AP Calculus AB AP Calculus BC

Transcription

1 AP Calculus AB AP Calculus BC Free-Response Questions and Solutions Copyright 3 College Entrance Eamination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Vertical Teams, APCD, Pacesetter, Pre-AP, SAT, Student Search Service, and the acorn logo are registered trademarks of the College Entrance Eamination Board. AP Central is a trademark owned by the College Entrance Eamination Board. PSAT/NMSQT is a registered trademark jointly owned by the College Entrance Eamination Board and the National Merit Scholarship Corporation. Educational Testing Service and ETS are registered trademarks of Educational Testing Service. Other products and services may be trademarks of their respective owners. For the College Board s online home for AP professionals, visit AP Central at apcentral.collegeboard.com.

2 989 BC Let f be a function such that f () = (a) Find f () if the graph of f is tangent to the line 3 y = at the point (, ). (b) Find the average value of f () on the closed interval [,]. Copyright 3 by College Entrance Eamination Board. All rights reserved.

3 989 BC Solution f = C (a) ( ) f ( ) = 3 C = 3 ( ) 3 f d = ( ) = d 3 f = (b) ( ) 3 ( ) d 4 3 = = = 3 Copyright 3 by College Entrance Eamination Board. All rights reserved.

4 989 BC Let R be the region enclosed by the graph of y = (a) Find the area of R., the line =, and the -ais. + (b) Find the volume of the solid generated when R is rotated about the y-ais. Copyright 3 by College Entrance Eamination Board. All rights reserved.

5 989 BC Solution (a) Area = d + = d + = arctan π = 4 (b) Volume = π d + = π d + π ln = + ( ln) = π or / y Volume = π dy y ( y ln y ) = π + ( ln) = π / Copyright 3 by College Entrance Eamination Board. All rights reserved.

6 989 BC3 Consider the function f defined by f () = e cos with domain [, π ]. (a) Find the absolute maimum and minimum values of f (). (b) Find the intervals on which f is increasing. (c) Find the -coordinate of each point of inflection of the graph of f. Copyright 3 by College Entrance Eamination Board. All rights reserved.

7 989 BC3 Solution f = e sin + e cos (a) ( ) [ cos sin ] e = π 5π f ( ) = when sin = cos, =, 4 4 π 4 5π 4 π f e ( ) π e π /4 e 5 π /4 π Ma: e ; Min: e 5π /4 (b) f ( ) + π 5π π Increasing on π 5π,,, π 4 4 f = e sin cos + e cos sin (c) ( ) [ ] [ ( ) = e sin f = when =, π, π Point of inflection at = π ] Copyright 3 by College Entrance Eamination Board. All rights reserved.

8 989 BC4 Consider the curve given by the parametric equations 3 3 = t 3 t and y = t t (a) In terms of t, find dy d. (b) Write an equation for the line tangent to the curve at the point where t =. (c) Find the - and y-coordinates for each critical point on the curve and identify each point as having a vertical or horizontal tangent. Copyright 3 by College Entrance Eamination Board. All rights reserved.

9 989 BC4 Solution dy (a) = 3t dt d = 6t 6t dt dy 3t t 4 = = = d 6t 6t t t t t (b) = 5, y = dy 3 = d 4 3 y = or 3 9 y = y+ 3= 9 ( ) ( t+ )( t ( ) ) (c) t ( y), type ( ) ( ) ( ) ( ) 8,6 horizontal, vertical, vertical 4, 6 horizontal Copyright 3 by College Entrance Eamination Board. All rights reserved.

10 989 BC5 At any time t, the velocity of a particle traveling along the -ais is given by the d differential equation dt = 6e4 t. (a) Find the general solution (t) for the position of the particle. (b) If the position of the particle at time t = is = 8, find the particular solution (t) for the position of the particle. (c) Use the particular solution from part (b) to find the time at which the particle is at rest. Copyright 3 by College Entrance Eamination Board. All rights reserved.

11 989 BC5 Solution (a) Integrating Factor: e dt = e t d 4 ( e t t t ) 6 e e = dt = + or h p () () t 6t e e C () t = Ce = Ae t = e + Ce 4t A = t Ae Ae = e 4t 4t 4t 4 6 t = Ce e t 4t 4t t (b) (c) 8= C ; C = () t = e e t 4t d = e 4e dt e e = t 4t t 4t 4 t = or ln 6 d 4t t ( e + e ) = 6e dt 4t t 4t + e e = 6e t = ln 6 4t Copyright 3 by College Entrance Eamination Board. All rights reserved.

12 989 BC6 Let f be a function that is everywhere differentiable and that has the following properties. (i) f ( + h) = f () + f (h) f ( ) + f ( h) (ii) f () > for all real numbers. (iii) f () =. for all real numbers h and. (a) Find the value of f (). (b) Show that f ( ) = f () for all real numbers. (c) Using part (b), show that f ( + h) = f () f (h) for all real numbers h and. (d) Use the definition of the derivative to find f ( ) in terms of f( ). Copyright 3 by College Entrance Eamination Board. All rights reserved.

13 989 BC6 Solution (a) (b) Let = h= f ( ) f ( ) Let h = ( ) + f ( ) ( ) + f ( ) f = + = = f ( ) ( ) f ( ) + f ( ) ( ) + ( ) f + = f = f f ( ) ( ) Use f = and solve for f = f ( ) or Note that f( + ) = f ( ) + f() f( ) + f() is the reciprocal of f(). + = (c) f ( h) = = ( ) + ( ) f f h + f f h ( ) ( ) ( ) + ( ) ( ) + ( ) ( ) ( ) f f h f f h f h f f f h ( ) ( ) (d) f ( ) ( + ) ( ) f h f = lim h h f f h f = lim h h f ( h) = f ( ) lim h h = f f = f ( ) ( ) ( ) ( ) ( ) ( ) Copyright 3 by College Entrance Eamination Board. All rights reserved.

14 99 BC A particle starts at time t = and moves along the -ais so that its position at any time t is given by (t) = (t ) 3 (t 3). (a) Find the velocity of the particle at any time t. (b) For what values of t is the velocity of the particle less than zero? (c) Find the value of t when the particle is moving and the acceleration is zero.

15 99 BC Solution (a) v t () = () t ( t ) ( t ) ( t ) 3 = ( t ) ( 8t ) = (b) v() t ( t )( t ) < when 8 < Therefore 8t < and t or t < and t 8 Since t, answer is t <, ecept t = 8 (c) a() t = v () t ( t )( t ) ( t ) ( t )( t ) = = a() t = when t =, t = 4 5 but particle not moving at t = so t = 4

16 99 BC Let R be the region in the y-plane between the graphs of y = e and y = e = to =. from (a) (b) Find the volume of the solid generated when R is revolved about the -ais. Find the volume of the solid generated when R is revolved about the y-ais.

