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SP10_MAC2311_E1ab - MAC 231 Spring 2010 Exam 1A A Sign your...

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Unformatted text preview: MAC 231]. Spring 2010 Exam 1A A. Sign your scantron sheet in the White area on the back in ink. B. Write and code in the spaces indicated: 1) Name (last name, first initial, middle initial) 2) UP ID number 3) Discussion Section number C. Under “special codes”, code in the test number 1, 1. e 2 3 4 5 6 7 8 9 O o 2 3 4 5 6 7 8 9 O D. At the top right of your answer sheet, for “Test Form Code” encode A. e B C D E E. This test consists of 10 five—point and 2 two-point multiple choice ques— tions, three one-point bonus questions and two sheets (4 pages) of partial credit questions worth 26 points. The time allowed is 90 minutes. F. WHEN YOU ARE FINISHED: 1) Before turning in your test check for transcribing errors. Any mistakes you leave in are there to stay. 2) 'You must turn in your scantron and tearoff sheets to your discussion leader or proctor. Be prepared to Show your picture ID with a legible signature. 3) The answers will be posted on the MAC 2311 homepage after the exam. Your discussion leader will return your tearoff sheet with your exam score in discussion. Your score will also be posted on Elearning within one . week. ‘ NOTE: Be sure to bubble the answers to questions 1—15 on your scantron. Questions 1 — 10 are worth 5 points each. l _ .7: _ 2 a: 1. If f(:c) — a: _4, evaluate irrflfla: 3:). 1 1 . . . a. 1 b. g c. E d. 0 e. The limit does not ex1st. 2. Find each value of a: on the interval [0, 27F) for which tang: 2 seem. a. (gwl o (3?” 2%) I b. [(19% ,3?) _ c. (32537”) f d. [0, g) U (3%, 271') e. Thereare no such values of 3:. 3. A population of rare insects on an isolated island has been observed to double every three years. If researchers counted 400 insects at the beginning of the study, how many years would it take for the population to grow to 1800? Hint: First find a formula which gives the population of insects on the island after 15 years of study a 31n2 ears b. i ears ln(9/2) ears ' ln(9/2) y 1:19y C‘ 31112 y 31n(9/2) — I ln(3/2) d. W years e. ln 2 years 2A 4. Let f(:z:): 33:1 and g(a:)= $231 2 51- (9013902; az+1 ,b. (90f)(93)=$_1 az+1 6- (90f)(93)=$_1 2 01- (90f)(93)=; 2; e. (gem): xi, 3, 6 5. Let f(:z:) = 42:22. Which of the following statements is/ are true? P. lim f(1:)= oo. x—>2- Q- f(93 R f domain: (—00, 0) U (0,00) domain: (—00, 1) U (1,00) domain: (—00, —1) U (—1, 1) U (1,00) domain: (—O0,0) U (0,1) U (1,00) domain: (—00, —1) U (—1, 1) U (1,00) ) (1:) has horizontal aSymptote y = 0. can be made continuous at CI: = —2 by defining, f(—2) S. f (as) has vertical asymptotes a: = —2 and a: = 2. a. P, Q and R d. P and R only b. R and S only e. P, Q and S 3A c. Q and R only . Find (9 o f)(:z:) and its domain. § 4. 6. An object moves in a straight line so that its position in feet after 75 seconds is given by 3(73) = 27524—3. Find a formula for the average velocity of the object on the interval [1, 1 + h] for h 7Q O, and the instantaneous velocity after 1 second. Average Velocity Instantaneous Velocity a. 4 + 2h feet / sec 6 feet/ sec ‘ b. 2h2 + 4 feet / sec 4 feet / sec c. 4 + 2h feet/sec 4 feet/sec d. 2 + 4h + 2h2 feet/sec ‘ 2 feet/sec e. 2h2 + 4 feet / sec 6 feet/ sec em‘3 a: < 2 7. Let f (as) = . Find the value of 7" which will make f (as) 7" — 2a: a: Z 2 continuous for all real numbers. 7" = 4 1 ‘ a.2+e b.— c.2——— d.4—e e.4—{—l e e e . ‘22: + 4| . 3 . L t = = _ _ 8 e p $133100 33—1 andq $1220 2-1-63: Then a.p:—ooa11dq=oo b. 102—2 andq=0 c. p=2andq=g d.p=—2andq=g e.p=2andq=0 4A 9. Let f (56) = x/4— 56. Find f’1(sc) and its domain. Hint: consider the relationship between range and domain of inverse functions. You may want to sketch your functions. a. f—1(x) = 4 — 562 domain: (—00, oo) 1 b. f‘1(sc) = \/4__ domain: (—oo,4) — :5 c. f—1(sc) = V4 — 562 domain: [—2, 2] d. f‘1(5c) = 4 — 552 domain: [0, oo) e. f_1(x) 2 v4 —- 562 domain: [0, 2] 10. According to the Intermediate Value Theorem, the function f (x) = 2 cossc — 2x — 1 must have a zero on which one of the following intervals? aw 13- cw) we w e. none of these The following problems are worth 2 points each. I 1 —cos:c 11. Evaluate lim —~.