AP Exam Practice Questions for Chapter 6

Transcription

1 AP Eam Practice Questions for Chapter 6 AP Eam Practice Questions for Chapter 6. To find which graph is a slope field for, 5 evaluate the derivative at selected points. At ( 0, ),.. 3., 0,. 5 At ( ) At ( 5, 0 ),, So, the answer is B. dp dt kp dp P k dt ln P kt + C P kt Ce Let C when t 0. 3 e ln 3 5k ln 3 5 Use this and ( ) k 5k So, the answer is B. k k k ln k + C k Ce Use ( ) ( ) f 0 8 and f 6 to find C and k. k( 0 8 Ce ) C 8 k( 6 Ce ) 6k 8e 6k e 4 ln k 6 4 k Because ( 6) ln( 4) ( ) 5, 3 to find k. f Ce 8 e, the answer is A C 5. Use ( ) to find C. ( ) + C C ( ) So, the answer is C ln 3 ln + C ln 3 ln + C 3 C Use () to find C. () 3 C C ( ) The solution of the differential equation is 3. So, the answer is B. 08 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

2 AP Eam Practice Questions for Chapter 6 6. Evaluate each differential equation for selected values of. 0 A: When,. C: When,. 3 3 When 3, 0. When 3, When 4,. When 4,. 3 3 B: When,. D: When 3 When 3, 0. When When 4,. When 3 So, the answer is A. 7. Use Euler s Method with ( 0) 3, and h ( ) 3 ( ) ( ) ( ) () ( ) So, the answer is A. to find (). 3 0, ,. 0 4, (a) dt dt 0.5 dt ln 0.5t + C 0.5t Ce Use ( 0, 00) to find C. 0.5( 0 00 Ce ) C t So, 00 e. Notes: 3 points ma if no constant of integration present 0 0 e dt e dt t e t 0.5t (b) ( 0.5 ) 5 ( e ) bacteria Notes: Be sure to write down the appropriate definite integral before numericall approimating it on our calculator. Be sure to round the answer to at least three decimal places to receive credit on the eam. Importing an incorrect particular solution frompart (a) can still earn the first two points in part (b), but not the answer point. 08 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

3 AP Eam Practice Questions for Chapter (a) Use Euler s Method with, f(), h to approimate f (.4 ). and 0. () f (.) ( ). f (.4) So, f ( ) Note: The answer does not need to be simplified. Leaving the answer as recommended. Be sure to write Because this is an approimation, a point ma be deducted if an equal sign is used. In general, equating two quantities that are not trul equal will result in a one point deduction on a free-response question. is (b) + + C C Use (, ) to find C. ( ) ( ) + C 4 + C C Because +, the solution is +, where the domain is (, ). Notes: points ma if no constant of integration present 08 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

4 4 AP Eam Practice Questions for Chapter 6 0. (a) Use Euler s Method,, f (), h to find f (. ). and 0. ( ) ( )( ) f (.). 0. (.)(.) f + f So, ( ) ( )( ) Notes: The answer does not need to be simplified. Leaving the answer as is recommended. Be sure to write Because this is an approimation, a point ma be deducted if an equal sign is used. In general, equating two quantities that are not trul equal will result in a one point deduction on a free-response question. (b) d + ( ) + + () On the interval [,. ], is positive. Because, and have the same sign on [,. ]. Because f() > 0, and are both positive on [,. ] and is increasing. Therefore, d > 0, so the solution is concave upward on this interval and an tangent line approimation will be below the curve. So, the approimation f.. found in part (a) is less than ( ) (c) ln + C ( ) C e e ( ) Ce Use (, ) to find C. ( )( Ce ) Ce C e So, the solution is ( ) ( ) e e e e Notes: points ma if no constant of integration present 08 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

5 AP Eam Practice Questions for Chapter 6 5. (a) d ( 0) + ( ) At the point ( ) + ( ) ()( ) d 5,,. 3 8 pts: implicit differentiation to find (in terms of and ); evaluates (b) Find at the point ( ),. ()( ) So, the equation of the tangent line is ( ) f (.) (.) +.05 Because f (.) < 0, the approimation f (.).05 is greater than f (. ). (c) ln + C Use f () to find C. ( ) ln ( ) + C 0 + C C So, the solution is ln + ln + 4 ln + 4. Notes: points ma if no constant of integration present 08 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

6 6 AP Eam Practice Questions for Chapter 6. (a) (0, 40) (4, 00) t pts: solution curves through given points (following slope field) (b) 0.9 dt 00 f 0 40, h 0.5, and Euler s Method to Use ( ) approimate f (). 40 f ( 0.5) ( 40) f () ( 8.4) So, f () Note: Be sure to write rather than Because this is an approimation, a point ma be deducted if an equal sign is used. In general, equating two quantities that are not trul equal will result in a one point deduction on a free-response question. (c) When t 0, the range of f is ( 00, 40 ). pts: answer 08 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

7 AP Eam Practice Questions for Chapter (a) At 00, At 00, dt ( 00) dt ( 00) Because 9 when 00 is less than 6 dt dt when 00, the disease is spreading faster when 00 people have the disease. Note: To avoid the risk of an arithmetic mistake, these answers do not need to be simplified. (b) k, where 000 dt L L and k. 0 So, L 000. kt ( 0 ) t + Ce + Ce Use ( 0, 00) to find C ( 0)( 0 + Ce ) + C 0 C 9 So, a model for the population is 000. ( 0 9 ) t + e Note: It is ver unlikel that ou would be asked to actuall solve a logistic differential equation on a free-response question. (This equation is separable, but finding its solution involves a partial fraction decomposition.) However, ou ma be asked to approimate a solution to a logistic differential equation using Euler s method, a Talor polnomial, or a slope field (c) lim t () lim 000 t t ( 0 9 ) t + e + 0 pt: answer 08 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.

8 8 AP Eam Practice Questions for Chapter 6 4. (a) pts: slopes of line segments (b) d + () pts: implicit differentiation to find (in terms of and ) (c) C Use ( 0) to find C. 3 8 ( ) ( 0) + C C 3 3 So, the solution is Notes: points ma if no constant of integration present 08 Cengage Learning. All Rights Reserved. Ma not be scanned, copied or duplicated, or posted to a publicl accessible website, in whole or in part.