17 99 BC Solution (a) V = π ( e e ) = π e + e π e e π e e 4 4 = = + d (b) V = π e e d ( ) = π e + e e + e d ( ) ( ) = π e + e e e ( e e ) ( e e ) ( ) = π + = π e + 3e

18 99 BC3 Let f () = for and f (). (a) The line tangent to the graph of f at the point ( k, f( k)) intercepts the -ais at = 4. What is the value of k? (b) An isosceles triangle whose base is the interval from (,) to (,) c has its verte on the graph of f. For what value of c does the triangle have maimum area? Justify your answer.

19 99 BC3 Solution (a) ( ) ( ) f = ; f = slope of tangent line at ( k f ( k) ), = k ( ) ( ) ( ) k f ( k) line through 4, &, has slope f k k = k 4 k 4 k k 4 k = or k = 6 but f 6 = 4 so k = k 8k+ = ( ) so 6 is not in the domain. k = (b) c c A= c f = c 4 3 c = 6c on,4 3 8 da 3c 3c = 6 ; 6 = when c=4. dc 8 8 Candidate test c A Ma First derivative A second derivative d A = 3 < so c = 4 gives a relative ma. dc c= 4 c = 4 is the only critical value in the domain interval, therefore maimum

20 99 BC4 Let R be the region inside the graph of the polar curve r = and outside the graph of the polar curve r = ( sinθ). (a) Sketch the two polar curves in the y-plane provided below and shade the region R. y (b) Find the area of R.

21 99 BC4 Solution (a) y y O 5 R 5 (b) π A= π ( ( θ) ) sin ( ) = sinθ sin θ dθ π ( ) = 4 sinθ dθ cosθ dθ π = 4cosθ θ sinθ ( ) ( ) [ π ] = = 8 π π dθ π

22 99 BC5 Let f be the function defined by f () =. (a) Write the first four terms and the general term of the Taylor series epansion of f () about. = (b) Use the result from part (a) to find the first four terms and the general term of the series epansion about = for ln. (c) Use the series in part (b) to compute a number that differs from ln 3.5. Justify your answer. by less than

23 99 BC5 Solution (a) Taylor approach Geometric Approach f ( ) = ( ) ( ) f = = ( ) = + ( ) 3 n 3 f = u+ u u + + u + f ( ) = ( ) = ; =! where u = 4 f ( ) f ( ) = 6( ) = 6; = 3! 3 n n Therefore = ( ) + ( ) ( ) + + ( ) ( ) + (b) Antidifferentiates series in (a): n n+ ( ) ( ) 3 4 ln = C+ ( ) + ( ) ( ) n + = ln C = ( ) n Note: If C, first 4 terms need not include ( ) 4 4 (c) ln = ln = + 3 = since <, =.375 is sufficient. 4 8 Justification: Since series is alternating, with terms convergent to and decreasing in absolute value, the truncation error is less than the first omitted term. n+ 5 Rn =, where C n+ < < Alternate Justification: C n + ( ) < n n + + < when n

24 99 BC6 Let f and g be continuous functions with the following properties. (i) g ( ) = A f( ) where Ais a constant (ii) (iii) 3 f ( d ) = gd ( ) 3 f ( d ) = 3A (a) Find 3 f ( d ) in terms of A. (b) Find the average value of g() in terms of A, over the interval [, 3]. (c) Find the value of k if f ( + ) d = k A.

25 99 BC6 Solution 3 3 (a) ( ) = ( ) + ( ) f d f d f d ( ) ( ) 3 3 = g d + f d ( ( )) ( ) 3 3 = A f d+ f d ( ) ( ) 3 3 = A f d+ f d = A 3 3 (b) A verage value = ( ) = ( ( )) (c) = ( + ) = ( ) g d A f d 3 = A f ( ) d = [ A A] = A ka f d f d = 3 ( ) g d = A+ 3A= 4A Therefore k = 4

26 99 BC5 Let f be the function defined by f () =. (a) Write the first four terms and the general term of the Taylor series epansion of f () about. = (b) Use the result from part (a) to find the first four terms and the general term of the series epansion about = for ln. (c) Use the series in part (b) to compute a number that differs from ln 3.5. Justify your answer. by less than Copyright 3 by College Entrance Eamination Board. All rights reserved.

27 99 BC5 Solution (a) Taylor approach f ( ) = ( ) ( ) f = = ( ) 3 f f ( ) = ( ) = ; =! 4 f ( ) f ( ) = 6( ) = 6; = 3! Geometric Approach = + ( ) where u = ( ) 3 n n = u u u u 3 n n Therefore = ( ) + ( ) ( ) + + ( ) ( ) + (b) Antidifferentiates series in (a): n n+ ( ) ( ) 3 4 ln = C+ ( ) + ( ) ( ) n + = ln C = Note: If C, first 4 terms need not include ( ) 4 4 (c) ln = ln = + 3 = since <, =.375 is sufficient. 4 8 Justification: Since series is alternating, with terms convergent to and decreasing in absolute value, the truncation error is less than the first omitted term. Alternate Justification: R n ( C ) < n n + + < when n Copyright 3 by College En trance Eamination Board. All rights reserved. n+ 5 =, where C n+ < < n +

28 99 BC6 Let f and g be continuous functions with the following properties. (i) g ( ) = A f( ) where Ais a constant (ii) 3 f ( d ) = gd ( ) (iii) 3 f ( d ) = 3A (a) Find 3 f ( d ) in terms of A. (b) Find the average value of g() in terms of A, over the interval [, 3]. (c) Find the value of k if f ( + ) d = k A. Copyright 3 by College Entrance Eamination Board. All rights reserved.

29 99 BC6 Solution 3 3 (a) ( ) = ( ) + ( ) f d f d f d ( ) ( ) 3 3 = g d + f d ( ( )) ( ) 3 3 = A f d+ f d ( ) ( ) 3 3 = A f d+ f d = A g d A f d 3 = A f ( ) d = [ A A] = A 3 3 (b) Average value = ( ) = ( ( )) ( ) ( ) (c) k A= f + d= f d = 3 ( ) g d = A+ 3A= 4A Therefore k = 4 Copyright 3 by College Entrance Eamination Board. All rights reserved.

30 99 BC A particle moves on the -ais so that its velocity at any time v(t) = t 36t +5. At t =, the particle is at the origin. (a) Find the position (t) of the particle at any time t. t is given by (b) Find all values of t for which the particle is at rest. (c) Find the maimum velocity of the particle for t. (d) Find the total distance traveled by the particle from t = to t =. Copyright 3 by College Entrance Eamination Board. All rights reserved.

