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

1. (a) \(y(1)\approx1.38,\ y(2)\approx6.25\)
2. (b) \(y(1)\approx1.66,\ y(2)\approx7.56\)
3. (c) \(y=x^3+1\); furthest: the \(h=0.5\) estimate of \(y(2)\) (6.25 against 9)

1. (a) \(y(2)\approx1.90\)
2. (b) \(y=6e^{-0.5x},\ y(2)=2.21\)
3. (c) \(14\%\)
4. (d) Use a smaller step length (for example \(h=0.25\)).

\(y(2)\approx0.779\)

1. (a) \(y(1)\approx0.883\)
2. (b) \(y=\dfrac{1}{2-x},\ y(1)=1\)
3. (c) The solution curve is concave up, so each tangent line lies below the curve and the estimate is too small.

\(y(8)\approx2.97\)

11 weeks

\(x\approx110,\ y\approx31.8\)

1. (a) 33.3 m
2. (b) Too small.

1. (a) Starts at \((0,3)\), then a minimum of about 2.47 at \(x=0.7\), then a maximum of about 3.1 at \(x=2.3\), then a minimum of about −1.6 at \(x=5.9\), and ends at \(y=1.06\) when \(x=8\).
2. (b) \(y_{\max}\approx3.1\)