14.18. Numeric Methods Homework¶

Save all of the commands entered into MATLAB and the output to a text file to submit on Canvas.

Use the fzero function to find where $f(x) = 5 + 10x - x^2$ is equal to zero in the range $-5 \leq x \leq 15$ .
Use the following code to make 8 $(x, y)$ points. Then use the interp1 function to expand the data to 32 data points. Generate a $2{\times}2$ subplot with the smooth function, 8 point plot, 32 point plots with linear and pchip interpolation.
```
>> f = @(x) cos(x) + 0.7*sin(x);
>> x = linspace(-pi,pi,8);
>> y = f(x);
>> plot(x,y)
```
Using the trapezoid rule (20 subintervals), Simpson rule (20 subintervals), the adaptive integral algorithm presented in Recursive Adaptive Integral, and MATLAB’s integral function compute the definite integral $\int_0^{10} x^2 e^{-x} dx$
Use the anonymous function f = @(t,y) t.^2 .* exp(-t); let $y'(t) = f(t)$ , $y(0) = 0$ define a differential equation. Use ode45 to plot $y(t)$ , $0 \leq t \leq 20$ .