(A) 0 (B) (C) (D) (E) 2.703

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1 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? 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) π ( + h ) cos ) limit = h 0 h Li McMulli, All Rights Reserved. of 6

2 (A) Noexistet (B) (C) 0 (D) h (E) Released Exam, 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 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, All Rights Reserved. of 6

3 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 Form B AB 4 ad may others. See Type Questio hadout. Li McMulli, All Rights Reserved. of 6

4 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 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? 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) See otes i TAPC /e p Li McMulli, All Rights Reserved. 4 of 6

5 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) Riema sums See otes i TAPC /e p 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 is a Riema Sum for (A) x dx 0 50 (B) x dx (C) 0 x dx Li McMulli, All Rights Reserved. 5 of 6

6 (D) 50 x dx (E) 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) 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) (B) 0.44 (C) 5.87 (D).08 (E), Li McMulli, All Rights Reserved. 6 of 6

7 . 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 PARAMETRIC AND POLAR EQUATIONS. From the 00 Released Exam BC 4, 7, 5, 7, 84 FR 00 BC, 007 BC Li McMulli, All Rights Reserved. 7 of 6

8 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, All Rights Reserved. 8 of 6

9 d. Fill i the table for these fuctios x 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 A ( ) 4 See TAPC /e p. 6 9 Li McMulli, All Rights Reserved. 9 of 6

10 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 Released Exam Applicatios of itegrals AB 8, 84, 86, 88; BC 5, 80, 8, 88, 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,, AB6(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, All Rights Reserved. 0 of 6

11 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, All Rights Reserved. of 6

12 (d) The slope field for the give differetial equatio is provided. Sketch the particular solutio that passes through the poit (, ) (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, All Rights Reserved. of 6

13 (A) 5 (B) (C) (D) (E) 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 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, All Rights Reserved. of 6

14 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 < (B) < E < (C) < E < (D) < E < (E) < 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, All Rights Reserved. 4 of 6

15 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 ) = (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, All Rights Reserved. 5 of 6

16 Mathematics ad Calculus Related Web Sites: My Web site: ad College Board AP Cetral NCAAPMT Newsletter: Sed $5 to Jeff Lucia NCAAPMT Treasurer, 78 Lasdowe Road, Charlotte, NC 870 Two issues late summer, early sprig. Or o-lie at THE BEST $5 YOU LL SPEND! Wiplot Wiplot Istructios are at Best graphig program aroud. FREE. Have your studets dowload it ad use it too. Geeral Math Resources NCTM Homepage Math Forum Iteret Collectio Calculators ad TI Texas Istrumets: ad Calculus i Motio for Geometers Sketchpad ad for Algebra i Motio D&S Review Books (Calculus, New York A ad B Exams) ad Teachig AP Calculus (/e) O adaptig free-respose questios by Dixie Ross O assessmet by Da Keedy ad other stuff Li McMulli, All Rights Reserved. 6 of 6

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig: