dy x a. Sketch the slope field for the points: (1,±1), (2,±1), ( 1, ±1), and (0,±1).

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1 Chapter 6. d x Given the differential equation: dx a. Sketch the slope field for the points: (,±), (,±), (, ±), and (0,±). b. Find the general solution for the given differential equation. c. Find the solution of this differential equation that satisfies the initial condition ( 0).

2 Chapter 6 d x. Given the differential equation: dx a. Sketch the slope field for the points: (,±), (,±),(, ±), and (0,±). + slopes of 0 along - axis + relative steepness and correct slant of others b. Find the general solution for the given differential equation. d x dx d d xdx xdx + + x C x k x + k c. Find the solution of this differential equation that satisfies the initial condition ( 0) 8 k (0) x + k correct antiderivatives + correct use of constant of integration + correct solution for + correct substitution of point (0, ) + correct solution

3 Chapter 6. Given the differential equation: d dx x. a. Sketch the slope field for the points (,±), (,±), and (-,±). b. Find the general solution for the given differential equation. c. Find the solution of this differential equation that satisfies the initial condition ( )

4 Chapter 6 4. Given the differential equation: d dx x. a. Sketch the slope field for the points (,±), (,±), and (-,±). + relative steepness in quadrants I & IV + negative slopes in quadrants II & IV; positive slopes in quadrants I & III b. Find the general solution for the given differential equation. d dx x d dx x d dx x ln x + C ln x + k ± ln x + k c. Find the solution of this differential equation that satisfies the initial condition (). 4 k ln + k ln x correct antiderivatives + correct use of constant of integration + correct solution for + correct substitution of point (, ) + correct solution

5 Chapter 6 5. Given the differential equation x. a. Use Euler s Method to find the first three approximations of the particular solution passing through (,). Use a step of h 0. b. Find the general solution of the differential equation. c. Find the particular solution of the differential equation that passes through (, ).

6 Chapter 6 6. Given the differential equation x a. Use Euler s Method to find the first three approximations of the particular solution passing through (, ). Use a step of h 0. () (.) (.4) b. Find the general solution of the differential equation. d dx x d x dx d x x x dx + C + k ± x + k c. Find the particular solution of the differential equation that passes through (, ). 4 + k k () x + k + + for. + for.48 + for.650 +: separates variables +: correct antiderivatives +: constant of integration +: uses initial condition +: solves for

7 Chapter Given the differential equation: x a. Use Euler s Method to find the first three approximations of the particular solution passing through (, ). Use a step of h 0.. b. Find the general solution of the differential equation. c. Find the particular solution of the differential equation that passes through (, ).

8 Chapter Given the differential equation: x a. Use Euler s Method to find the first three approximations of the particular solution passing through (, ). Use a step of h [()() ]. [(.)(.) ]. + 0.[ (.)(.) ]. 675 b. Find the general solution of the differential equation. d x dx d x dx ln e x Ae x x e C + C c. Find the particular solution of the differential equation that passes through (, ). Ae A e x e + for. + for. + for.68 +: separates variables +: correct antiderivatives +: constant of integration +: uses initial condition +: solves for

9 Chapter The number of bacteria in a culture is increasing according to the law of exponential growth. There are 50 bacteria in the culture after hours and 400 bacteria after 5 hours. a. Write an exponential growth model for the bacteria population. b. Identif the initial population and the continuous growth rate. c. Use logarithms to determine after how man hours the bacteria count will be approximatel 5,000

10 Chapter The number of bacteria in a culture is increasing according to the law of exponential growth. There are 50 bacteria in the culture after hours and 400 bacteria after 5 hours. a. Write an exponential growth model for the bacteria population. 50 Ce 50 Ce k Therefore, e 8 k e 8 ln k k e ( ) So, C ; 400 Ce t 5k k e b. Identif the initial population and the continuous growth rate. The initial population is approximatel and the continuous growth rate is 49.04%. c. Use logarithms to determine after how man hours the bacteria count will be approximatel 5,000? e e.4904t ln t t.9 hours.4904t 5k + correct value for k + correct value for C + exponential equation for + initial population growth rate 49.04% + set 5,000 + correct use of logs + correct answer

11 Chapter 6 6. The number of bacteria in a culture is increasing according to the law of exponential growth. There are 7,768 bacteria in the culture after 5 hours and 6,576 bacteria after 8 hours. a. Write an exponential growth model for the bacteria population. b. Identif the initial population and continuous growth rate. c. After how man hours will the bacteria count be approximatel 5,000? Use logarithms to find our answer.

12 Chapter 6 6. The number of bacteria in a culture is increasing according to the law of exponential growth. There are 7,768 bacteria in the culture after 5 hours and 6,576 bacteria after 8 hours. a. Write an exponential growth model for the bacteria population Ce.4 e 7768 Ce C e 5k Therefore, e k ln.4 k; ; t 6576 Ce ( 5) 8k 5k e 8k so, k b. Identif the initial population and continuous growth rate. The initial population is approximatel 000 bacteria and the continuous growth rate is 4%. c. After how man hours will the bacteria count be approximatel 5,000? Use logarithms to find our answer. +: correct value for k +: correct value for C + exponential equation for +: initial population is 000 +: growth rate is 4% +: set 5,000 +: correct use of logs + correct answer e 5 e.4t.4t ln 5.4t t 7.85 hours

13 Chapter 6 7. The rate of the spread of an epidemic is jointl proportional to the number of infected people and the number of uninfected people. In an isolated town of 4000 inhabitants, 60 people have a disease at the beginning of the week and 00 have it a week later. a. Write a model for this population. b. How man will be infected after 0 das? c. How long does it take for 80% of the population to become infected?

