Class Questios 007 BC Calculus Istitute Questios for 007 BC Calculus Istitutes CALCULATOR. How may zeros does the fuctio f ( x) si ( l ( x) ) Explai how you kow. = have i the iterval (0,]? LIMITS. 00 Released Exam Limit Questios, 6, 79. l lim x x 5 ( x ) 0.0 = 4. How may times do the graphs of y = x ad y 4 = x itersect? 5. 998 AB DERIVATIVES si( + h) si( ) = h π π 6. limit 7 7 h 0 (A) 0 (B) 0.44 (C) 0.90 (D).0 (E).70 7. π ( + h ) cos ) limit = h 0 h Li McMulli, 006. 007 All Rights Reserved. of 6
(A) Noexistet (B) (C) 0 (D) h (E) 8. 00 Released Exam, 9. 00 Released Exam Derivative theory ad MVT a. AB:, 6, 80, b. BC: 6, 9, 7,8,9 ad c. 005 AB (d) a very formal use of the MVT but oe your studets should see. 0. 00 Released Exam Computig Derivatives: a. Basic rules AB:, 4, 9, 4, BC:, 9, 7 b. Iverses AB: 7, BC: 7, c. FTC: AB, 9, BC: 8 d. Implicit: AB 6, also 004 AB4/BC4 e. BC, 9,, 79 (compare with 007 AB free-respose where x = 0.95 out of 9 poits). TAPC /e p. 7 7 Table ad see 007 AB ( x = 0.95 out of 9 poits. See 00 BC 79); 006 AB 6. What is the value of the derivative of the iverse of the fuctio (0,0)? x x y e e = at Li McMulli, 006. 007 All Rights Reserved. of 6
DERIVATIVE APPPLICATIONS. 995 BC 5 (Suitable for AB) Let f (x) = x, g(x) = cosx, ad h(x) = x + cos x. From the graphs of f ad g show above i Fig. ad Fig., oe might thik the graph of h should look like the graph i Fig.. (a)sketch the actual graph of h i the viewig widow provided below, ( 6 x 6 ad 6 y 40). (b)use h (x) to explai why the graph of h does ot look like the graph i Fig. (c)prove that the graph of y = x + cos(kx) has either o poits of iflectio or ifiitely may poits of iflectio, depedig o the value of the costat k. 4. Cosider the fuctio f ( x) = x x 4x + k (a) Fid the x-coordiate of the fuctio s relative maximum. Justify your aswer. (b) Fid the x-coordiate of the fuctio s relative miimum. Justify your aswer. (c) Fid the x-coordiate of the fuctio s poit of iflectio. (d) Give that f has exactly real roots, fid both possible values of k. 5. 007 Form B AB 4 ad may others. See Type Questio hadout. Li McMulli, 006. 007 All Rights Reserved. of 6
6. The figure shows a small sectio of the graph of the derivative of a fuctio. Which of the choices best describes the correspodig part of the graph of the fuctio? a. Decreasig ad cocave up oly b. Decreases ad chages from cocave dow o the left to up o the right. c. Decreases ad does ot chage cocavity d. Icreasig ad cocave up e. Icreasig ad chages from cocave up o the left to dow o the right. 7. 00 Released Exam BC, 78, 8, 86, 87, 90, 9 OPTIMIZATION 8. A wire feet log is cut ad formed ito a square ad a circle. Where should the wire be cut so that the total area of the square ad a circle is a maximum? 9. 98 AB 6: A tak with a rectagular bottom ad rectagular sides is to be ope at the top. It is costructed so its width is 4 meters ad its volume is 6 cubic meters. If buildig the tak costs $0 per square meter for the bottom ad $5 per square meter for the sides, what is the cost of the least expesive tak? INTEGRATION 0. Exploratio : A tak is beig filled with water usig a pump that is old, ad slows dow as it rus. The table below gives the rate at which the pump pumps at te-miute itervals. If the tak is iitially empty, approximate how much water is i the tak after 90 miutes? Elapsed time (Miutes) Rate (gallos / miute) 0 0 0 0 40 50 60 70 80 90 4 40 8 5 5 8 0 9 0 See otes i TAPC /e p. 0-04 Li McMulli, 006. 007 All Rights Reserved. 4 of 6
