224StudyGuide - Purdue University Study Guide for MA 224...

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Unformatted text preview: Purdue University Study Guide for MA 224 for students who plan to obtain credit in MA 224 by examination. Textbook: Applied Calculus For Business, Economics, and the Social and Life Sciences, Expanded ninth edition. Author: L. D. Hoffman and G. L. Bradley Publisher: McGraW Hill, 2007 When you are ready for the examination, obtain the proper form from your academic advisor. Follow the instructions on the form. To prepare for the exam, you should obtain from the MA 224 course web page, or from the Undergraduate Services Office (MATH 242): 1. the assignment sheet 2. the final exam practice problems 3. the final exam formula pages The url for the course web page is: http:/ /www. math. purdue. edu/ courses/ma224 The assignment sheet lists the sections of the text that are covered in the course. (A copy of the text is on reserve in the Hicks Undergraduate Library.) The homework problems from the assignment sheet, along with the final exam practice problems, provide good preparation for the exam. The final exam formula pages will be attached to the final exam. A calculator with logarithmic and exponential functions will be needed for the exam. Any brand of one-line calculator may be used. However, no multi-line calculator are allowed. Cell phones and PDA’s may not be used as a calculator. MA 224 Assignment Sheet Text: Text: Applied Calculus by S.T. Tan, Sixth Edition, Brooks/Cole, 2005. A calculator with logarithm and exponential functions is required. Calculators may be used when appropriate on the assignments below. Graphing calculators or programmable calculators may not be used on quizzes or exams. Calculators which are capable of numerical or symbolic differentiation or integration are considered programmable and are not allowed on exams. Lessons Sections Assignments 1 8.1 p544: 1,6,13,15,17,19,20, 22, 23 2 8.2 p557 : 3,6, 8,9, 12, 13, 15, 16, 17, 23,30 3 8.2 p558 : 33, 37, 38,39,40, 42, 43, 47 4 8.3 p570: 1,2,3, 6, 7,8, 13, 14, 20 5 8.3 p570: 22, 23, 24, 26, 27, 29, 30 6 8.4 p578: 1,4, 7, 8, 9, 10 7 8.5 19594 : 1,2,3,4,5,6,7,8, 12 8 8.5 p594 : 17, 18, 19, 20, 21, 25 9 8.6 _ p600 : 2,5,8, 11, 14, 20,21, 23, 28 10 8.6 p600 : 31, 32, 33,34, 35, 37 11 6.1 p407 : 9,12,15,22,31,38,39,52,55,60 12 6.2 p419: 1,2,6,9,10,13,18,19, 21, 24, 28 13 6.2 p419 : 29,30,32,35, 38, 39,42,51,55 14 6364 p430: 1,2;p439 : 3,6,8,9,10,13, 14,15 15 6.5 p449 : 2,3,9,10,14,15,18,21,22 16 6.5 p449 : 24, 26, 28,31, 34, 35, 40,42 17 6.6 p462 : 4, 11, 17, 18,23,28, 30, 35,40 18 7.1 19499 : 1,2,8,10,11,12, 14, 19,21 19 7.1 p499 : 27,30, 34, 35, 36, 37,39 20 7.2 p507: 1,9, 12,14, 17, 18, 21,27, 30 21 7.3 p520 : 3, 12, 16, 19, 25, 28(Trapezoida] Rule only) 22 7.4 p531 : 15,17,18,20,21,24,25,28 ' 23 7.4 p532 :30, 31, 34, 36, 37,40, 41,45,46 24 9.1 p627: 1,3,6, 14, 15, 19,20, 23 25 9.2 p633: 2,3,5,6,8,15,17,19,20,31,34 26 9.3 p641 : 2,3,4,5,6,7 27 9.3 p642 : 9,10,11,14,15 28 10.1 p663 : 4,6,14,15,18,19,22,25,26,27 29 10.1 . p666 : 44,45,46,49,50,51 30 11.1 p716: 1,4,7,]2,13,15,22,27 31 11.2 p716 : 11,12,14,30,31,34,35,36,40,41 32 11.3 p727 : 6,7,10,11,16,17,19,32,35,36 33 11.5 p747 : 2,3,5, 6,8, 11, 14, 17 34 11.6 p756: 1,2,3,4,5 35 11.6 p756:9,11,12,15,16 36 11.6 19756 : 23,24,26, 28, 29 MA 224 