31 99 BC Solution 3 (a) () t = 4t 8t + 5t+ C () () = = C Therefore C = t = t t + t (b) = v() t = t 36t+ 5 ( t )( t ) 3 5= t = 5, (c) dv = 4t 36 dt dv 3 = when t = dt v = 5 ( ) 3 v = v = 9 ( ) Maimum velocity is 5 / (d) Total distance = v() t dt () = = / v t dt ( ) ( ) 5 5 ( ) = 7 Copyright 3 by College Entrance Eamination Board. All rights reserved.

32 99 BC Let f be the function defined by f () = e for all real numbers. (a) Find each interval on which f is increasing. (b) Find the range of f. (c) Find the -coordinate of each point of inflection of the graph of f. (d) Using the results found in parts (a), (b), and (c), sketch the graph of f in the yplane provided below. (Indicate all intercepts.) y Copyright 3 by College Entrance Eamination Board. All rights reserved.

33 99 BC Solution (a) ( ) = ( ) + = ( ) f increases on (,] f e e e (b) () ( ) f = ; lim f = ( ] Range:, (c) f ( ) = e ( ) + ( ) e ( ) = ( ) e Point of inflection at =. (d) y 3 Copyright 3 by College Entrance Eamination Board. All rights reserved.

34 99 BC3 Let R be the shaded region in the first quadrant enclosed by the y-ais and the graphs of y = sin and y = cos, as shown in the figure above. (a) Find the area of R. (b) (c) Find the volume of the solid generated when R is revolved about the -ais. Find the volume of the solid whose base is R and whose cross sections cut by planes perpendicular to the -ais are squares. Copyright 3 by College Entrance Eamination Board. All rights reserved.

35 99 BC3 Solution (a) π /4 Area = cos sin d ( sin cos ) = + π /4 = + + = ( ) π /4 (b) V = π cos sin = π π /4 π = sin cos d π /4 π π = ( ) = d ( ) π /4 (c) V = cos sin π /4 ( sin ) = sin cosd = π = 4 π = 4 π /4 ( ) d Copyright 3 by College Entrance Eamination Board. All rights reserved.

36 99 BC4 Let F() = t + t dt. (a) Find F (). (b) Find the domain of F. (c) Find lim F(). (d) Find the length of the curve y = F() for. Copyright 3 by College Entrance Eamination Board. All rights reserved.

37 99 BC4 Solution F = 4 + (a) ( ) ( ) ( (b) t + t ; therefore t or t Since, want Therefore F m = = (c) li F( ) ) ( ) (d) = + ( ) L F d 6 8 = + + = 4+ d 7 = + = d Copyright 3 by College Entrance Eamination Board. All rights reserved.

38 99 BC5 4 Let f be the function given by f (t) = and G be the function given by + t G() = f (t) dt. (a) (b) (c) Find the first four nonzero terms and the general term for the power series epansion of f (t) about t =. Find the first four nonzero terms and the general term for the power series epansion of G() about =. Find the interval of convergence of the power series in part (b). (Your solution must include an analysis that justifies your answer.) Copyright 3 by College Entrance Eamination Board. All rights reserved.

39 99 BC5 Solution 4 (a) f () t =, geometric with a = 4, r + t () ( ) = t 4 6 n n f t = 4 4t + 4t 4t + + 4t (b) G( ) = = ( ) dt t t t dt + t (c) By Ratio Test, ( ) n n t = 4t t + t t n + ( ) n n = n + ( ) n+ n+ 3 4 n+ n+ = n n+ n+ 3 4 n+ 3 ( ) n + lim = ; < for < < n n Check endpoints: G () = Converges by Alternating Series Test G ( ) = Converges by Alternating Series Test 3 5 Converges for Copyright 3 by College Entrance Eamination Board. All rights reserved.

40 99 BC6 A certain rumor spreads through a community at the rate dy = y( y), where y is the dt proportion of the population that has heard the rumor at time t. (a) (b) (c) What proportion of the population has heard the rumor when it is spreading the fastest? If at time t = ten percent of the people have heard the rumor, find y as a function of t. At what time t is the rumor spreading the fastest? Copyright 3 by College Entrance Eamination Board. All rights reserved.

41 99 BC6 Solution y y = y y is largest when 4y = (a) ( ) so proportion is y = (b) y ( y) y ( y) dy = dt + dy = y y ( ) t dt dt ln y ln y = t+ C y ln = t+ C y y = ke y dy = y( ) =. k = 9 t e y = t 9 + e (c) = e 9 t = e 9 t t = ln 9 = ln 3 Copyright 3 by College Entrance Eamination Board. All rights reserved.

42 99 AB4/BC Consider the curve defined by the equation y + cosy = + for y π. (a) Find dy d in terms of y. (b) Write an equation for each vertical tangent to the curve. (c) Find d y in terms of y. d Copyright 3 by College Entrance Eamination Board. All rights reserved.

43 99 AB4/BC Solution (a) dy dy sin y = d d dy ( sin y) = d dy = d sin y (b) dy undefined when sin y = d π y = π + = + π = (c) d y d d sin y = d dy cos y d = ( sin y) cos y sin y = = ( sin y) cos y ( sin y) 3 Copyright 3 by College Entrance Eamination Board. All rights reserved.

44 99 AB5/BC Let f be the function given by f () = e, and let g be the function given by g() = k, where k is the nonzero constant such that the graph of f is tangent to the graph of g. (a) Find the -coordinate of the point of tangency and the value of k. (b) (c) Let R be the region enclosed by the y-ais and the graphs of f and g. Using the results found in part (a), determine the area of R. Set up, but do not integrate, an integral epression in terms of a single variable for the volume of the solid generated by revolving the region R, given in part (b), about the -ais. Copyright 3 by College Entrance Eamination Board. All rights reserved.

45 99 AB5/BC Solution (a) y ( ) ; ( ) f = e g = k = e k e = k = and k = e f ( ) = e g( ) = k ( ) = ( + ) e e d e e d (b) ( ) (c) ( π ( e ) ( e) ) e = e + e = ( + ) e + e = d Copyright 3 by College Entrance Eamination Board. All rights reserved.

46 99 BC3 At time t, t π, the position of a particle moving along a path in the y-plane is given by the parametric equations = e t sint and y = e t cost. (a) Find the slope of the path of the particle at time (b) Find the speed of the particle when t =. π t =. (c) Find the distance traveled by the particle along the path from t = to t =. Copyright 3 by College Entrance Eamination Board. All rights reserved.