Exploratio : The speed of a airplae i miles per hour is give at half-hour itervals i the table below. Approximately, how far does the airplae travel i the three hours give i the table? How far is it from the airport? Elapsed time (miutes) Speed (miles per hour) 0 0 60 90 0 50 80 75 90 400 90 85 50 45 Riema sums See otes i TAPC /e p. 0-04. Approximate 4 + x dx usig () a left Riema sum with 6 equal subdivisios ad () a right Riema sum with 6 equal subdivisios.. 00 Released Exam Riema sums AB 85; BC 8, 5, 85, 88;. Let T be ay Trapezoidal rule approximatio to is true? b S = x x dx a. Which statemet I. If a < b < 0, the T > S. II. If a < 0 < b, the T = S. III. If 0 < a < b, the T < S. (A) I oly II. II oly (C) III oly (D) I ad III oly (E) I, II ad III 4. Let f be a cotiuous fuctio defied for all x such that f ( x) possible value for ay Riema sum for f o the iterval [,5] is (A) (B) 7 (C) (D) 6 (E) 8 7. The largest 5. (997 AB 4) The expressio 50 + + + + is a Riema 50 50 50 50 50 Sum for (A) x dx 0 50 (B) x dx (C) 0 x dx 50 50 0 Li McMulli, 006. 007 All Rights Reserved. 5 of 6
(D) 50 x dx (E) 0 50 50 0 x dx 6. limit k = e k = 7. limit k + = k = 8. If the closed iterval [0, b] is divided ito equal parts each of legth b, the ( ) b f x dx = 0 ( )( ) b I. f ( b ) II. limit b f ( k ) III. f ( b) f ( 0) k = (A) I oly (B) II oly (C) III oly (D) I ad III oly (E) II ad III oly. 9. If t is measured i hours ad f ( t) ( ) f t dt? (Note: kot = autical mile per hour) 0 is measured i kots, what is the value of (A) f ( ) kots (B) f ( ) f ( 0) kots (C) ( ) (D) f ( ) f ( 0) autical miles (E) f ( ) f ( 0) 0. 998 AB #88: Let ( ) F ( 9) = F x be a atiderivative of ( l x ) x f autical miles kots per hour.. If F ( ) = 0, the (A) 0.048 (B) 0.44 (C) 5.87 (D).08 (E),640.50 Li McMulli, 006. 007 All Rights Reserved. 6 of 6
. The table below gives the velocity i the vertical directio of a rider o a Ferris wheel at a amusemet park. The rider moves smoothly ad the table gives the values for oe complete revolutio of the wheel. (This is similar to 998 AB ) a. Durig what iterval of time is the acceleratio egative? Give a reaso for your aswer. b. What is the average acceleratio durig the first 5 secods of the ride? Iclude uits of measure. c. Approximate 0 v( t) dt usig a Riema Sum with six itervals of 0 equal legth. d. Approximate the diameter of the Ferris Wheel. Explai your reasoig. t secods v feet/secod 0 0 5.6 0.7 5. 0.7 5.6 0 0 5 -.6 40 -.7 45 -. 50 -.7 55 -.6 60 0 PARAMETRIC AND POLAR EQUATIONS. From the 00 Released Exam BC 4, 7, 5, 7, 84 FR 00 BC, 007 BC Li McMulli, 006. 007 All Rights Reserved. 7 of 6
ACCUMULATION. Ivestigatio : a. O the axes provided graph f ( t ) =. Let [0, x ], be a iterval o the t- axis. Write the equatio of the fuctio A ( ) x that gives the area of the regio i the first quadrat uder the graph of y = f ( t), above the t- axis, betwee t = 0 ad t = x. Idicate where this regio appears o the graph by shadig a typical regio ad idicatig where x is. f ( t) f ( t) t b. O the axes provided graph f ( t) = t. Let [0, x ], be a iterval o the t- axis. Write the equatio of the fuctio A ( ) x that gives the area of the regio i the first quadrat uder the graph of y = f ( t), above the t- axis, betwee t = 0 ad t = x. Idicate where this regio appears o the graph by shadig a typical regio ad idicatig where x is. f ( t) t c. O the axes provided graph f t = t +. Let [0, x ], be a ( ) iterval o the t-axis. Write the equatio of the fuctio A ( x ) that gives the area of the regio i the first quadrat uder the graph of y = f ( t), above the t-axis, betwee t = 0 ad t = x. Idicate where this regio appears o the graph by shadig a typical regio ad idicatig where x is. t Li McMulli, 006. 007 All Rights Reserved. 8 of 6
