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,

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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) B) C) D) E) 0.551

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) B) C) D) E) (Graph derivative and find where y = 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.

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

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

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.

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.

At t = 3

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 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 ⇒ ∞

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 P(t), where the constant of proportionality is k. a) If P(0) = 500, find P(t) in terms of t and k. implies

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.

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

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

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

c) Find lim P(t). t ⇒ ∞

c) Find lim P(t). t ⇒ ∞