Final Exam for SM121/131, 1330-1630, 15-DEC-2015

NAME: ~~~~~~~~~~

VERSION 4 ALPHA __ _

NO CALCULATORS ALLOWED ON TlilS PART SHOW ALL WORK IN THIS EXAM PACKAGE

19. a) State the limit definition of the derivative.

f't)t.) :::- ,R;rv.., ft~-+-~ -{f>t!) h-'JO

b) Use the definition of the derivative to show that ;fr(2x2 + x) = 4x + 1.

k'W\ i\? 0

k'wi ~ ...>;> 0

2 (x+t.-.)2.-r(x+h)- ( 21l-tx)

h

2(xi. +z.xh + h ~) ~ ~ h - 2x'"-~ =

h.

l~ t\I\ h?O

~ + ~}(h+..2..hth-~ =fi ~v\. ~( l-tx+z.h t-1)

· k h~o ~

it.Vl!l l 'tx+2-h +I);:;.. 4x-r I. i,~o

20. State and prove ONE of the following: Option 1: Product rule for computing the derivative of f (x)g(x). Option 2: The derivative of y = sin x is ~ = ... . Note: You may benefit from looking up the results listed on page 14.

2..

d$_ d.x.

Lf+i"" l ~bc.)~(Jc.~ I::. t~iC) jCx.) +tC~~'L.ic)

Final Exam for SM121/ 131, 1330-1630, 15-DEC-2015

NAME:

~- Page 9of14 VERSION 4

ALPHA

NO CALCULATORS ALLOWED ON THIS PART SHOW ALL WORK IN THIS EXAM PACKAGE

21. Consider the function f(x) = {ix+ 8.

a. Find the equation for the tangent line toy= f (x) at a= 0. 1j

-rt o)-= ~ g - =- 9 -u~ {-( :><) =-G+ g) 3

I - 2/3 .f Cx) -=- ~ (x+ &') ~ ....!., ?> I 2 ---- - 3 \J x...\-~ \j~2+~ x]

•r ( I

·, -r ( 0) -

Lrx) ;:;;.. 2..+ L \: 12-

b. Use your linear approximation from a. to estimate .ys.24.

c,. Sketch the graph off (mark the intercepts on the graph), and the graph of the tangent line from a.

L[x)

_..:::::.::::=---- J ~ ( (k)

9

=- J_ 12.

Final Exam for SM121/131, 1330-1630, 15-DEC-2015

NAME: -------------------

Page 10 of 14

ALPHA ____ _

YOU SHOULD TURN IN PARTS I, II, and the SCANTRON FORM PRIOR T O USING THE CALCULATOR ON THIS PART Show details of your work and box your answers.

PART III: LONG ANSWER. Calculators allowed AFTER TURNING IN PARTS I, II and the SCANTRON FORM. Parts I , II and the bubble sheet are due 20 min prior to the end of the exam period.

22. Consider the function f(x) = x2 + 2}!1

a. Use the first derivative test to identify the intervals where f is increasing or decreasing. If you use the calculator to assist your computations, make sure to show the relevant results.

f'(x) = 2x - 5'0A = (x2-+ 1 ):z..

'

z X(K-z)(x.rz.) ( ~ '!+G) c x '2.-+r) 2-

I

... -... - 2.-

/ ++

D

/, +-l- 12 {x

I ;x

x 2+1 =:.. .t 5

x2 +1 = 5

\x : ±2 , x~!

~ (-3) =. -1/z._ ( .f (- 1) :;_ ~'

I 2.

+ l Vt t.r(Q-0i l1.d oY\..

( - 2,o) o..'t\.d ( 2, ?0)

+ ( 1) ::. - 2..1 /z.. b. Find the x and y coordinates of the local minima and local maxima for f

lo~ YY\~ri VvtA.

+:Cf + =- 'l

c. Use the formula for f to determine whether f is even, odd, or neither.

{ (-,x)::. ~x)~ -1-

f iS .. e-i,tC-11\.

(-x(~ I

d. Sketch f (incorporate your results from a,b,c into your sketch).

e. Find the most general antiderivative for f.

10

+ ~~·~~ C5}\_

( - oe>1 - 2.) GL ~ • ( ~ ;?.

