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 graphof f at (x, f(x)) equal to 3?

2

A) 0.168B) 0.276C) 0.318D) 0.342E) 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 graphof f at (x, f(x)) equal to 3?

2

A) 0.168B) 0.276C) 0.318D) 0.342E) 0.551

(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.

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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 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 ⇒∞

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

a) If P(0) = 500, find P(t) in terms of t and k.

P'(t) = k(800 - P(t))

-ln|800 - P| = kt + Cln|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

800 - 500 = Ce0

300 = C

P(t) = 800 - 300e-kt

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

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

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c) Find lim P(t). t ⇒∞

c) Find lim P(t). t ⇒∞