1969, 1973, 1985, 1993, 1997, 1998 AP Multiple Choice Sections, AB and BC, Solutions (620)

These two facts imply that f is differentiable at x 3

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Unformatted text preview: ² ( x ) = x 3 is not defined at x = 0 . 3 17. B d2 (x ) 2 x2 f β€² ( x ) = (1) β‹… ln( x 2 ) + x β‹… dx 2 = ln( x 2 ) + 2 = ln( x 2 ) + 2 x x 18. C ∫ sin(2 x + 3) dx = 2 ∫ sin(2 x + 3) (2dx) = βˆ’ 2 cos(2 x + 3) + C 19. D g ( x) = e f ( x ) , g β€²( x) = e f ( x ) β‹… f β€²( x) , g β€²β€²( x) = e f ( x ) β‹… f β€²β€²( x) + f β€²( x) β‹… e f ( x ) β‹… f β€²( x) 1 ( 1 ( g β€²β€²( x) = e f ( x ) f β€²β€²( x) + f β€²( x) 2 20. C )) = h( x)e f ( x) β‡’ h( x) = f β€²β€²( x) + ( f β€²( x)2 ) Look for concavity changes, there are 3. AP Calculus Multiple-Choice Question Collection Copyright Β© 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 189 1985 Calculus BC Solutions 21. B Use the technique of antiderivatives by parts: u = f ( x) dv = sin x dx du = f β€² ( x ) dx v = βˆ’ cos x ∫ f ( x) sin x dx = βˆ’ f ( x) cos x + ∫ f β€²( x) cos x dx and we are given that 2 2 3 ∫ f ( x) sin x dx = βˆ’ f ( x) cos x + ∫ 3x cos x dx β‡’ f β€²( x) = 3x β‡’ f ( x) = x 22. A A = Ο€ r 2 , A = 64Ο€ when r = 8 . Take the derivative with respect to t. dA dr dr dr = 2Ο€ r β‹… ; 96Ο€ = 2Ο€(8) β‹… β‡’ =6 dt dt dt dt 1+ h 23. C lim ∫1 x5 + 8 dx h h β†’0 F (1 + h) βˆ’ F (1) =F β€²(1) where F β€²( x) = x5 + 8 . F β€²(1)=3 h β†’0 h = lim 1+ h Alternate solution by L’HΓ΄pital’s Rule: lim h β†’0 24. D ∫1 x5 + 8 dx h = lim (1 + h )5 + 8 1 h β†’0 1Ο€ 1Ο€1 1βŽ› 1 ⎞ Area = ∫ 2 sin 2 (2ΞΈ) d ΞΈ = ∫ 2 (1 βˆ’ cos 4ΞΈ ) d ΞΈ = ⎜ ΞΈ βˆ’ sin 4ΞΈ ⎟ 20 20 2 4⎝ 4 ⎠ Ο€ 0 2 = = 9=3 Ο€ 8 1 only. 2 25. C At rest when v(t ) = 0 . v(t ) = e βˆ’2t βˆ’ 2teβˆ’2t = eβˆ’2t (1 βˆ’ 2t ) , v(t ) = 0 at t = 26. E Apply the log function, simplify, and differentiate. ln y = ln ( sin x ) = x ln ( sin x ) yβ€² cos x x = ln ( sin x ) + x β‹… β‡’ yβ€² = y ( ln ( sin x ) + x β‹… cot x ) = ( sin x ) ( ln ( sin x ) + x β‹… cot x ) sin x y 27. E Each of the right-hand sides represent the area of a rectangle with base length (b βˆ’ a) . I. Area under the curve is less than the area of the rectangle with height f (b) . II. Area under the curve is more than the area of the rectangle with height f (a) . III. Area under the curve is the same as the area of the rectangle with height f (c) , a < c < b . Note that this is the Mean Value Theorem for Integrals. 