47 99 BC3 Solution d (a) = t t e sin t + e cost dt dy t t = e cost e sin t dt t dy dy / dt e t t = = t d d / dt e sint cost ( cos sin ) ( + ) π /( ) π / ( + ) π dy e at t =, = = d e t t t t (b) spee = ( + ) ( d e sin t e cost + e cost e sin t when t = speed is ( ) ( ) esin+ ecos + ecos esin = e ) (c) dist ance is t t t t ( sin + cos ) + ( cos sin ) ( ) ( e ) e t e t e t e t dt t t = e sin t+ cos t dt = e dt t = e = Copyright 3 by College Entrance Eamination Board. All rights reserved.

48 99 BC4 Let f be a function defined by f () = for, + k + p for >. (a) For what values of k and p will f be continuous and differentiable at =? (b) (c) For the values of k and p found in part (a), on what interval or intervals is f increasing? Using the values of k and p found in part (a), find all points of inflection of the graph of f. Support your conclusion. Copyright 3 by College Entrance Eamination Board. All rights reserved.

49 99 BC4 Solution (a) For continuity at =, ( ) = f () = ( + k+ p) lim lim + Therefore = + k+ p Since f is continuous at = and is piecewise polynomial, left and right derivatives eist. () () f = and f = + k + For differentiability at =, = + k. Therefore k =, p = f =, (b) ( ) > ( ) < f =, > > > Since f increases on each of (,) and (, ) and is continuous at =, f is increasing on (, ). (c) f ( ) =, < f ( ) =, > f ( ) < ( ) f ( ) > on (, ) and f () is defined, ( f ()) = ( ) Since on, and,, is a point of inflection. Copyright 3 by College Entrance Eamination Board. All rights reserved.

50 99 BC5 The length of a solid cylindrical cord of elastic material is 3 inches. A circular cross section of the cord has radius inch. (a) (b) (c) What is the volume, in cubic inches, of the cord? The cord is stretched lengthwise at a constant rate of 8 inches per minute. Assuming that the cord maintains a cylindrical shape and a constant volume, at what rate is the radius of the cord changing one minute after the stretching begins? Indicate units of measure. A force of pounds is required to stretch the cord inches beyond its natural length of 3 inches. How much work is done during the first minute of stretching described in part (b)? Indicate units of measure. Copyright 3 by College Entrance Eamination Board. All rights reserved.

51 99 BC5 Solution (a) V = π r h= π 3 = 8 π (b) dv dr dh rh r dt dt dt at t =, h= 5 and = = π + π ; so 8π = π 5, so r = 5 r dr Therefore = π dt 5 dr 7 = π 4 + dt 5 dr 9 = in/min dt 5 ( ) π ( ) or V = 8 π = πr h, so r = 8 h dr 8 8 dh Therefore = dt h h dt at t =, h= 5 so dr 8 8 = dt = in/min 5 ( 8) (c) 8 8 Work = d= = 8 = 34 in-pounds = 7 foot-pounds Copyright 3 by College Entrance Eamination Board. All rights reserved.

52 99 BC6 Consider the series n=, where p. n p ln(n) (a) Show that the series converges for p >. (b) Determine whether the series converges or diverges for p =. Show your analysis. (c) Show that the series diverges for p <. Copyright 3 by College Entrance Eamination Board. All rights reserved.

53 99 BC6 Solution (a) < < for ln ( n p p ) >, for n 3 n ln n n ( ) by p-series test, converges if p > p n and by direct comparison, converges. ln ( ) p n= n n ln (b) L et f ( ) =, so series is f ( n) n= b d = lim ln ln = lim[ln(ln( b)) ln(ln )] = ln b b Since f () monotonically decreases to, the integral test shows diverges. n nln n = (c) > > for p <, p n ln n nln n so by direct comparison, diverges for p p n n ln n < = Copyright 3 by College Entrance Eamination Board. All rights reserved.

54 993 AB3/BC Consider the curve on the curve. y = 4 + and chord AB joining the points A( 4,) and B(, ) (a) Find the - and y-coordinates of the point on the curve where the tangent line is parallel to chord AB. (b) Find the area of the region R enclosed by the curve and the chord AB. (c) Find the volume of the solid generated when the region R, defined in part (b), is revolved about the -ais. Copyright 3 by College Entrance Eamination Board. All rights reserved.

55 993 AB3/BC Solution (a) B(,) sl ope of AB = = 4 A ( 4, ) Method : Method : dy y = 4 + ; = ; = d so = 3, y = dy y = ; so y = and y =, = 3 d (b) 3/ Method : 4 ( 4 ) d= / 6 4 = ( 4) ( 4+ 8) = 4= Method : ( 4) ( 4) 8 4 = 4 = 3 3 y y dy y = 3 y 3 Method 3: d= ; Area of triangle = Area of region = 4 = 3 3 Copyright 3 by College Entrance Eamination Board. All rights reserved.

56 993 AB3/BC Solution, continued (c) Method : π ( ) d = π d 4 6 8π = π 8 = Method : ( 4) ( 4) Method 3: π ( ) 8π π y y y dy = d= π 4+ ( ) = π 6+ 8 = 8π 4 6π Volume of cone = π ( ) ( 4) = 3 3 6π 8π Volume = 8π = 3 3 Copyright 3 by College Entrance Eamination Board. All rights reserved.

57 993 AB4/BC3 Let f be the function defined by f () = ln( + sin ) for π π. (a) Find the absolute maimum value and the absolute minimum value of f. Show the analysis that leads to your conclusion. (b) Find the -coordinate of each inflection point on the graph of f. Justify your answer. Copyright 3 by College Entrance Eamination Board. All rights reserved.

58 993 AB4/BC3 Solution f = cos ; + sin 3π In [ π, π], cos = when = ; (a) ( ) π π 3π f ( ) ( ) () ln =.693 ln ( ) ln = absolute maimum value is ln absolute minimum value is = (b) f ( ) ( )( + ) ( + sin) sin sin cos cos sin = ; ( + sin) f ( ) = when sin = 7π π =, 6 6 sign of f concavity π down + up down 7π π π 6 6 7π π = and since concavity changes as indicated at these points 6 6 Copyright 3 by College Entrance Eamination Board. All rights reserved.

59 993 BC The position of a particle at any time t is given by (t) = t 3 and y(t) = 3 t3. (a) Find the magnitude of the velocity vector at t = 5. (b) Find the total distance traveled by the particle from t = to t = 5. (c) Find dy d as a function of. Copyright 3 by College Entrance Eamination Board. All rights reserved.