Final Exam for SM121/131 , 1330-1630, 15-DEC-2015

NAME: -----------------------

Page 11 of14

ALPHA ________ _

YOU SHOULD TURN IN PARTS I, II, and the SCANTRON FORM PRIOR TO USING THE CALCULATOR ON THIS PART Show details of your work arid box your answers.

23. a. Find (show all work) limx-" f(x), where

· \' 5e°'+3er b. Fmd lffix --+oo aox+e2r .

0

) 0

. l +cos :z: {~

f(x) = sin(2x) tanx

for 0 < x ~ 2,

for 2 < x ~ 4,

Uvi·L ~ U>2f ~x :L. X-7TC" ~c...\x )

I. f re t.ci. le.. . x~vo ,....zx r I + ~~ 1

1... L e 1 )CJ

:::. 5 I

2. L'H ix ><

l i0 e +- 3<:

. ' )

3.o+ 2e2 ><

24. A rocket R is launched vertically from a pad P on the ground located 3 miles from the ground radar station S. Find the altitude and speed of the rocket at the moment when the rocket is 5 miles from the radar station, if the angle between SR and the horizontal is increasing at a rate of 12 radians per hour.

j -= j (i.)-'- ~ oJ:n'fi.,Je.. , iVI trJ I~<; .' t :::. tlv\ ~ } tit\ h_trtLfJ

f7-;. elt) 0...~ le t f<._ 'S f\ I h l{a.cJ..l'o...11 S

J 2 + 3 2-;: ~ L/' cj =- lj- t:0,:'e Y>~ ~1~\\: J

c.~-ze

q

2-5

)

Final Exam for SM121/ 131, 1330-1630, 15-DEC-2015

NAME: ----- - --------

Page 12 of 14

ALPHA _ __ ___ _

YOU SHOULD TURN IN PARTS I, II, and t he SCANTRON FORM PRIOR T O USING THE CALCULATOR ON THIS PART Show details of your work and box your answers.

25. A rectangular storage container with an open top and a square base is to have a volume of 800 m3. Material for the base costs $10 per square meter. Material for the sides costs $2 per square meter. Find the d imensions for the cheapest such container.

ld: X -:: -4C.e ~-(J.l la# , ~ ~ baS(:

( i ri Yy\ ~+-.rs ) ~ ::.. +PLP h~j h:f &f ~ box i( WLtfers)

c

&oo -)

de.. 2o X - b lf-oO X_.,_ ;: Zox. - 6;:o JX

J) _ ~oo I 3 r.::­h -- =- O '\lS ><. a..

rvC~). ) $ ) 2-ox~-::. 6'fo ::>

x:. :... '6 2.o , ,'llf x

X-? ~ .,.-> oo

c ....,QI) e ~ O<)

x::. ~2-~ ~ [ . B~ 'X ;;.. tr 'VS Mto .

Do not writ e b e low this line.

Question 1 u s

Question 2 u s E

Question 3 u s E

Question 4 u s E

Question 5 u s E

E DrtA...wS o... p!"du.rt> !l")iv-n cluc.t~ Va.r1~le.s lc..Sds 'Pi"c fu •

( Vl~ U.l't.t./-<":.) )

Sc. h I.Ar -i-k " "k.Lw\ e. rc.11..u: r e IN\ et"\.. -t-

s.e.-h ~ ~ cost ( o~ j ecff i--e) t.t<:. ii...:'.) ~"t~ l1d( s Sds \..tr +£..t l"t+ as s;"'-'jle vQ .. r1·aJ/e &vie--fl'Ov\..

h Vt.&S -1{.e_ c..r ; ti' etr.e po i vd recd.-0~1 s (\-1<:. Kc n~ ~1 C<VU l~tVl'S u..(l olt\f\'\ e v. ,,·~"t . V>f~ t.u'lt'fs)

O: ci'V lh M t\C ) lX" J~ ~-e'\\1.l r;4 ~ ~ k b~+ verJ 5lof pd ~K '5 : cl&e.s t')IOJ~ r! +e.e vnr.'k..J p:>SS ·· b~ wi.-f-{1 ')!11'-_a.ll N rv1$l ~u.f ( c)l\ ccpt Ok