14 Chapter The rate of the spread of an epidemic is jointl proportional to the number of infected people and the number of uninfected people. In an isolated town of 4000 inhabitants, 60 people have a disease at the beginning of the week and 00 have it a week later. a. Write a model for this population. d k( )( 4000 ) K( ) dx 4000 d K dt ln d K dt 4000 Kt C Ae Kt Kt K ( ) A Ae + Ae K ( ) K e t + 4e b. How man will be infected after 0 das? ( 0) + 4e Approximatel 5 people will be infected after 0 das. c. How long does it take for 80% of the population to become infected? t + 4e 0.965t + 4e t ln 4 t.7 It will take approximatel 4 das for 80% of the population to become infected. +: k( )(4000 ) +: separation of variables +: correct solution for +: substitution for t 0 +: answer +: substitution for 00 +: answer with units

15 Chapter Given the logistic equation: d ( ) dt 5 a. Determine the general solution of this equation. b. Find the particular solution satisfing ( 0). lim t c. Evaluate and explain its meaning.

16 Chapter 6 6 d 8. Given the logistic equation: ( ) dt 5 a. Determine the general solution of this equation. d dt 5 d dt 5 d dt 5 + d 5 5 ln 5 e t C t C dt 5 Be t 5 t + Be b. Find the particular solution satisfing ( 0). 5 ( 0) + Be + B 5 ( ) B 5 t + e c. Evaluate and explain its meaning. lim t t 5 Since e 0 as t, lim 5. t + e t This represents the carring capacit of the population modeled b the given logistic differential equation. +: separation of variables +: antiderivatives +: constant of integration +: solution for +: use of initial condition +: answer +: limit equals 5 +: explanation

17 Chapter A 00-gallon tank contains 00 gallons of water with a sugar concentration of 0. pounds per gallon. Water, with a sugar concentration of 0.4 pounds per gallon, flows into the tank at a rate of 0 gallons per minute. Assume the mixture is mixed instantl and water is pumped out at a rate of 0 gallons per minute. Let (t) be the amount of sugar in the tank at time t. a. Set up the differential equation which models the given information. b. Solve the differential equation for (t). c. What is the amount of sugar in the tank when it overflows?

18 Chapter A 00-gallon tank contains 00 gallons of water with a sugar concentration of 0. pounds per gallon. Water, with a sugar concentration of 0.4 pounds per gallon, flows into the tank at a rate of 0 gallons per minute. Assume the mixture is mixed instantl and water is pumped out at a rate of 0 gallons per minute. Let (t) be the amount of sugar in the tank at time t. a. Set up the differential equation which models the given information. Rate of sugar coming in 0.4lb 0 gal min 8lb gal min ( )( ) Rate of sugar going out lb 0 gal lb 00 0t gal min t min d Therefore, 8 dt 0 + t b. Solve the differential equation for (t). d + 8 dt 0 + t α( t) e 0+ t ( t) t dt e + 0 ln ( t+ 0) t + 0 ( t + 0)( 8) 00 ( t) 4t + 40 t + 0 c. What is the amount of sugar in the tank when it overflows? Since the tank contains t gallons at time t, when the tank overflows t 00. Therefore, the tank overflows in 0 minutes. 00 (0) 4(0) ( 0) pounds of sugar are in the tank when it overflows. dt +: rate of sugar coming in of 8 lb/min +: rate of sugar going out of lb / min 0 + t +: integrating factor of t +0 +: solution for (t) +: setting t 00 +: answer

19 Chapter A 00-gallon tank contains 00 gallons of water with a salt concentration of 0. pounds per gallon. Water, with a salt concentration of 0. pounds per gallon, flows into the tank at a rate of 0 gallons per minute. Assume that the mixture is mixed instantl and water is pumped out at a rate of 0 gallons per minute. a. Set up the differential equation which models this information. b. Solve the differential equation for (t). c. What is the amount of sugar in the tank when it overflows?

20 Chapter A 00-gallon tank contains 00 gallons of water with a salt concentration of 0. pounds per gallon. Water, with a salt concentration of 0. pounds per gallon, flows into the tank at a rate of 0 gallons per minute. Assume that the mixture is mixed instantl and water is pumped out at a rate of 0 gallons per minute. a. Set up the differential equation which models this information. Rate of salt coming in ( 0.lb )( 0 gal min) 6lb gal min Rate of salt going out t lb gal 0 gal lb min 0 + t min d Therefore, 6 dt t + 0 b. Solve the differential equation for (t). d + 6 dt 0 + t α ( t) e dt 0+ t e ln ( t+ 0) ( t) + t + 0 t + 0 ( t 0)( 6) dt Therefore, ( t) t + 60 t c. What is the amount of salt in the tank when it overflows? Since the tank contains t gallons at time t, when it overflows t 00. Therefore, the tank overflows in 0 minutes. 400 (0) (0) (0) 0 pounds of salt are in the tank when it overflows +: rate of salt coming in of 6 lb/min +: rate of salt going out of min t + 0 lb / +: integrating factor of t + 0 +: solution for (t) +: setting t 00 +: answer

( + ) 3. AP Calculus BC Chapter 6 AP Exam Problems. Antiderivatives. + + x + C. 2. If the second derivative of f is given by f ( x) = 2x cosx

( + ) 3. AP Calculus BC Chapter 6 AP Exam Problems. Antiderivatives. + + x + C. 2. If the second derivative of f is given by f ( x) = 2x cosx Chapter 6 AP Eam Problems Antiderivatives. ( ) + d = ( + ) + 5 + + 5 ( + ) 6 ( + ). If the second derivative of f is given by f ( ) = cos, which of the following could be f( )? + cos + cos + + cos + sin