Final Exam Practice Problems The Table of Integrals (pages 502-503 of the text) and the Formula Page may be used. They will be attached to the final exam. 1. If f(x,y) 2 (xy + 1)2 — My2 — x2, evaluate f(—2, 1). A. 1B. 1—\/50. NotdefinedD. —1—\/5E. —1—\/§ 2. A paint store carries two brands of latex paint. Sales figures indicate that if the first brand is sold for $1 dollars per gallon and the second for $2 dollars per gallon, the demand for the first brand will be D1(x1, x2) = 100 + 5x1 — 10$2 gallons per month and the demand for the second brand will be D2(a;1,$2) = 200 — 102:1 + 152:2 gallons per month. Express the paint store’s total monthly revenue, R, as a function of $1 and 352. A. R = $1D1(.’L'1,.’L'2) + $2D2<$1,.’L‘2) B. R = D1(.’L‘1,.’L‘2) + D2(.’L‘1,.’II2) C. R = D1($1,$2)D2($1,$2) D. R 2 x2D(:I;1, x2)+$1D2($1,$2) E. R = $1$2+D1($1, $2)D2(a;1, $2) 3. Compute %, where z = ln(xy). 1 1 1 1 1 A.—B.—C.—D.~+—E. x y xy x y 4. Compute fm, iff = M) + e“+2“ A. 0 B. u + 2eu+2v C. v + 2eu+2v D. 1 + 2eu+2v E. 1 + e“+2” 81‘: 5. Find and classify the critical points of f(:c, y) = (cc — 2)2 + 2y3 — 6y2 — 18y + 7. A. (2,3) saddle point; (2,—1) relative minimum B. (2,3) relative maximum; (2,—1) relative minimum C. (2,3) relative minimum; (2,—1) relative maximum D. (2,3) relative maximum; (2,—1) saddle point E. (2,3) relative minimum; (2,—1) saddle point 6. A manufacturer sells two brands of foot powder, brand A and brand B. When the price of A is x cents per can and the price of B is y cents per can the manufacturer sells 40 — 8x + 5y thousand cans of A and 50 + 9x — 7y thousand cans of B. The cost to produce A is 10 cents per can and the cost to produce B is 20 cents per can. Determine the selling price of brand A which will maximize the profit. A. 40 cents B. 45 cents C. 15 cents D. 50 cents E. 35 cents 82 82 7. Use the total differential to estimate the change in 2 at (1,3) if a— = 2x — 4, 5— : 2y + 7, the 37 2/ change in a; is 0.3 and the change in y is 0.5. A. 7.1 B. 2.9 C. 4.9 D. 5.9 E. 6.3 8. Using 33 worker—hours of skilled labor and y worker—hours of unskilled labor, a manufacturer can produce f (cc, y) = xzy units. Currently 16 worker—hours of skilled labor and 32 worker—hours of unskilled labor are used. If the manufacturer increases the unskilled labor by 10 worker—hours, use calculus to estimate the corresponding change that the manufacturer should make in the level of skilled labor so that the total output will remain the same. A. Reduce by 4 hours. B. Reduce by 10 hours. C. Reduce by % hours. D. Reduce by g hours. E. Reduce by 5 hours. MA 224 Final Exam Practice Problems 9. Find the maximum value of the function f (:12, y) : 20:123/ 2y subject to the constraint x+y = 60. Round your answer to the nearest integer. A. 84,654 B. 188,334 C. 4,320 D. 259,200 E. 103,680 2 10. Evaluate / (— — x/E) dar. (I: A.1n|x|—%+C B. —:—2— fill/2+0 C.21n|x|—2x3/2+C D —%—2$:/2+C swim—5%“) 11. Evaluate/ngi—lyldx. ‘fiJFC B.—Em:7+c emu) —W+C E. —W+C 12. Evaluate /e3_2$d:r. 64—23: 63—20: D_l3—-2a: . 4_ 2112+C 36 +0 E 3—23: 13. Find a function f whose tangent line has slope {Ex/5 — {1:2 for each value of :1: and whose graph passes through the point (2,10). A- M) = —%<5 — W” B f<x >= §<5 — $213” + a 0. f(x) = %<5 — x2)” + % D. f<x> = —§<-5 — $213” + 3—31 E f<x> = 3(5 — $213” + 12—7 A.