d. Fill i the table for these fuctios x 0 4 5 A ( ) A ( x ) A ( x ) Do these umbers agree with your idea of area? Why does A = A+ A? Show graphically why this is true. e. Fill i the table for these values: x A ( ) A ( x ) A ( x ) Explai your reasoig; specifically tell how does this relates to the area? f. Calculate: da ( x) da ( x) da ( x) = ; = ; = dx dx dx What do you observe about the derivatives? Why do you thik this is? g. Cosider a ew fuctio ( ) 4 A x that gives the area uder y = t + o the da4 ( x) iterval [, x ]. Complete the table below ad fid dx. Why does da4 ( x) da ( x) =? dx dx x 0 4 5 A ( ) 4 See TAPC /e p. 6 9 Li McMulli, 006. 007 All Rights Reserved. 9 of 6
4. O the iterval [ 0, π ] which fuctio has a average value that is ot 0? I. cos( x ) II. si ( x) π III. π x 5. Let f ad g be cotiuous fuctios with f ( x) g ( x) =. Which statemet is true? I. O the iterval [0, 0] the average value of f is 0 more tha the average value of g. II. O the iterval [0,0] the average value of g is less tha the average value of f. 6 6 III. ( ) ( ) f x dx g x dx = 5 5 (A) I oly (B) II oly (C) III oly (D) I ad III oly. (E) II ad III oly 6. 00 Released Exam Applicatios of itegrals AB 8, 84, 86, 88; BC 5, 80, 8, 88, 89. 7. 00 Released Exam: Methods of itegratio AB, 5, 8, ad BC, 8,, 6 DIFFERENTIAL EQUATIONS 8. Slope Fields from past exams; 998 BC mc:4 ad BC4, 000 BC6, 00 BC 5, 00 BC mc:4, 004 AB 6, form B AB5, 005 AB6, BC4, 006 AB5, 9. Other BC Differetial Equatio Questios (icludig Euler s Method) 00 Released Exam 5, 4,, 80. 40. 005AB6(c) Give dy dx x y =, fid the particular solutio y f ( x) differetial equatio with the iitial coditio f ( ) = to the give = (Part (a) was draw a slope field, ad part (b) approximate f (.) with taget lie at (, ). ) Li McMulli, 006. 007 All Rights Reserved. 0 of 6
4. 00 AB 5 4. Cosider the differetial equatio dy y y = for all x 0. dx x x (a) Verify that y =, x C is a geeral solutio for the give x + C differetial equatio. (b) Write a equatio of the particular solutio that cotais the poit (, ) ad fid the value of dy dx at (0,0) for this solutio. (c) Write a equatio of the vertical ad horizotal asymptotes of the particular solutio foud i (b). Cotiued ext page Li McMulli, 006. 007 All Rights Reserved. of 6
(d) The slope field for the give differetial equatio is provided. Sketch the particular solutio that passes through the poit (, ). 4 4 4 5 4 (Note: This is a good problem for Wiplot. Graph the geeral solutio ad use a slider for C. Notice the slope at (0,0) is idetermiate, but each solutio has a slope there. Also ivestigate the vertical ad horizotal asymptotes ad the solutio curve whe C is close to 0.) 4. The slope field for y ' = x y is show below. Which graph could be a solutio of the differetial equatio show? Cotiued ext page Li McMulli, 006. 007 All Rights Reserved. of 6