60 993 BC Solution (a) () () ( ) y ( ) t = t y t = t v 5 = 5 = 5 ( ) = + = = (b) 5 5 = t + = t + 4t dt ( t ) t dt 5 3/ 3/ = ( 6 ) () () (c) dy y t t = = = t d t t = 3; = + 3 t t t = + 3 dy = d + 3 or = 3; = + 3 t t 3 y = t ; y = dy = + 3 d ( ) 3/ Copyright 3 by College Entrance Eamination Board. All rights reserved.

61 993 BC4 Consider the polar curve r = sin(3θ) for θ π. (a) In the y-plane provided below, sketch the curve. y O (b) Find the area of the region inside the curve. (c) Find the slope of the curve at the point where θ = π 4. Copyright 3 by College Entrance Eamination Board. All rights reserved.

62 993 BC4 Solution y (a) O A= 4sin 3θ dθ cos6θ dθ θ sin6θ π = = 6 = π π (b) ( ) 3 π /3 or 4sin 3 6 π /6 or 4sin 3 θ dθ = = π θ dθ = = π π (c) = sin3θ cosθ y = sin 3θsinθ d = sin 3θsinθ + 6cos3θ cosθ dθ dy = sin 3θ cosθ + 6cos3θsinθ dθ π dy d At θ =, = and = 4, so 4 dθ dθ dy = = d 4 or ( ) 3 + y = 6 y y dy dy dy ( + y ) + y = 6 + y 6y d d d π At θ =, = and y = so 4 dy dy dy 4 + = d d d Copyright 3 by College Entrance Eamination Board. All rights reserved. dy dy 8+ 8 = = d d

63 993 BC5 Let f be the function given by f () = e. (a) Write the first four nonzero terms and the general term for the Taylor series epansion of f () about =. (b) Use the result from part (a) to write the first three nonzero terms and the general term of the series epansion about = for g() = e. n (c) For the function g in part (b), find g () and use it to show that 4(n +)! =. n= 4 Copyright 3 by College Entrance Eamination Board. All rights reserved.

64 993 BC 5 Solution (a) / 3 n e = ! 3! n! e 3 ( / ) ( / ) ( /) = ! 3! n! 3 n! 3 3! n n! = n 3 n (b) e / n =! 3! n! n = n! 3! n! ( ) n (c) n g ( ) = n! 3! n! n ( n ) = n 8 4 n! n ( n ) g ( ) = n! 3! n! n = n! n = 4 n +! n= ( ) / e Also, g( ) = e e g ( ) = e ( e ) g ( ) = = 4 4 ()( ) / / Copyright 3 by College Entrance Eamination Board. All rights reserved.

65 993 BC6 Let f be a function that is differentiable throughout its domain and that has the following properties. f () + f (y) (i) f ( + y) = for all real numbers, y, and + y in the domain of f f () f (y) (ii) lim h f (h) = (iii) lim h f (h) h = (a) Show that f () =. (b) Use the definition of the derivative to show that f ( ) = + [ f( )]. Indicate clearly where properties (i), (ii), and (iii) are used. (c) Find f () by solving the differential equation in part (b). Copyright 3 by College Entrance Eamination Board. All rights reserved.

66 993 BC6 Solution (a) Method : f is continuous at, so f ( ) lim f ( ) or Method : f ( ) f ( ) (b) f ( ) ( ) ( ) ( ) + f ( ) f ( ) f ( ) ( ) = ( ) ( f ) = ( ) ( ) ( ) f = + = f f f f f = ( + ) ( ) ( ) ( ) ( ) = = f h f = lim h h f ( ) + f ( h) f ( ) f ( ) f ( h) = lim [By (i)] h h f ( h) + f ( ) = lim h h f ( ) f ( h) + f = [By (iii) & (ii)] f = + f dy t y = f ; = + y d dy = d + y (c) Method : Le ( ) tan y = + C = tan ( + ) ( ) = = [ or = π, Z] ( ) = tan or ( ) = tan ( + π ) y C f C C n n f f n Copyright 3 by College Entrance Eamination Board. All rights reserved.

67 993 BC6 Solution, continued or Method : Guess that ( ) f = tan f ( ) f ( ) ( ) = tan( ) = + = + tan = sec = f Since the solution to the D.E. is unique f ( ) = tan is the solution. Copyright 3 by College Entrance Eamination Board. All rights reserved.

68 994 AB -BC Let R be the region enclosed by the graphs of y = e, y =, and the lines = and = 4. (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the -ais. (c) Set up, but do not integrate, an integral epression in terms of a single variable for the volume of the solid generated when R is revolved about the y-ais. Copyright 3 by College Entrance Eamination Board. All rights reserved.

69 994 AB - BC y (a) 4 Area = e d OR = e e 4 ( e ) 4 = 4 = e Using geometry (area of triangle) R 4 4 e d 44 or Using y-ais e ydy+ y ln ydy+ 4 ln ydy 4 = ( ) ( ) 4 (b) V = π e d = π e d 3 π e 3 64 π e e = π e π = 4 Copyright 3 by College Entrance Eamination Board. All rights reserved.

70 994 AB - BC (continued) [or] Using geometry (Volume of the cone) π ( e ) 4 4 = π e d π 3 8 = π e 4 4 π π 3 Using y- ais 4 4 e π y ydy+ y( y ln y) dy+ y( 4 y) dy 4 ln 4 (c) Vy = π ( e )d or 4 4 e Vy = π y dy + y ( ln y) dy + 6 ( ln y) dy 4 Copyright 3 by College Entrance Eamination Board. All rights reserved.

71 994 AB 5-BC A circle is inscribed in a square as shown in the figure above. The circumference of the circle is increasing at a constant rate of 6 inches per second. As the circle epands, the square epands to maintain the condition of tangency. (Note: A circle with radius r has circumference C = π r and area A= π r ) (a) Find the rate at which the perimeter of the square is increasing. Indicate units of measure. (b) At the instant when the area of the circle is 5π square inches, find the rate of increase in the area enclosed between the circle and the square. Indicate units of measure. Copyright 3 by College Entrance Eamination Board. All rights reserved.

72 994 AB 5-BC (a) P = 8R dp dr = 8 dt dt dc dr 6= = π dt dt dr 3 dp 4 = ; = inches/second dt π dt π inches/second R (b) Area = 4R π R ( Area) ( ) d dr dr = 8R π R dt dt dt dr = ( 4 π ) R dt Area of circle = 5π = πr d R = 5 Area 3 inches /second dt = π ( π ) 3 π 4 inches /second = 8.97 inches /second Copyright 3 by College Entrance Eamination Board. All rights reserved.

