Euler's method: accuracy and contexts Problem set B Answers

1. (a) \(y(2)\approx8,\ y(4)\approx86\)
2. (b) \(y(2)\approx12.5,\ y(4)\approx107\)
3. (c) \(y=2x^3+2\); furthest: the \(h=1\) estimate of \(y(4)\) (86 against 130)

1. (a) \(y(4)\approx19.5\)
2. (b) \(y=8e^{0.25x},\ y(4)=21.7\)
3. (c) \(10.2\%\)
4. (d) Use a smaller step length (for example \(h=0.5\)).

\(y(2)\approx1.85\)

1. (a) \(y(1)\approx0.442\)
2. (b) \(y=\dfrac{1}{4-2x},\ y(1)=0.5\)
3. (c) The solution curve is concave up, so each tangent line lies below the curve and the estimate is too small.

\(y(10)\approx5.71\)

6 days

\(x\approx40.9,\ y\approx29.1\)

1. (a) 582 m
2. (b) Too large.

1. (a) Starts at \((0,2)\), then a minimum of about 1.73 at \(x=0.4\), then a maximum of about 3.53 at \(x=2.4\), then a minimum of about −2.51 at \(x=5.8\), and ends at \(y=-2.46\) when \(x=6\).
2. (b) \(y_{\max}\approx3.5\)