—2e32$+CB—l32T+CC. +0 14. Evaluate /:1:ln(:1:2)d:1:. A. x2lnx—%x2+C B.%x21nx2—%x+C C. %x2lnx2—%x3+0 D.xlnx2+%+C E. %x21nx—%x2+C 15. The area of the region bounded by the curves y = 11:2 + 1 and y = 3:1: + 5 is 12_5 E a a _3_2 A.6B.3C.2D.6E.3 2 16. Find the average value of f (:12) = :1: over the interval 1 g :1: g 4. Ag 3% 0.21 D.%5 E? 17. A calculator manufacturer expects that :1: months from now consumers will be buying 1000 calculators a month at a price of 20 + 3\/E dollars per calculator. What is the total revenue the manufacturer can expect from the sale of calculators over the next 4 months? A. $8,000 B. $16,000 C. $96,000 D. $192,000 E. $116,000 (1 18. The general solution of the differential equation d—y_ — 2y + 1 is: :1: A.:1:=y2 +y+C B. 2y+1=Ce2$ C.y=2:1:y+;1;+C D.y=Ce2$—2y—1 E.y=Ce2$ MA 224 Final Exam Practice Problems 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. The value, V, of a certain $1500 IRA account grows at a rate equal to 13.5% of its value. Its value after t years is: _ A. V = 1500f"-135t B. V = 1500 + 0.135t C. V = 1500e0-135t D. V = 1500(1 + 0.135t) E. V = 15001n(0.135t) Evaluate' /d_x ' x/9x2—4 A.ln|3:1:+\/9x2—-4|+C B.%In|x+Vx2—4|+C C.%ln|x+\/:B2+(4/9)|+C D. §ln|x+x/x2—(4/9)|+C E.ln|:1:+‘/a:2—(4/9)I+C. 2 Evaluate / xln(x2)dx. Round your answer to two decimal places. 1 A. 1.27 B.—0.30 C. 2.27 D. 0.77 E. 1.77 It is estimated that t years from now the population of a certain town will be increasing at a rate of 5 + 37.2/3 hundred people per year. If the population is presently 100,000, by how many people will be population increase over the next 8 years? A. 100 B. 9,760 O. 6,260 D. 24,760 E. 17,260 The probability density function for the life span of light bulbs manufactured by a certain company is f (:12) = 0.01e‘0‘01$ where :1: denotes the life span in hours of a randomly selected bulb. What is the probability that the life span of a randomly selected bulb is less than or equal to 10 hours? Round your answer to three decimal places. A. 0.009 B. 0.095 C. 0.905 D. 0.090 E. 0.303 Calculate the following improper integral /' xe‘z2da: 0 A. —% B. 1 C. % D. g E. The integral diverges. An object moves so that its velocity after t minutes is given by the formula 72 = 206‘0'0”. The distance it travels during the 10th minute is 10 10 10 A. / 20e“°'°“dt B. / (—20e—0-0“)dt C. / (—20e-0-01t)dt 0 9 0 10 10 D. / 20e"°'°“dt E. / (—0.2e-0-0“)dt 9 9 Find the value of k such that f (:12) : k(3 — a2) is a probability density function on the interval [07 3]- A.k=%B.k=—§C.k=—§D.k=§ E.k=% Records indicate that t hours past midnight, the temperature at the West Lafayette airport was f (t) = —0.3t2 +4t + 10 degrees Celsius. What was the average temperature at the airport between 2:00 A.M. and 7:00 A.M.? Round your answer to the nearest degree. A. 3° B. 27° C. 21° D. 5° E. 18° 3 Let f (:12) be the probability density function on the interval [1,00) defined by f (:12) = :B—4. Calculate P(:1: 2 2). A1133- C.%D.% E.§ MA 224 Final Exam Practice Problems 29. 30. 31. 32. 33. 34. 35. 