MAC 2233 Final Exam Review

Transcription

1 MAC 2233 Final Exam Review Instructions: The final exam will consist of 10 questions and be worth 150 points. The point value for each part of each question is listed in the question, and the total point value of each question is listed below. Only a basic calculator or scientific calculator may be used, although a calculator is not necessary for the exam. NO GRAPHING CALCULATORS OR CALCULATORS ON A DEVICE (SUCH AS ipod, CELL PHONE, ETC.) WHICH CAN BE USED FOR ANY PURPOSE OTHER THAN AS A CALCULATOR WILL BE ALLOWED! The concepts and types of questions on the final exam will be similar to the previous tests, previous reviews, and this review, although the numbers and functions may be different on each question on the exam. The questions on the exam will be taken from the previous four exams. Each of the 10 questions can be found in the following places: 1. Exam #1, Problem #2 or Section 1.5 (18 points) 2. Exam #1, Problem #6 or Section 2.4 (16 points) 3. Exam #2, Problem #1 or Section 2.5 (8 points) 4. Exam #2, Problem #2 or Section 2.6 (12 points) 5. Exam #2, Problem #5 or Section 3.1 & 3.2 (24 points) 6. Exam #2, Problem #6 or Section 3.3 (8 points) 7. Exam #3, Problem #4 or Section 4.3 & 4.5 (12 points) 8. Exam #4, Problem #1 or Section 5.1 (16 points) 9. Exam #4, Problem #2 or Section 5.2 (16 points) 10. Exam #4, Problem #4 or Section 5.4 (20 points)

2 1. Find each limit. If a limit does not exist, explain why it does not exist. (6 points each) (a) vx lim x-tl x-i X~.?, - Y ~ ~+b -;>. ~}~ - II'M ),..., \ -- ( \ iv~ \i M ~ti -t;) ~ 'X.~, ():- I) Ch+? t:;).) x-'}, Jkt3 +-~ ~+-~ fi"t-d fjh - J. Jl{-~ _ ~_Q kil"\ X 4 / l - \ r ~(JiH +:l) -L [l... -.j+;l - () 0 0! INkJuMI~ (b) " x2-6 I1m x-> - 4 X - I - ~ - 10 ~ I-J:1-5 L:.l (c) " x IIm-- x->-2 (x + 2)2 lim )( - ~ - J :::-00 X~-2 - t!ny pos ~ lif\') '(..., - 2+ ~t2):l ~ X - -?1~Alj ::..-00

3 2. Find each derivative. (8 points each) (a) h(t) = (t 3-4) (4t 3-2t2 + 5) (b) f(x) =!~=~ t'(l();:: (~lc -5') (3) - (3 \( -SV'I) (~X, s)2 ::.- 12x - l~ - (/2'( - W) ~\ y-s-)'" - '~ - IS--.,0" (h:- S-V

4 3. Find slope of the tangent line to the graph of f(x) = Jx2-6x + 9 at the point (5,2). (8 points) -f (X)-= ('/--G,x+4)'" -r' (X\= ~ (~.-Co x-r~(. (2'K -ij,) -f' (~') ::: ~ (S'--"s +1r i. (2'S-- ~) \ - ~ (~~-30+~)-~ - ~ (yr t. y, (10 -Co) -- ~f\i. l{ - y - ~ ' d- 'i ":::: --.. ~ -- ill

5 4. (a) Find the second derivative of the function h(3) = 3 3 ( ). (6 points) ~t{~)~cos5-i(os~- (os [!" (s)= 305~ -lfgs~- Co] (b) Find the third derivative of the function f (x) = 2x 5 + 4x 3 2x. (6 points) ~(~\~ J'f..'b +~'{3-2x.f '(X) ~ lox x:z. - 2 til (X):: 4Dx ~ +Jl\x

6 (a) Find the critical numbers. (6 points) -f'(x)~ '<~'-("~')-{2X'-)j'){~.) =- /Dlx~-(qX4-32.l _ /2x 4 -g yuzk (:4~)"l- ~kl.l l(k'-l _ "K'-4+32~ _ W(k 3 T9) :" X3T~ ~ (X.J.,()(X2.J~X"'~) X 3 4'(ll - J4'x.>f3 )(3 -f '(K)~~t,'"J J0=fO (x=oj (b) Find the open intervals on which the function is increasing or decreasing. (6 points) r ', ri(k x:::: --3: +'(-3)~ m:, pos ;y- O.;.:g =pos II : V'IV\. 1(':: - I', +' t (-I)::: -f; :: :J lit : fig~ x~ I'. +' I (1'')::! ::: pos '\ fllreasi1 '. (~(XJI - d) U(0) (0) J.tc.CUl s', ~ : (- i}. ( 0 )

7 (c) Find the local maximum and local minimum values of the function. (6 points).("(x) ~ ~ t"'1>\~ -b ~Vt of x.~ -J, So tel<) ~s ~ 'lv\tr~i1 -t. ckc.((~~\"j oj-- X--()., 80 ~ j'i> mJ ~ at x-=--j. ~ (-ir= a(-l\~ -g ~(- 8) -~ -,- rfa--<6 _ -}-y ~ _~ J[-d-)'Z - ~('-t) <6 ~ ilur.j VWN( =(-Jr -3)J )< ::00 i:; 0. ~CJ 1l.~t"fM 50 ',+ ~ I.. c. I~ r>4nt: or [~""'VI. n~ MM~v\~] (d) Find the extreme maximum and extreme minimum of the function on the interval [-4, -1]. (6 points) txtrl~ ~ : (-;),-~) ~+rtjn. ~\'" '. (- \t, -.!})

8 6. Find the inflection points and the open intervals on which the function f(x) = x4-6x x2 + 7x - 6 is concave upward or concave downward. (8 points).f II (X-) =- 12x z - 3(0)( +-~~ :: I~ [X'- 3X tj.) ~ 1:2. (X -02.')C)(-1) t~(x-~') (X-I \.:: D 1. :?ic.t. '1<::::0 " -\' \I (0)= (po5)(~)(jj-= r.s Jr 'f Pid x:: ;',-f II (;)~ CfOS} ("j)(pes) =:-J ][: 'Piel x= 3', fl C3! -.: ( FJ(pos)Ceos) ~p05 CiN\CQ.vt ~ '. (- )0 J,) u((;l, ~') ~CAve. ~', (/,,,)

9 7. Differentiate the function. (6 points each) (a) f(x) = in (V4x + 6).f(Xk I~ ((l{~ho) I /~) I \ -. --~. y - - i 4tH" ;; ~ ttt~

10 8. Find the indefinite integral. (8 points each) (a) J(4x 3-5x + l)dx ~J X~~ - s JXrk. +JItNx i(4) -s(tl H +C [ X~- ~X~rltCJ (b)

11 9. Find the indefinite integral. (8 points each) (a) (b) _ 'l \.;\-~ - ~ J\A ~ 3:; "Lol:; Ju? oi~" _ Ju-s;)~ -l\ J:L.. +c. -y

12 10. Evaluate each definite integral. (10 points) (a) 3L>toW -Dd0< i(~)- k I_~ X~-l< I : :: [l -:21-( I~-Il ~[~ - J1-[H1 (b) 13 (y + 3)2dy U::CJ+:S ~ iu ::: ~ ~r:' 3 : Ct -:: ~-t 3 ::. ~ d-=-o : U:=-o -\-1~ ~ r u\k =- "~ 1 -=- - Co 3 O ~ :3 dlb ~+ - ~-3 :::.. ~;;t - q :: ~3]