—2~——. Hint: use a trig identity to rewrite the function. : m—)O+ 8111 {E a.oo‘ b.0 c.1 I d.—oo a; 5A 2 12. Evaluate lim 63:1. Note that 32:1 is the exponent. $—)—OO 1 a. 0 b. ——00 c. — 6 Bonus! ! State Whether each of following is true or false (one point each). is an odd function. sin :1: a. True b. False 14. lirn ln(5 — ct) 2 ~00. (5—)5— a. True b. False 15. The graph of the function f (2:) = ICC—l has a hole at :6 = 0. .7: a. True b. False 6A MAC 2311 TEST IA PART 11 SPRING 2010 Sect # Nanie UFID Signature SHOW ALL WORK TO RECEIVE FULL CREDIT. 1. Let f(a:) = V218 + 3. a. Use limits to find the slope of the tangent line to f (as) at a: = 3. b. Write the equation of the tangent line to f (as) at a: 2 3. 7A 2. Solve each of the following: a. Find the solution set: Zlna: — ln(a: + 2) = 0 b. Let f(:1:) = 005 2:1: and let g($) = Being: + 2. Find each value of :1: on the interval [0, 271'] at which f(:1:) 2 9(33). 3. Evaluate the following: 2 a. cot (arcsin g) = . Hint: draw a triangle. 8A Sect # Name 4. A farmer wishes to enclose a rectangular piece of land to create three equal garden areas as shown below using 160 yards of fencing. a. Find a function A(:I;) which gives the total area enclosed in terms of 1:. Find the domain of A(J:) domain(interval notation) b. Use your function to find the dimensions that will enclose an area of 800 square yards. 1; : y = 5. It can be shown that for 0 in (0, g), 0cos9 g sin9 S 6’. . . t . . sin6’ Use the Squeeze Theorem and thls inequality to find 011151+ 9 _) 9A tan—1:1: a: < O 6. Consider the function f(:1:) : 3 — lasl O S a: < 3 :1:2 — 9 a: > 3 (a) Sketch the graph of f (:r) (b) Find the following limits if possible. my}; f(a:) = 33% f<x) = lini f(a:) : lirn “93) = a:—>0+ m——>3+ (c) List all discontinuities of ‘ f (as) and state Whether they are jump, infinite or removable. (d) Find each interval on which f (as) is continuous. 10A MAC 2311 Spring 2018 Exam 1B A. Sign your scantron sheet in the white area on the back in ink. ~ B. Write and code in the spaces indicated: 1) Name (last name, first, initial, middle initial) 2) UF IDnumber 3) Discussion Section number C. Under “special codes”, code in the test number 1, 2. . e 2 ' 3 4 5 6 7 8 9 0 1 e 3 4 5 6 7 8 9 0 D. At the top right of your answer sheet, for “Test Form Code” encode B. A e C D E E. This test consists of 10 five—point and 2 two—point multiple choice ques— tions, three one—point bonus questions and two sheets (4 pages) of partial credit questions worth 26 points. The time allowed is 90 minutes. F. WHEN YOU ARE FINISHED: 1) Before turning in your test check for transcribing errors.‘ Any mistakes you leave in are there to stay. 2) You must turn in your scantron and tearoff sheets to your discussion leader or proctor. Be prepared to Show your picture 11) with a legible signature. 3) The answers will'be posted on the MAC 2311 homepage after the exam. Your discussion leader will return your tearoff sheet with your exam score in discussion. Your score will also be posted on Elearning within one week. NOTE: Be sure to bubble the answers to questions 1—15 on your scantron. Questions 1 - 10 are worth 5 points each. 1 1 4 1. If f(a:)= :r _ 16’ evaluate 33131116f(m). . . . 