5 Steps to a 5 AP Calculus BC 2019

More Applications of Definite Integrals 319

13.4 Differential Equations

Main Concepts:Exponential Growth/Decay Problems, Separable Differential

Equations

Exponential Growth/Decay Problems

1. If
   dy
   dx
   =ky, then the rate of change ofyis proportional toy.


2. Ifyis a differentiable function oftwithy> 0
   dy
   dx
   =ky, theny(t)=y 0 ekt; wherey 0 is
   initial value ofyandkis constant. Ifk>0, thenkis a growth constant and ifk<0,
   thenkis the decay constant.

Example 1---Population Growth

If the amount of bacteria in a culture at any time increases at a rate proportional to the

amount of bacteria present and there are 500 bacteria after one day and 800 bacteria after

the third day:

(a) approximately how many bacteria are there initially, and

(b) approximately how many bacteria are there after 4 days?

Solution:

(a)Since the rate of increase is proportional to the amount of bacteria present,

then:

dy

dx

=ky, whereyis the amount of bacteria at any time.

Therefore, this is an exponential growth/decay model:y(t)=y 0 ekt.

Step 1. y(1)=500 andy(3)= 800

500 =y 0 ekand 800=y 0 e^3 k

Step 2. 500 =y 0 ek ⇒ y 0 =

500

ek

= 500 e−k

Substitutey 0 = 500 e−kinto 800=y 0 e^3 k.

800 =(500)

(

e−k

)(

e^3 k

)

800 = 500 e^2 k⇒

8

5

=e^2 k

Take the ln of both sides :

ln

(

8

5

)

=ln

(

e^2 k

)

ln

(

8

5

)

= 2 k

k=

1

2

ln

(

8

5

)

=ln

√

8

5

.