Mathematics

Problem

A population

P(t)

(measured in hundreds of organisms) satisfies

\dfrac{dP}{dt}=0.03P\left(1-\dfrac{P}{50}\right)

with

P(0)=10

. A numerical algorithm will approximate

P(10)

using a uniform step size

h

over the time interval

0\le t\le10

. Experience with this algorithm on similar problems shows that the global error at

t=10

does not exceed

4h

. You must design a computation whose approximation of

P(10)

is guaranteed to be within

0.02

of the exact value. a. Determine the largest step size

h

that meets the stated error requirement. Then state the corresponding minimum number of time steps that must be carried out over

0\le t\le10

. Show all work clearly.