Question

1. (calculator not allowed)

eulers method and logistic growth (bc only)
student study session
assume that ( f ) and ( f ) have the values given in the table. use eulers method with two equal
steps to approximate the value of ( f(4.4) ).

Explanation:

Step1: Calculate the first - step approximation

Euler's method formula is \(y_{n + 1}=y_{n}+f^{\prime}(x_{n})\Delta x\).
Here, \(x_{0}=4\), \(y_{0}=f(4) = 2\), \(\Delta x=\frac{4.4 - 4}{2}=0.2\).
For \(n = 0\):
\(y_{1}=y_{0}+f^{\prime}(x_{0})\Delta x\)
\(y_{1}=2+(- 0.5)\times0.2\)
\(y_{1}=2 - 0.1=1.9\)

Step2: Calculate the second - step approximation

Now, \(x_{1}=4.2\), \(y_{1}=1.9\)
For \(n = 1\):
\(y_{2}=y_{1}+f^{\prime}(x_{1})\Delta x\)
\(y_{2}=1.9+(-0.3)\times0.2\)
\(y_{2}=1.9 - 0.06=1.84\)

Answer:

\(1.84\)