MAC 1147 Exam #4/ Final Exam Review

Instructions: Exam #4 and the final exam will each consist of 14 questions. The first six questions will each be worth 6 points and the last 8 questions will each be worth 8 points. There are four questions from Exam #1 and five each from Exam #2 and Exam #3. Only a basic calculator or scientific calculator may be used, although a calculator is not necessary for either exam. NO GRAPHING CALCULATORS WILL BE ALLOWED! For Exam #4, you will be allowed to use any resource you wish other than another person (no cell phones or com­puters, even if they are not being used to communicate with another person). So you may bring your book, notes, note cards, reviews, old exams, etc. For the final exam, you WILL NOT be allowed to use a note card, although you may use the Trigonometric Identities and Unit Circle which I posted online for Exam #3, provided that neither is written on. I will NOT provide these for you on the exam, so if you would like to use them, you must bring them your­self! Both exams will be similar to the previous tests and this review, although the numbers and functions may be different on each question on the exams. The questions on both exams will be taken from the previous three exams. Each of the 14 questions can be found in the following places:

1. Exam #1, Problem #7 or Section 5.4 (6 points)

2. Exam #2, Problem #2 or Section 6.2 (6 points)

3. Exam #2, Problem #5 or Section 7.5 (6 points)

4. Exam #3, Problem #3 or Section 8.4 or 8.5 (6 points)

5. Exam #3, Problem #9 or Section 8.7 (6 points)

6. Exam #3, Problem #2 or Section 8.8 (6 points)

7. Exam #1, Problem #2 or Section 3.4 (8 points)

8. Exam #1, Problem #3 or Section 3.5 (8 points)

9. Exam #1, Problem #9 or Section 5.5 (8 points)

10. Exam #2, Problem #7 or Section 6.1 (8 points)

11. Exam #2, Problem #9 or Section 6.6 (8 points)

12. Exam #2, Problem #14 or Section 7.8 (8 points)

13. Exam #3, Problem #11 or Section 8.1 (8 points)

14. Exam #3, Problem #5 or Section 8.3 (8 points)

1. Solve each inequality.

(a) X4 - 8x2 < 9 (3 points)

X'1-<tx2 - qL: 0

(",7. _~)(",' +1) L.. 0

( x +3) ex -~)(i -H) L.. 0

('(~ ~) (x -s) (y!-f-() ::- 0

x=- "'~I 3

(b) ~~; > 3 (3 points)

~X-+·-3~O x-3

'i'(- t - (3/f\- '')------:>0

'(-~

X-\-'2... .. . '>- 0 ')(-3

x..\-'l-=--o '"X - ~ ~u .­

X-=--;). X-=--~

2. Suppose f is the one-to-one function

4xf(x) =­

x-4

(a) Find the inverse of f. (4 points)

y7­

(y -y). ~ (';J-) Cr'r)

(b) Find the domain and range of f and f-l. (2 points)

1)o~W\ +~ RCW\~ .f -I == 1R\ ~'\}

UbI'NIA" f -' =~ += 1R\ ~ '11

3. The point ef) -~) is a point on the unit circle. Find the

value of the six trigonometric functions corresponding to this point. Be sure to simplify your answers as much as possible and rationalize the denominators.

, 5·,t'\ e-= - "3

~~ Cos e= - :;

\ --.. .

~ . ­G

4. Find the exact value of each expression. You should simplify the fractions as much as possible, but you do not need to rationalize them. (3 points each)

(a) sec345°

(b) sin ei;) \S"

.!Jl( . ltft =ass ~ ,R r SI~ as~ ::: ~I(\ ( '&)0° - 4~') ~ il'" ?/JOO (0S 4~ - (.f)S 30cf bin '-(SO

5. Solve each equation on the interval 0 < () < 27T. (3 points each)

(a) sin (3()) = - V;

38:: ~n- ' (- ~0

r -.,

e::- 1(£0 8,()..so 3'-1SC

) J

(b) 1 - 2 sin2 () = 0

6. Solve the equation on the interval 0 < fJ < 21f.

sin (2fJ) = cos fJ

Z~I"e lO~ ~ - cos & =-0

Cos e (.;tSiV\ 0 -I) ~ 0

(D~ e=-0 ()r () ~\~ G- I ::: 0

1&-:: ~OO, J.1oJ ()r ~I;'" It =-/

~Irdt =~

7. Let f (x) be the piecewise defined function

f(x) = {~ x 3 if - 2 <x< 0

if x> 0

(a) Find f( -1). (1 point)