1 Approximate / emzdx using the trapezoidal rule with n = 4. Round your answer to two 0 decimal places. A. 1.49 B. 2.98 C. 5.96 D. 1.73 E. 1.96 The slope of the least—squares line for the points (1,2), (2,4), (4,4), (5,2) is I A.0 B.1C.2 D.3 E.4 oo 2 n Find the sum of the series 2 (—3) 71:1 B. —§ C. % D. —% E. The series diverges. A. (film Use a Taylor polynomial of degree 2 to approximate v3.8. Round your answer to five decimal places. A. 1.94936 B. 1.94938 C. 1.94940 D. 1.94947 E. 1.95000 100 m2 + 1 0.1 Use a Taylor polynomial of degree 2 to approximate / dac. Round your answer to 0 five decimal places. A. 9.96687 B. 10.00000 C. 9.96677 D. 9.66667 E. 9.96667 Find the radius of convergence of the power series :0: ”5171f:- A.§B.1C.%D.§E.oo ”:0 Find the Taylor series of f(x) = 2:12 at at = 0. A. ii: 3 i—“liiw a BM D. °° 2:2: E.i—(*1;::f"“ 71:0 n=0 n=0 n=0 n=0 Answers 1. C; 2. A; 3. A; 4. D; 5. E; 6. A; 7. D; 8. D; 9. E; 10. C; 11. B; 12. B; 13. D; 14. A; 15. A; 16. E; 17. C; 18. B; 19. C; 20. D; 21. A; 22. B; 23. B; 24. C; 25. D; 26. D; 27. C; 28. E; 29. A; 30. A; 31. B; 32. B; 33. E; 34. A; 35. E MA 224 FORMULAS THE SECOND DERIVATIVE TEST Suppose f is a function of two variables :1; and y, and that all the second- order partial derivatives are continuous. Let D = fzzfyy _ (fury)2 and suppose (a, b) is a critical point of f. 1. If D(a, b) < 0, then f has a saddle point at (a, b), 2. If D(a, b) > 0 and fm(a, b) < 0, then f has a relative maximum at (a, b). 3. If D(a, b) > O and fm(a, b) > 0, then f has a relative minimum at (a, b) 4. If D(a ,b)— — O, the test is inconclusive. LEAST-SQUARES LINE The equation of the least-squares line for the n points (x1,y1), (x2,y2), , (men), is y = ma: + b, where m and b are solutions to the system of equations (x§+x§+---+x§,)m+(x1+$2p+---+xn)b=x1y1+x2y2+~-+xnyn (x1+x2+---+$n)m+nb=y1+y2+-'-+yn TRAPEZOIDAL RULE / m 2A2$[f(xo) +21%) +2f<x2>+ ~+ wen—1) + f(%) , b—a n . where a = x0, x1, x2, . . . , zn = b subdivides [(1, b] into n equal subintervals of length Ax = ERROR ESTIMATE FOR THE TRAPEZOIDAL RULE If M is the maximum value of | f " (m)[ on the interval a S :1: S b, then M(b——a)3 E < '"l- 12n2 GEOMETRIC SERIES IfO < Irl < 1, then TAYLOR SERIES The Taylor series of f (x) about a: = a is the power series f (”(0) 2! (:2:—a)2+... 0° (11),, 2f ()(r—a)”=f(a)+f’(a)($-a)+ Examples: (with a = 0) 00 n 00 +1 _1n 63:5 %,for—oo<x<oo; lnx=E ;($—1)",for0<x32 Forms Involving a + bu u du 1 - = — — + + 1 Ja+bu b2 [a+bu alnla bul] C u2 du 2 2 2' Ia + bu 22—b3 [(0 + bu)— 4a(a + bu) + 2a lnla + bul] + C udu 3 I(a +bu)2 4- JuVa+budu= =b_1[a+abu+mla+b“I]+C (3bu — 2a)(a + bur!2 + c 152b1 5- — V + + IVE: 3720’“ 2“) a b" C du 1 a+bu— _ 6- —=— +C 1fa>0 Iu a+bu Win Va+bu+—_—\/: ( ) 7- J-Val+uzdu = 8. [nix/(11+ uzdu=§—(az+2u1)\/az+u'2—§ ln|u+ Va2+u1| + C Forms Involving Va2 + uz ' Z gVaz+uz+a§ Inlu+Va2+u1|+C du 9. I——,m=lnlu+ Va2+uz|+C 10 IL: __1nV—a2+u +a +C . u (12+uz a u Vaz+uZ =———+C du 11. ——————— . J.ui Vaz + u2 ' a2u du u WWW—“Tm” Forms Involving m 2 lam—mgr —« gmwmw 14 I” V“ “2"“: §(2uz~aZ)W——lnlu+m|+c 15.]? 1° N: du= ——' “I: +1n|u‘+ W|+C =In|u+Vuz —a2|+C y/ul _. a2 du . ——___= + 17 juz uz—az “I” C __d" _ _i“__ “kw—mm— WW2” Forms Involving Vaz —- u2 du = + 28. [ulna 1n|1nu| C_ 29. Ifln u)" du = u(ln u)" — "I (in u)"l du \/ _ 2 ‘/ _ 2 D.J#du=\/_az—uz—aln£%—u— +C _ Z m.Ii_:_l,nL_ Wu +C u a2—-u2 a u ‘1 1— 2 21' L:_u+c u2V¢12—uz 07'“ du u _22. [(az_uz)m~——al ”_a1—u?+C Forms Involving e“andlnu m 1 23.Iue du=;(au-1)e‘"+C 24. Iu"e“"du = lu"e"“ — EIu‘He‘" du a a du 1 _25. 1+be-""" Zln(l+be"‘)+C 26.]1nudu=u1nu—u+c_ 27 "In d un+l .Ju u u—(n+1)z[(n+l)lnu—1]+C (nsé-l) ...
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