1. Problem of the Day - Calculator Let f be the function given by f(x) = 2e4x. For what value of x is the slope of the line tangent to the graph of f at (x, f(x)) equal to 3? 2 A) 0.168 B) 0.276 C) 0.318 D) 0.342 E) 0.551

2. Problem of the Day - Calculator Let f be the function given by f(x) = 2e4x. For what value of x is the slope of the line tangent to the graph of f at (x, f(x)) equal to 3? 2 A) 0.168 B) 0.276 C) 0.318 D) 0.342 E) 0.551 (Graph derivative and find where y = 3)

3. You have learned to analyze visually the solutions of differential equations using slope fields and to approximate solutions numerically using Euler's Method. You have solved equations of the form y' = f(x) and y'' = f(x) Now you will learn to solve using the separation of variables method.

4. Separation of Variables Method Rewrite equation so that each variable occurs on only one side of the equation. -

5. Growth and Decay Application of separation of variables where rate of change of y is proportional to y Cekt

7. Find the particular solution for t = 3 if the rate of change is proportional to y and t = 0 when y = 2, and t = 2 when y = 4.

8. Find the particular solution for t = 3 if the rate of change is proportional to y and t = 0 when y = 2, and t = 2 when y = 4.

9. At t = 3

10. Let P(t) represent the number of wolves in a population at time t years, when t > 0. The population P(t) is increasing at a rate directly proportional to 800 - P(t), where the constant of proportionality is k. a) If P(0) = 500, find P(t) in terms of t and k. b) If P(2) = 700, find k. c) Find lim P(t). t ⇒ ∞

11. Let P(t) represent the number of wolves in a population at time t years, when t > 0. The population P(t) is increasing at a rate directly proportional to 800 - P(t), where the constant of proportionality is k. a) If P(0) = 500, find P(t) in terms of t and k. implies

12. a) If P(0) = 500, find P(t) in terms of t and k. P'(t) = k(800 - P(t)) -ln|800 - P| = kt + C ln|800 - P| = -kt + C |800 - P| = ekt + C |800 - P| = ekt eC |800 - P| = Cekt.

13. a) If P(0) = 500, find P(t) in terms of t and k. |800 - P| = Cekt 800 - 500 = Ce0 300 = C P(t) = 800 - 300e-kt

14. b) If P(2) = 700, find k.

15. b) If P(2) = 700, find k. - - - - - - - - -

16. c) Find lim P(t). t ⇒ ∞

17. c) Find lim P(t). t ⇒ ∞