_ (_1)3 :: _ (-I) :: QJ (b) Find f(2). (1 point)

\ 2

(c) Find the domain of f(x). (2 points)

(d) Sketch the graph of f(x). (4 points)

-10

-4 -2 o 2 4

-5

8. Graph the function.

I I t

r I

-~

-I

.,

.J

r

--- i' - ---- i 1

o .-,,

x

I

-, ~- - -' - ­--t­"" I

i II­I

9. Let f(x) be the polynomial function below. Find all the real zeros of f(x) and use them to factor f over the reals.

f(x) = X4 - 4x3 + 3x2 + 4x - 4

f\)~"\\..,-t =3 or l

.fe-X) -:: xl.! +~x-; +3x'l. - ~\ - 4

f'( - \) -= 0

~ I - '-( '-{

l ~

~

-~ -( $""4 , - 5 f - '1 0

~G~ ~m', + (X) =(~.j-I)('t-\)(~~l)l

=tuo~ ·. - \I 'j 2

3(\} ::: 0

JJ I - :; ~ - \..I ,t -~ ~ - \{ y D

-f (~) ~ (~+\) (X-t} (t - '{\ +\{)

+L'f..\:: (~+I) (x -\) (~ -2') (,,-2-)

10. Suppose 2x 2

f(x) = ; g(x) = ­2x + 1 x

(a) Find fog and state the do/ ain. (4 points)

-tG(.~~ (:r)' ~ .l~ jx . ~ ~ ~"tl (~+IX ~

Uo~" ..Ii J =1R\to}

Ot,~" J -l' = W\'-i} =l> -} '* - i . ~"> 1=> X ~ - II ((ross "",h'P7)

[1)0:''' .f -1'. j =1R\ \,O, -I{} J (b) Find g 0 f and state the domain. (4 points)

\ (). (;?k~) Ibx+,) pxtJ3(f (~)) :: ( ;)x' - - -" ~.(:wf,) - ~x

1)()~'I~ {)f f = l\~-~ 1 V~V\ .t j:: w.\fo} ~ 2:: to

11. Solve each equation.

(a) 22x - 3 . 2x+2 + 25 = 0 (4 points)

(a 'ICy - ~ . ~ . JX +-~= 0 \OJ 2Y:::)( I foJ:2 ~~x

(ixY - \2 ~ J'X -4-?>2 ::. 0

)(

y-=- ;;

,2 - 127+32 '::0

oCy - !j)(r~) ~

Y~Y I '1::-~

2x ::~ I 1. '( =8

G~ l , x =3 1

(b) log3 (x + 10) = 2 -log3 (x + 2) (4 points)

1DJ/K-I'/O) +\bJ :/Xf2) ==- '2.

(OjJ r~~o) (M~~"=- 2.

(~+Ir,) 6:'1 il = S 1

Xl +Ilx..Jz.o =: 9

'/ + 12~+l l =- ()

(x~1t) (~·H)-= 0

1- ll , G=-I] h ..-\­ J1rw.J

12. Graph the function. Be sure to label the intervals on the x - and y-axes and show at least two cycles.

y = - 2 cos ( 7rX + ; )

(~ /'\ /~ ( \ I \ I I \\

~ ~ 0 ..1 I ~ 2. 5­ 3 % 'j

i. J. 2­ 2­

I I

\) \ ) \ ~ \ IJ

13. Est ablish the identity.

1 + sin e 1 - - --1 = --­

cos2 e esc e- 1

I -t ~1\ G - (l- ~,n't e) /i1>'iV\ e-fi 51h~G --

~2.e \- ~\V\"2.e

~'t\.1T~~e~ - ~1t'\&-

-L _ 50;&­(I-&j~~)~ ~i't\& &;~9

-

14. Find the exact value of each expression. (2 points each)

(a) sin -1 [sin (4;) ] ~ _ '!r- .1[ ~ II - '3

(b) cos- 1 [cos (i;)]

~rr----. 1'2

(c) cos [cos -1 ~ ]

(d) cos [csc -1 (i~) ] ~

QvJ r N2

l'L Llli :;

CJYS 6 c: r~}

Extra Blank Graphs.

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