28. E x+e e x u x x+e e ∫ e dx = ∫ e (e dx) . This is of the form ∫ e du, u = e , so ∫ e dx = e + C x x x AP Calculus Multiple-Choice Question Collection Copyright Β© 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. x x 190 1985 Calculus BC Solutions 29. D Ο€βŽž βŽ› sin ⎜ x βˆ’ ⎟ sin t Ο€ 4⎠ ⎝ Let x βˆ’ = t . lim = lim =1 Ο€ Ο€ t β†’0 t 4 xβ†’ xβˆ’ 4 4 30. B 3 dy dy = dt = 2 3t + 1 At t = 1, dx dx 3t 2 βˆ’ 1 dt 31. D The center is x = 1 , so only C, D, or E are possible. Check the endpoints. At x = 0: At x = 2: ∞ βˆ‘ n =1 ∞ ( βˆ’1)n n 1 βˆ‘n 3 3 =4= t =1 3 βˆ’ 1 8 converges by alternating series test. which is the harmonic series and known to diverge. n =1 32. E y (βˆ’1) = βˆ’6, yβ€²(βˆ’1) = 3 x 2 + 6 x + 7 x =βˆ’1 = 4 , the slope of the normal is βˆ’ 1 and an equation 4 1 for the normal is y + 6 = βˆ’ ( x + 1) β‡’ x + 4 y = βˆ’25 . 4 33. C This is the differential equation for exponential growth. 1 1 βŽ›1⎞ 1 βŽ›1⎞ y = y (0) eβˆ’2t = e βˆ’2t ; = e βˆ’2t ; βˆ’2t = ln ⎜ ⎟ β‡’ t = βˆ’ ln ⎜ ⎟ = ln 2 2 2 ⎝2⎠ 2 ⎝2⎠ 34. A This topic is no longer part of the AP Course Description. 1 Surface Area = ∫ 2Ο€ y 2 3 0 35. B βˆ‘ 2πρ βˆ†s where ρ = x = y3 () 1 βŽ› dx ⎞ 1 + ⎜ ⎟ dy = ∫ 2Ο€ y 3 1 + 3 y 2 0 ⎝ dy ⎠ 2 1 dy = 2Ο€ ∫ y 3 1 + 9 y 4 dy 0 Use shells (which is no longer part of the AP Course Description) βˆ‘ 2Ο€ rh βˆ†x where r = x and h = y = 6 x βˆ’ x 2 6 ( ) Volume = 2Ο€βˆ« x 6 x βˆ’ x 2 dx 0 AP Calculus Multiple-Choice Question Collection Copyright Β© 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 191 1985 Calculus BC Solutions 1 3 1 3 3 L x2 3 x 1 ∫ βˆ’1 x 2 dx = 2 ∫ 0 37. A This topic is no longer part of the AP Course Description. y = yh + y p where yh = ce βˆ’ x is x2 dx = 2 lim+ ∫ 1 36. E L β†’0 dx = 2 lim+ βˆ’ L β†’0 L which does not exist. ( ) dy + y = 0 and y p = Ax 2 + Bx e βˆ’ x is a particular dx solution to the given differential equation. Substitute y p into the differential equation to the solution to the homogeneous equation 1 determine the values of A and B. The answer is A = , B = 0 . 2 ( ) ( lim ln 1+5e x ) 1 x ( ln 1+5e x ( x β†’βˆž 39. A Square cross sections: 40. A ) 5e x x 1 u = , du = dx ; when x = 2, u = 1 and when x = 4, u = 2 2 2 lim 1 + 5e x ∫2 βŽ›x⎞ 1βˆ’ ⎜ ⎟ ⎝2⎠ x yβ€² = 1 x2 4 41. C 42. E 43. E 44. A ) x = xlim e β†’βˆž 1 38. C 1 ln 1+5e x x = e x β†’βˆž lim = e xβ†’βˆž lim x = e xβ†’βˆž1+5e = e x 3 1 3 βˆ‘ y 2 βˆ†x where y = eβˆ’ x . V = ∫ 0 e βˆ’2 x dx = βˆ’ 2 e βˆ’2 x 0 = ( 1 1 βˆ’ e βˆ’6 2 ) 2 , L=∫ dx = ∫ 2 1 3 0 2 1βˆ’ u2 1βˆ’ u2 β‹… 2 du = ∫ du 1 2u u 1 + ( yβ€² ) dx = ∫ 2 3 0...
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This note was uploaded on 12/29/2010 for the course MATH 214 taught by Professor Smith during the Fall '10 term at Oregon Tech.

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