73 994 BC 3 A particle moves along the graph of y = cos so that the -coordinate of acceleration is always. At time t =, the particle is at the point ( π, ) and the velocity vector of the particle is (,). (a) Find the - and y-coordinates of the position of the particle in terms of t. (b) Find the speed of the particle when its position is ( 4,cos4 ). Copyright 3 by College Entrance Eamination Board. All rights reserved.

74 994 BC 3 () () ( ) ( ) () = +, ( ) = π = () = t + π (a) t = t = t+ C = C = ; t = t t t k k t () = cos( t + π ) y t (b) dy = sin t ( t + π ) dt () s t d dy = + dt dt ( ) ( t) tsin( t π ) = + + ( π ) = + + 4t 4t sin t when 4, π 4; 4 = t + = t = π ( π) ( π) s = sin 4.34 Copyright 3 by College Entrance Eamination Board. All rights reserved.

75 994 BC 4 Let f () = 6. For < w < 6, let A(w) be the area of the triangle formed by the coordinate aes and the line tangent to the graph of f at the point ( w,6 w ). (a) Find A(). (b) For what value of w is A(w) a minimum? Copyright 3 by College Entrance Eamination Board. All rights reserved.

76 994 BC 4 (a) ( ) ( ) f () = f = 6 ; f = y 5= ( )or y = int: y int: A() = ( 7) = 4 (b) f ( w) = w; y ( 6 w ) = w( w) 6 + w int: y int: 6 + w w ( ) A w = ( 6 + w ) 4w ( ( + )( )) ( + ) 4w 6 w w 4 6 w A ( w) = 6w A w = when 6 + w 3w 6 = ( ) ( )( ) w = A + w 6 Copyright 3 by College Entrance Eamination Board. All rights reserved.

77 994 BC 5 Let f be the function given by f () = e. (a) Find the first four nonzero terms and the general term of the power series for f () about =. (b) Find the interval of convergence of the power series for f () about =. Show the analysis that leads to your conclusion. (c) Let g be the function given by the sum of the first four nonzero terms of the power series for f () about =. Show that f () g() <. for.6.6. Copyright 3 by College Entrance Eamination Board. All rights reserved.

78 994 BC 5 (a) e e u 3 n u u u = + u ! n! ( ) n n n = ! n! (b) The series for u e converges for < u So the series for converges for < < < e And, thus, for < < Or lim n n+ n+ ( n+ ) ( ) ( + ) ( ) a n+ n! lim = lim n a n n! n = = < n + So the series for (c) f ( ) g( ) e = + 4! 5! n n n converges for < < This is an alternative series for each, since the powers of are even. Also, value. + n = < a a n n+ for.6.6 so terms are decreasing in absolute ( ) Thus f ( ) g( ) 4! 4! =. <. Copyright 3 by College Entrance Eamination Board. All rights reserved.

79 994 BC 6 Let f and g be functions that are differentiable for all real numbers and that have the following properties. (i) f () = f () g() (ii) g () = g() f () (iii) f () = 5 (iv) g() = (a) Prove that f () + g() = 6 for all. (b) Find f () and g(). Show your work. Copyright 3 by College Entrance Eamination Board. All rights reserved.

80 994 BC 6 (a) f ( ) g ( ) f ( ) g( ) g( ) f ( ) + = + = so f + g is constant. ( ) ( ) ( ) ( ) f + g = 6, so f + g = 6 Or (b) f ( ) = 6 g( ) so ( ) ( ) 6 ( ) ( ) g = g + g = g 6 dy dy = y 6; = d d y 6 ln y 6 = + C ln y 6 = + K y 6 = e y 6= Ae + K = y = so 4= A 4 6 y = e + = 3 = ( ) ( ) ( ) y e g f = 6 g = 3+ e Copyright 3 by College Entrance Eamination Board. All rights reserved.

81 995 AB4/BC Note: Figure not drawn to scale. The shaded regions R and R shown above are enclosed by the graphs of and g ( ). f( ) (a) (b) (c) Find the - and y-coordinates of the three points of intersection of the graphs of f and g. Without using absolute value, set up an epression involving one or more integrals that gives the total area enclosed by the graphs of f and g. Do not evaluate. Without using absolute value, set up an epression involving one or more integrals that gives the volume of the solid generated by revolving the region R about the line y 5. Do not evaluate. Copyright 3 by College Entrance Eamination Board. All rights reserved.

82 995 AB4/BC Solution (a), 4 4,6.767,.588 or (.766,.588) (b) 4 or.767 d d ln y ln y y dy y dy y dy.588 ln 4 ln (c) 5 5 or d.588 ln y y ln 5 y y dy 5 y dy Copyright 3 by College Entrance Eamination Board. All rights reserved.

83 995 AB5/BC3 As shown in the figure above, water is draining from a conical tank with height feet and diameter 8 feet into a cylindrical tank that has a base with area 4 square feet. The depth h, in feet, of the water in the conical tank is changing at the rate of ( h ) feet per minute. (The volume V of a cone with radius r and height h is V r h.) 3 (a) Write an epression for the volume of water in the conical tank as a function of h. (b) At what rate is the volume of water in the conical tank changing when h 3? Indicate units of measure. (c) Let y be the depth, in feet, of the water in the cylindrical tank. At what rate is y changing when h 3? Indicate units of measure. Copyright 3 by College Entrance Eamination Board. All rights reserved.

84 995 AB5/BC3 Solution (a) (b) r 4 r h h 3 3 h V h h dv h dh dt 9 dt h h V is decreasing at 9 ft /min (c) Let W = volume of water in cylindrical tank dw dy W 4 y; 4 dt dt dy 4 9 dt 9 y is increasing at ft/min 4 Copyright 3 by College Entrance Eamination Board. All rights reserved.

85 995 BC Two particles move in the y-plane. For time t, the position of particle A is given by 3t t and y ( t ), and the position of particle B is given by 4 and 3t y. (a) Find the velocity vector for each particle at time t 3. (b) (c) (d) Set up an integral epression that gives the distance traveled by particle A from t = to t = 3. Do not evaluate. Determine the eact time at which the particles collide; that is, when the particles are at the same point at the same time. Justify your answer. In the viewing window provided below, sketch the paths of particles A and B from t until they collide. Indicate the direction of each particle along its path Viewing Window [ 7,7] [ 5,5] Copyright 3 by College Entrance Eamination Board. All rights reserved.

86 995 BC Solution (a) V, t4 ; V 3, V A B , ; VB3, (b) distance t4 A 3 dt 3t (c) Set t 4; t 4 When t 4, the y-coordinates for A and B are also equal. Particles collide at (,4) when t 4. (d) Viewing Window [ 7,7] [ 5,5] Copyright 3 by College Entrance Eamination Board. All rights reserved.