(A) 5 (B) 5 4 4 4 4 5 4 4 5 (C) (D) (E) 5 4 4 5 5 4 4 5 5 4 4 4 5 4 4 5 4 4 5 4 4 4 5 5 5 dy 44. The slope field for a differetial equatio = f ( y) is show i the figure above. dx Which statemet is true about y( x )? I. If y(0) > the limit y( x) x II. If 0 < y(0) < the limit y( x) x III. If y(0) < the limit y( x) x (A) I oly (B) II oly (C) III oly (D) I ad II oly (E) I, II ad III dy + y 45. 006 AB 5(b): Cosider the differetial equatio = where x 0. Fid the dx x = to the differetial equatio with the iitial equatio particular solutio y f ( x) f ( ) = ad state its domai. For more o the domai of the solutio of a differetial equatio see the articles by L. Riddle ad D. Loma i the Articles File of the Participats File 007. Li McMulli, 006. 007 All Rights Reserved. of 6
Powers Series Questios. Write the first four ozero terms i the Maclauri series for x xe.. π ( ) (A) k = 0 π k = π (B) π (C) + π (D) π + π (E) The series does ot coverge. x. Let E be the error whe the Taylor polyomial T ( x) = x is used to approximate! f x = x at x = 0.5. Which of the followig is true? ( ) si ( ) (A) E < 0.000 (B) 0.000 < E < 0.000 (C) 0.000 < E < 0.005 (D) 0.005 < E < 0.007 (E) 0.007 < E 4. The Taylor series of a fuctio f(x) about x = is give by ( x ) ( x ) ( + )( x ) 5 7 f ( x) = + ( x ) + + + +!!! What is the value of ( 7 ) ( ) f () ad f? 4 x x x 5. What are all values of x for which the series x + + coverges? 4 (A) x (B) x < (C) < x (D) < x < (E) All real umbers x. 6. k = ( ) ( π ) ( )! = 7. Let f(x) be the fuctio defied by the power series ( ) = ( ) ad ( 0) = the ( ) g x f x g g x = f ( x) = x. If k = 0 Li McMulli, 006. 007 All Rights Reserved. 4 of 6
8. Let f(x) be a fuctio with the followig properties: (i) f ( 0) = (ii) f ( x) = f ( x) (iii) The th derivative of f, ( ) ( ( ) = ) ( ) f x f x (a) Give the first four ozero terms ad the geeral term of the Maclauri series for f. (b) Fid f(x) by solvig the differetial equatio i (ii) with the iitial coditio i (i). (c) Graph ad label both f ad the third degree Maclauri polyomial of f o the axes below ad label each. [Widow is [,] by [ 40, 0]) Power Series Aswers:. x x + x x + ;. (E);. B; 4. 7, 5; 5. C; 5! 4! 6. cos( π ) = 7. 8. (a) x + the is the costat of itegratio. k = x 6x 9x 9x!, (b) f ( x) e = x, (c) below From the 00 Released Exam Covergece tests BC 6, 0,, 4 Series BC, 0, 8, 77 Li McMulli, 006. 007 All Rights Reserved. 5 of 6
Mathematics ad Calculus Related Web Sites: My Web site: www.limcmulli.et ad E-mail: lmcmulli@aol.com College Board AP Cetral http://apcetral.collegeboard.com NCAAPMT Newsletter: Sed $5 to Jeff Lucia NCAAPMT Treasurer, 78 Lasdowe Road, Charlotte, NC 870 (luciaj@pds.charlotte.c.us). Two issues late summer, early sprig. Or o-lie at www.cctt.org/ncaapmt/ THE BEST $5 YOU LL SPEND! Wiplot Wiplot http://math.exeter.edu/rparris/default.html Istructios are at http://matcmadiso.edu/alehe/wiptut/wipltut.htm Best graphig program aroud. FREE. Have your studets dowload it ad use it too. Geeral Math Resources NCTM Homepage http://www.ctm.org/ Math Forum Iteret Collectio http://mathforum.org/ Calculators ad TI Texas Istrumets: ad http://educatio.ti.com/ Calculus i Motio for Geometers Sketchpad www.calculusimotio.com ad for Algebra i Motio D&S Review Books (Calculus, New York A ad B Exams) ad Teachig AP Calculus (/e) www.dsmarketig.com O adaptig free-respose questios by Dixie Ross http://apcetral.collegeboard.com/members/article/,046,5-65-0-994,00.html O assessmet by Da Keedy http://baylor.chattaooga.et/%7edkeedy/assessmet ad other stuff http://baylor.chattaooga.et/%7edkeedy/home Li McMulli, 006. 007 All Rights Reserved. 6 of 6