1 1 a. The l1m1t does not ex1st. b. 1 c 0 d. a e 1 2. Find each value of :1: on the interval [0, 2%) for which tana: _<_ sec :13. a. [0%) u (33712”) b. (333”) c. [(1,725) o #33”) 3. A population of rare insects on an isolated island has been observed to double every four years. If researchers counted 600 insects at the beginning of the study, how many years would it take for the population to grow to 1500? Hint: First find a formula which gives the population of insects on the island after t years of study. ln(5/2) ln(5/2) , 4ln(5/2). a. 4ln 2 years b. ln2 years c. W years d. i e rs 4ln2 ln5 y a” V 6' ln(5/2) years 213 1 3:1: 4. Let f(x) = x _ 2 and g(x) = x + 1. Find (9 o f)(:c) and its domain. a. (g o f)(x) = :3 domain: (—oo, '2) o (2, oo) b (gofxn) : £1 domain: ('—oo,i)o(1,oo) o. (gof)(x) = FEE—:5 domain: (—oo,—1)U(—1,2)U(2,oo) d (g o f)(a:) = x 3 domain: (—oo, -1) o (—1,2) o (2,oo) o. (gof)(x) = “Ti—1 domain: (—oo,1)U(1,2)U(2,oo) 5. Let f(:c) : ECU—+3” Which of the following statements is / are true? P. lirn f(x) = 00. ctr—+3— Q. f (x) can be made continuous at x = —3 by defining f (—3) : R. f (x) has vertical asymptotes x = —3 and x = 3. S. f (11") has horizontal asymptote y = 0. a. P and S only b. Q and S only C. R and S only d. P,QandS e. P,Qan‘dR 3B 1 7" + 3x ac g 2 . . . Find the value of 7" which Wlll make f (ac) em’3 cc > 2 6. Let flan) ={ continuous for all real numbers. r = 1 1 a.6+e 113—6 c.—3—— , d.—6~—e e.4+- e e e . An object moves in a straight line so that its position in feet after 75 seconds is given by 5(t) = 3732 —|— 1. Findva formula for the average velocity. of the object on the interval [1, 1 + h] for h 7E 0, and the instantaneous velocity after 1 second. Average Velocity Instantaneous Velocity a. 6 + 3h feet / sec 6 feet / sec b. 3h2 + 6 feet/sec 6 feet/sec c. 362 + 6 feet / sec 3 feet / sec (:1. 6 + 3h feet / sec 3 feet / sec e. 3 + 6h + 3h2 feet/sec L 9 feet/sec 8. Let p = Aimee litjfl and q 211330 2 fem. Then 5 a.p=—4andq=§ b.p=—ooandq=oo e.p=4andq=g 4B 9. Let f(33) = E. Find f‘1(33) want to sketch your functions. a. f’1(as) : N domain: b. f‘1(3:) = N domain: c. f‘1(3:) = 1 ~ $2 domain: _1 1 . d. f (3:) = domain: 1 — a: e. f‘1(3:) = 1 — 3:2 domain: 10. According to the Intermediate Value Theorem, the function a zero on which one of the following f(:c) = 3 coszr — 2a: — 2 must have intervals? 37r 377 a. (—2—, 27F) b. (Tr, 3) e. none of these The following problems are worth 2 points each. w2+2 and its domain. Hint: consider the relationship between range and domain of inverse functions. You may 11. EValuate lim 6 . Note that $212 is the exponent. 31—) — oo '1 5B (DIN) 1—cosa: 12. Evaluate lim ——_—2———. Hint: use a trig identity to rewrite the function. 2H0+ sm :1: a. —oo b. 1 c. —00 d. 0 e. % Bonus“ State Whether each of following is true or false (one point each). 13. lim ln(2 — as) 2 —oo. a:—>2— a. True b. False 14. The graph of the function f (x) = E has a hole at a: = O. x a. True b. False 15. f (as) : , 1s an odd function. Sin :1: a. True b. False 6B MAC 2311 TEST IB PART 11 SPRING 2010 Sect # Name UFID Signature SHOW ALL WORK TO RECEIVE FULL CREDIT. 1. Let f(:c) : V350 -|— 1. a. Use limits to find the slope of the tangent line to f (2:) at (E = 1. b. Write the equation of the tangent line to f (cc) at a: = 1. 7B 2. Solve each of the following: a. Find the solution set: 2 lncr; — ln(2:c + 3) = 0 b. Let f(x) = cos 23: and let 9(513) : 2 — 3sin cc. Find each value of a: on the interval [0, 271’] at which f(a:) 2 9(23). IE :- ' 3. Evaluate the following: _1 < 4n) a. cos cos— : 3 . 2 i . . b. tan (arcsm g) : . Hint: draw a triangle. 8B Sect # ‘ Name . It can be shown that for 9 in (0, g), 6 cos 9 S sin6 S 9. . . . , sin 6 Use the Squeeze Theorem and this inequality to find ghrgir 6 _). . A farmer wishes to enclose a rectangular piece of land to create three equal garden areas as shown below using 240 yards of fencing. a. Find a function A(a:) which gives the total area enclosed in terms of 3:. Find the domain of 14(1). 4‘ \L A(a:) : domain(interval notation) b. Use your function to find the dimensions that will enclose an area of 1800 square yards. 9B 6. Consider the function f(a:) = V17 + 1 ——1 < :c < 0 (a) Sketch the graph of f (ac) (b) Find the following limits if possible. .133“;— f(r) : _ _... 1331 me) = $131+ W) =————-—— 33:23, W = 133+ f(1‘)=_.____'__. gym: (c) List all discontinuities of f (x) and state Whether they are jump, infinite or removable. (d) Find each interval on Which f (51:) is continuous. 10B ...
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