization of g(x) =p

x at a = 4.

b. Use your result in part a. to approximatep

3.9.

2

f' GI = I -2 cos = o

" o

Costa .

- I .: X÷÷÷¥÷sm⇒⇒.rs . . absm÷¥;

abs.mu?=ftaI-#g'Cx7=ztx'

n=zn*g ( 3. 9) =L C 3.97=2+4 (3.9-4)€2-¥o=7f


3. (10 points)

A rectangular photograph whose width is 3 times the height is being enlarged, and the height is increasingat a rate of 2 in/min. How fast is the diagonal of the rectangle increasing when the height is 4 in? Indicateunits.

4. (10 pts.)

Evaluate the following limits.

a. limx!1

ln xpx

b. lim✓!⇡/2

1 � sin ✓1 + cos(2✓)

3

W = 3h

\✓h CZ w 't h2

= @h5th 2

-

w I = to h-

c = To h

Castro¥Ironing

in.

Ein.

= Eo ¥

= LIE ¥ = OEiEsa

.. =E→⇒÷÷

it f÷±i:÷=÷ , =¥


5. (15 points)

Consider the curvey =

x2

x2 + 5.

a. Determine the domain, and symmetry (if any).

b. Find all asymptotes (horizontal and vertical). For each asymptote compute a limit which justifiesthat asymptote.

c. Find all intervals of increase and decrease.

4

noverhialasymptotesffsj.FI⇐ I so

A is horizontal asymptoteCan be

justified byL' Hop .tl 's rule

fG) ⇐2×6757-442×72⇐

( x 't 5) 2 ( x 75 )

=10×2 f 70€XZO62*5 )

'


Problem 5 continued....

d. Find all points of inflection.

e. Sketch the graph, labeling all important points.

5

f"

( x ) =t5OxhCx75 ) - 2x

CX¥5,4- =

⇒G¥]47534

= 106--3×7*7533

f"

( x ) = o ⇐ 5-2×2=0

x=t pts of m- 81.

Ft

- - -. - - -

.-

- - - . - .

I-

- . . . - - - --

-- -

>

- Ta• ×


6. (15 points)

An open-topped box with a square base is to be made so that its volume is 4000 cm3. Find the dimensionsof the box using the least materials. Indicate units.

6

a↳#Y 4000 =×2y

A =x2t4xya tbase sides

y= 40¥ :

A = It 160×1

A' Klux-

"

A 4×1=0 ⇐ 2×3=16000

⇐ X2=8000

⇐ x=2o ⇒ y -_=4%2=4%4--10

::::÷%:


7. (10 pts.)

A racetrack is described by the equation

2(x2 + y2)2 = 25(x2 � y2),

whose graph is shown. If an object falls o↵ a speeding car asit travels over the track, it will follow a tangent line.

a. Verify that (3, 1) is a point on the track, and mark it on the graph.

b. Draw the tangent line at (3,1) on the graph.

c. Find an equation of the tangent line at (3,1).

7

my✓

?

2 ( 371 ) 2=25 ( 32 - I )

2 ( 105 I 25C 8)

200 = 200 U

✓

implicit differentiator

4 ( x' ty ) C 2x try DEE) = 25/2×-2 y Iff)8×1×75 ) t 8g ( x2 tf ) off = 5-Ox - 50 y dff

dy 50 x - 8*1×2+42)Fx

= -

8yCx7y2 ) -150 y

m= Ext. ,=:%¥=t÷=i÷

y-i=-f


8. (10 points)

The graph of the derivative f 0 of a function f isshown.

a. On what intervals is f increasing or decreas-ing? Use interval notation.

1 2 3 4 5 6 7

�2

�1

1

2

3

x

f 0(x)

b. At what values of x does f have a local maximum or minimum?

Extra Credit. (3 points)

For x > 0, let g(x) = xx. Compute the first and second derivatives of g.

8

increasing : [ 1,33 U [ 6,7 ]

decreasing : [ o, D U [ 3,63

x. X =3

loc.mil X = I ,X= 6

g C x ) = xx

I n g Cx ) = xlnx

get, Stx ) = I - hx t X - ¥ = I thx

C I thx ) glx ) =(lthxIn g'Cx) = h ( Ith x ) t xhx

g 9"Cx ) = ÷ t I . hxtx - I

gf¥xt⇒IH7YI÷

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SOLUTIONS Name: Section: F01 (Rhodes) F02 (Bueler) Rules