5 Steps to a 5 AP Calculus BC 2019

330 STEP 4. Review the Knowledge You Need to Score High

Step 1: The rate of spread is

dP

dt

=kP

(

1 −

P

2000

)

.

Step 2: The solution of the differential equation isP(t)=

2000

Ae−kt+ 1

, and with one person

exposed,A=

2000 − 1

1

=1999, orP(t)=

2000

1999 e−kt+ 1

.

Step 3: Taking April 10 as day 10, P(10)= 500 =

2000

1999 e−k(10)+ 1

. Solving the equation

givesk≈.6502, soP(t)=

2000

1999 e−.^6502 t+ 1

.

Step 4: 98% of the population of 2000 is 1960 people. To determine the day when 1960

people are infected, solve 1960=

2000

1999 e−.^6502 t+ 1

. This givest≈ 17 .6749, so the
98% infection rate should be reached by April 18.

13.7 Euler’s Method

Main Concepts:Approximating Solutions of Differential Equations by Euler’s

Method

Approximating Solutions of Differential Equations by Euler’s

Method

Euler’s Method provides a means of estimating the numerical solution of differential equa-

tions by a series of successive linear approximations. Represent the differential equation by

y′=f′(x,y) and the initial conditiony 0 =f(x 0 ), and choose a small value,Δx, as the incre-

ment between estimates. Begin with the initial valuey 0 , and evaluatey 1 =y 0 +Δx·f′(x 0 ,y 0 ).

Continue withy 2 =y 1 +Δx·f′(x 1 ,y 1 ), and in general,yn=yn− 1 +Δx·f′(xn− 1 ,yn− 1 ).

Example 1

Given the initial value problem

dy

dt

=cos 2πtwithy(0)=1, approximatey(1), using five

steps.

Step 1: The interval (0, 1) divided into five steps gives usΔt= 0 .2.

Step 2: Create a table showing the iterations.

tycos 2πtyn=yn− 1 +Δx· f′(xn− 1 ,yn− 1 )

01 1 1 +.2(1)= 1. 2

0.2 1.2 .309016 1. 2 +.2(.309016)= 1. 261803

0.4 1.261803 −. 809016 1. 261803 +.2(−.809016)= 1. 1

0.6 1.1 −. 809016 1. 1 +.2(−.809016)= 0. 938196

0.8 .938196 .309016. 938196 +.2(.309016)= 1

11

y(1)≈ 1