Etra Practice Recovering C 1 Given the second derivative of a function, integrate to get the first derivative, then again to find the equation of the original function. Use the given initial conditions to help find C. 1) y 4 1 Find y if Find y if y 7 y = y 7 y = ) d y 1 d Find y if y 1 y = Find y if y 1 7 y = ) y sin Find y if y y = Find y if y() y = 4) d y 1 d Find y '' if y '' 1 y '' = Find y if Find y if y1 y = y 1 1 y = 5) y' e Find y if yln y = 6) y 1 1 Find y if y 85 y = Find y if y1 15 y =
Review of Section 6.1 1. Solve the initial value problem: 4 d, y(). Solve the initial value problem: y 4 1, y(1) 1, y() 18.. Solve the differential equation: d y 4. Solve the differential equation that satisfies the initial condition: yy e, y() 4 Sketch the slope field of the differential equation. Then use it to sketch a solution curve that passes through the given point. 5. 1, (-1,-1) 6. d y y, (1,)
Slopefield Drawing Practice
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6. Consider the differential equation y 1 d a) On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated. 5 b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the y-plane. Describe all points in the y-plane for which the slopes are positive. c) Find the particular solution y= f() to the given differential equation with the initial condition f() = 7. Let f be the function that is defined for all real numbers and that has the following properties. (i) f // () = 4 18 (ii) f / (1) = -6 (iii) f() = a. Find each such that the line tangent to the graph of f at (, f()) is horizontal. b. Write an epression for f(). c. Find the average value of f on the interval 1 < <.
6 NAME # U-Substitution Puzzle (Section 6.) Solve the following problems, then match the answers with the letters below. It should spell a message! You may or may not use u-substitution. 1... 4 ( 1) d 1. V) ( 1) e d. T) ln 1e 9 d. T) 4 C 1 8 / (9 ) d 4. L) C ( 4) 4. 5. 6. 4( 1) d 5. I) C 6 8/ C 6 ( ) C 4 4 (9 ) 6. N) ln 1 C 7. 18 cos( ) d 7. O) sin e C 1 sin( 4 ) d 8. I) 8. sin( ) C 5 5 9. d 9. E) 1 tan() C 1 1 1. sec ( ) d 1. R) sin C 4 4 5 11. 6sin (9 ) cos d 11. E) C 5 ( 1) 1. d 1. W) C 6 16 15 4 1. tan ( 4 )sec ( ) d 1. N) cos( ) C e 1 14. d 14. E) e C 1 e 5 15. ( ) d 6 15. G) sin C sin 1 16. cos e d 16. O) C 4( 4) 1 17. (1 ) d 17. A) 1 tan 5 ( ) C 15
7 Using U-Substitution in Separating Variables Find the general solution to each differential equation. If you are given an initial condition, find the specific solution that fits the given information. You may or may not use U-Substitution. 1. e 11. ( 1) ' d y y when y ( ) 1 tan 1tan sec. d HINT:. yy ' sin 4. 14 y ' d y cos y (cos ) y e d 4 yln d 5. 6. sin 7. du dv when ye ( ) 1 8. uvsin( v ) when u () 1 dr ds rs 9. e when r() 1. y(1 ) y' (1 y ) when y() 1 1. 1 ( 1) DAY 5 HW--Change the limits and evaluate in terms of u : d 6. 4 1 t( t 1) dt t ( 9) dt t 7. 4.. /6 sin cos d 8. 4. 1 cos( ) d 5. 1 t 1dt 5 (9 ) 9 d 9. / / sin 4 5 sin(cos ) d 1. cos( ) d d
NOTES: Eponential Growth and Decay 8 A quantity y increasing or decreasing at a constant rate proportional to the amount currently present can be written as a differential equation ky dt, with k as a constant and y y at t =.. E1) Write a differential equation to show the rate of change of N with respect to t is proportional to 5-t. Solve this for N at any time, t. E) Write a differential equation to show the rate of change of P with respect to t is proportional to ½ of P. Solve this for P at any time, t. E) Write a differential equation to show that a population grows at a rate of % per year (assume time, t is in years). Solve this for P at any time, t. E4) Value of a car decreases at a rate of 1% per year. Write a differential equation to model this. Solve for the value of the car at any given time. CLASSWORK 1. (a) Write a differential equation (one with a dm dt ) that shows a money increases at a rate of 5% per year. (b) Solve this for M, the money at any given time.. Earth s atmospheric pressure p is often modeled assuming that the rate dp at which p dh changes with the altitude h above sea level is proportional to p. a) Write a differential equation that shows this. b) Solve this equation for p, the pressure at any time. (Call p the pressure at sea level). c) Find the constant k, if the pressure at sea level is 11 millibars and the pressure at km is 9 millibars. d) My grandmother, who has trouble breathing, has been told she needs to stay where air pressure is more than 5 millibars. How high up can she go in the mountains, meeting this restriction?. The rate of change of y with respect to is proportional to 1-. a) Write the equation that shows this. b) Solve this equation for y.
Growth and Decay Word Problems 1. The rate at which a population increases is proportional to the number of people in the population, dp kp. In 19, the population was 5,. dt (a) Write an equation for the population in terms of k and t. 9 (b) If the population in 194 was 154,, find k. (c) If growth continued like this, what would the population be in the year 1? Round to the nearest person.. Let y represent the temperature (in F) of an object in a room whose temperature is kept at a constant 6. Newton s Law of Cooling says that the rate of change in y is proportional to the difference between y and 6. This can be written as k( y 6) dt when 8 y 1. (a) Solve this equation for y in terms of k and t. (b) Find C in the equation, given that the food is 1 when it comes out of the oven. (c) If the object cools from 1 to 9 in 1 minutes, how MUCH LONGER will it take for its temperature to decrease to 8?
Free Response Practice 1 Calculator Active 199-AB6 Let Pt () represent the number of wolves in a population at time t years, with t. The population Pt () is increasing at a rate directly proportional to 8 Pt ( ), where the constant of proportionality is k. (a) If P() 5, find Pt () in terms of t and k. (b) If P() 7, find k. (c) Find lim Pt ( ). t Non-Calculator 1997AB/BC-6 Let vt () be the velocity, in feet per second, of a skydiver at time t seconds, t. After her dv parachute opens, her velocity satisfies the differential equation v, with the initial dt 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?