87 995 BC4 Let f be a function that has derivatives of all orders for all real numbers. Assume f() 3, f(), f(), and f () 4. (a) Write the second-degree Taylor polynomial for f about and use it to approimate f (.7). (b) Write the third-degree Taylor polynomial for f about and use it to approimate f (.). (c) Write the second-degree Taylor polynomial for f, the derivative of f, about and use it to approimate f (.). Copyright 3 by College Entrance Eamination Board. All rights reserved.

88 995 BC4 Solution (a) T 3 f (b) T 3 f (c) T 3 f Copyright 3 by College Entrance Eamination Board. All rights reserved.

89 995 BC5 Let f( ), g( ) cos, and h( ) cos. From the graphs of f and g shown above in Figure and Figure, one might think the graph of h should look like the graph in Figure 3. (a) Sketch the actual graph of h in the viewing window provided below Viewing Window [ 6, 6] [ 6, 4] (b) Use h ( ) to eplain why the graph of h does not look like the graph in Figure 3. (c) Prove that the graph of y cos( k) has either no points of inflection or infinitely many points of inflection, depending on the value of the constant k. Copyright 3 by College Entrance Eamination Board. All rights reserved.

90 995 BC5 Solution (a) (b) 6 h sin ; h cos cos for all, so graph must be concave up everywhere (c) y k cosk If k, y for all, so no inflection points. If k, y changes sign and is periodic, so changes sign infinitely many times. Hence there are infinitely many inflection points. Copyright 3 by College Entrance Eamination Board. All rights reserved.

91 995 BC6 Graph of f Let f be a function whose domain is the closed interval,5. The graph of f is shown above. Let 3 h ( ) f( tdt ). (a) Find the domain of h. (b) Find h (). (c) At what is h ( ) a minimum? Show the analysis that leads to your conclusion. Copyright 3 by College Entrance Eamination Board. All rights reserved.

92 995 BC6 Solution (a) (b) h f 3 3 h f 4 (c) h is positive, then negative, so minimum is at an endpoint h 6 f t dt 5 h 4 f t dt since the area below the ais is greater than the area above the ais therefore minimum at 4 Copyright 3 by College Entrance Eamination Board. All rights reserved.

93 996 AB3/BC3 kt The rate of consumption of cola in the United States is given by St ( ) Ce, where S is measured in billions of gallons per year and t is measured in years from the beginning of 98. (a) (b) The consumption rate doubles every 5 years and the consumption rate at the beginning of 98 was 6 billion gallons per year. Find C and k. Find the average rate of consumption of cola over the -year time period beginning January, 983. Indicate units of measure. (c) Use the trapezoidal rule with four equal subdivisions to estimate 5 7 St () dt. (d) Using correct units, eplain the meaning of consumption. 5 7 St () dtin terms of cola Copyright 3 by College Entrance Eamination Board. All rights reserved.

94 996 AB3/BC3 Solution (a) S t S S Ce kt 6C 6 5 6e 5k 5k e ln k 5.38 or.39 (b) 3 ln 3 3.6ln.6ln 3 Average rate 6e e e (9.68 billion gal/yr) 3 ( ) t ln 5 dt billion gal/yr 7 (c) S t dt S 5S 5.5S 6 S 6.5S (d) This gives the total consumption, in billions of gallons, during the years 985 and 986. Copyright 3 by College Entrance Eamination Board. All rights reserved.

95 996 AB4/BC4 This problem deals with functions defined by f( ) b sin, where b is a positive constant and. (a) Sketch the graphs of two of these functions, y sin and y 3sin. y y (b) Find the -coordinates of all points,, where the line y b is tangent to the graph of f( ) b sin. (c) Are the points of tangency described in part (b) relative maimum points of f? Why? (d) For all values of b, show that all inflection points of the graph of f lie on the line y. Copyright 3 by College Entrance Eamination Board. All rights reserved.

96 996 AB4/BC4 Solution (a) y 6 y (b) y bcos bcos cos y b bsin b bsin sin 3 or (c) No, because f f or at -coordinates of points of tangency (d) f bsin sin f b at -coordinates of any inflection points Copyright 3 by College Entrance Eamination Board. All rights reserved.

97 996 BC Consider the graph of the function h given by h ( ) e for. (a) Let R be the unbounded region in the first quadrant below the graph of h. Find the volume of the solid generated when R is revolved about the y-ais. (b) Let Aw ( ) be the area of the shaded rectangle shown in the figure above. Show that Aw ( ) has its maimum value when w is the -coordinate of the point of inflection of the graph of h. Copyright 3 by College Entrance Eamination Board. All rights reserved.

98 996 BC Solution (a) Volume lim b e d b b lim e lim e e b b or b e d Volume ln y dy lim ln y dy (b) Maimum: A w we w A w e w e w w w e A w when,, w. w w w A w when, A w when. Therefore, ma occurs when w Inflection: h e, h e, h e e e. when, when, when. h h h a Therefore, inflection point when. Therefore, the maimum value of Aw ( ) and the inflection point of h ( ) occur when and w are. a Copyright 3 by College Entrance Eamination Board. All rights reserved.

99 996 BC The Maclaurin series for f( ) is given by (a) Find (7) f () and f (). 3 n! 3! 4! ( n )! (b) For what values of does the given series converge? Show your reasoning. (c) Let g ( ) f( ). Write the Maclaurin series for g, ( ) showing the first three nonzero terms and the general term. (d) Write g ( ) in terms of a familiar function without using series. Then, write f( ) in terms of the same familiar function. Copyright 3 by College Entrance Eamination Board. All rights reserved.

100 996 BC Solution (a) (b) a n 7 f n n! n! fa f 7! a7 7! 8! 8 n n! lim lim n n n n n! Converges for all, by ratio test (c) g f 3 n! 3! n! (d) n e! n! e g f e if f if Copyright 3 by College Entrance Eamination Board. All rights reserved.

101 996 BC5 An oil storage tank has the shape as shown above, obtained by revolving the curve 9 4 y from to 5 about the y-ais, where and y are measured in feet. 65 Oil weighing 5 pounds per cubic foot flowed into an initially empty tank at a constant rate of 8 cubic feet per minute. When the depth of oil reached 6 feet, the flow stopped. (a) (b) Let h be the depth, in feet, of oil in the tank. How fast was the depth of oil in the tank increasing when h 4? Indicate units of measure. Find, to the nearest foot-pound, the amount of work required to empty the tank by pumping all of the oil back to the top of the tank. Copyright 3 by College Entrance Eamination Board. All rights reserved.

102 996 BC5 Solution (a) h 5 V ydy 3 dv 5 dh h dt 3 dt 5 dh at h 4, dt dh ft/min dt 5 (b) W y ydy W 5 9y y dy W 5 9y y W 69, ft-lbs to the nearest foot-pound 69,58 ft-lbs 6 Copyright 3 by College Entrance Eamination Board. All rights reserved.

103 996 BC6 The figure above shows a spotlight shining on point P (, y ) on the shoreline of Crescent Island. The spotlight is located at the origin and is rotating. The portion of the shoreline on which the spotlight shines is in the shape of the parabola y from the point, to the point 5, 5. Let be the angle between the beam of light and the positive - ais. (a) For what values of between and does the spotlight shine on the shoreline? (b) Find the - and y-coordinates of point P in terms of tan. (c) If the spotlight is rotating at the rate of one revolution per minute, how fast is the point P traveling along the shoreline at the instant it is at the point 3,9? Copyright 3 by College Entrance Eamination Board. All rights reserved.

104 996 BC6 Solution (a) tan or tan tan 5 or Therefore, tan 5 4 (b) (c) y tan Therefore, tan y tan d dt d d dy d sec ; tansec dt dt dt dt d At 3,9 : dt dy 3 dt Speed 37 or 38.9 Copyright 3 by College Entrance Eamination Board. All rights reserved.

105 997 AB5/BC5 The graph of the function f consists of a semicircle and two line segments as shown above. Let g be the function given by g ( ) ftdt ( ). (a) Find g (3). (b) Find all the values of on the open interval,5 at which g has a relative maimum. Justify your answer. (c) Write an equation for the line tangent to the graph of g at 3. (d) Find the -coordinate of each point of inflection of the graph of g on the open interval,5. Justify your answer. Copyright 3 by College Entrance Eamination Board. All rights reserved.

106 997 AB5/BC5 Solution (a) (b) (c) g 3 3 f( t) dt 4 g ( ) has a relative maimum at because g' to negative at g(3) g' 3 f (3) y 3 f ( ) changes from the positive (d) graph of g has points of inflection with -coordinates = and = 3 because g f( ) changes from the positive to negative at and from negative to positive at 3 or because g f changes from increasing to decreasing at and from decreasing to increasing at 3 Copyright 3 by College Entrance Eamination Board. All rights reserved.

107 997 AB6/BC6 Let vt ( ) be the velocity, in feet per second, of a skydiver at time t seconds, t. After dv her parachute opens, her velocity satisfies the differential equation v 3, with dt initial condition v() 5. (a) Use separation of variables to find an epression for v in terms of t, where t is measured in seconds. (b) Terminal velocity is defined as lim vt ( ). Find the terminal velocity of the skydiver to the nearest foot per second. t (c) It is safe to land when her speed is feet per second. At what time t does she reach this speed? Copyright 3 by College Entrance Eamination Board. All rights reserved.

108 997 AB6/BC6 Solution dv dt dv dt v 6 ln v6 t A (a) v3v6 v6 e e e v6 Ce ta A t t Ce C 5 6 ; 34 v e t 34 6 t (b) v t e lim lim t t t (c) v t 34e 6 e t ; t ln Copyright 3 by College Entrance Eamination Board. All rights reserved.

109 997 BC During the time period from t to t 6 seconds, a particle moves along the path given by t ( ) 3 cos( t) and yt ( ) 5 sin( t). (a) Find the position of the particle when t.5. (b) On the aes provided below, sketch the graph of the path of the particle from t = to t = 6. Indicate the direction of the particle along its path. (c) How many times does the particle pass through the point found in part (a)? (d) Find the velocity vector for the particle at any time t. (e) Write and evaluate an integral epression, in terms of sine and cosine, that gives the distance the particle travels from t =.5 to t =.75. Copyright 3 by College Entrance Eamination Board. All rights reserved.

110 997 BC Solution (a) y.5 3cos.5.5 5sin.5 5 (b) y (c) 3 (d) () t 3 sin t y() t 5 cost vt 3 sin t,5 cost.75 (e) distance 9 sin t5 cos t dt Copyright 3 by College Entrance Eamination Board. All rights reserved.

111 997 BC 3 4 Let P ( ) 73( 4) 5( 4) ( 4) 6( 4) be the fourth-degree Taylor polynomial for the function f about 4. Assume f has derivatives of all orders for all real numbers. (a) Find f(4) and f (4). (b) Write the second-degree Taylor polynomial for f about 4 and use it to approimate f (4.3). (c) Write the fourth-degree Taylor polynomial for g ( ) ftdt ( ) about 4. 4 (d) Can f (3) be determined from the information given? Justify your answer. Copyright 3 by College Entrance Eamination Board. All rights reserved.

112 997 BC Solution (a) f P f !, f 4 (b) P P 3 f P g, P( t) dt (c) t4 5 t4 t4 dt (d) No. The information given provides values for 4 f 4, f 4, f 4, f 4 andf 4 only. Copyright 3 by College Entrance Eamination Board. All rights reserved.

113 997 BC3 Let R be the region enclosed by the graphs of y ln (a) Find the area of R. and y cos. (b) (c) Write an epression involving one or more integrals that gives the length of the boundary of the region R. Do not evaluate. The base of a solid is the region R. Each cross section of the solid perpendicular to the -ais is an equilateral triangle. Write an epression involving one or more integrals that gives the volume of the solid. Do not evaluate. Copyright 3 by College Entrance Eamination Board. All rights reserved.

114 997 BC3 Solution (a) y R ln cos.9586 let B.9586 B area cos ln B.68 d B B (b) sin (c) L d d B B base 3 base 3 area of cross section cos ln cos ln B 3 cos ln d 4 volume B Copyright 3 by College Entrance Eamination Board. All rights reserved.

115 997 BC4 Let ky, where k. (a) Show that for all k, the point 4, k is on the graph of (b) Show that for all k, the tangent line to the graph of 4, k passes through the origin. ky. ky at the point (c) Let R be the region in the first quadrant bounded by the -ais, the graph of ky, and the line 4. Write an integral epression for the area of the region R and show that this area decreases as k increases. Copyright 3 by College Entrance Eamination Board. All rights reserved.

116 997 BC4 Solution (a) k k y 4, k R 4 (b) ky dy ky d dy d y / k k the tangent line is y 4 k k y which contains (, ) k or slope of the line through, and 4, / k is which is the same as the slope of the tangent line / k 4 k / k (c) A 4 ky dy or 4 A d k A da dk 4 3 k.5 k 3.5 for all k thus the area decreases as k increases Copyright 3 by College Entrance Eamination Board. All rights reserved.