Exercise $1$: Stencil operations & slicing

$f(x)=(\frac{x^2+4x+5}{x+2e^x}{)}^{2x}$

Hint: Remember the use of the lambda function.

# YOUR CODE HERE

raise NotImplementedError$()$

b$)$ Calculate the derivative of $f(x)$ with the central difference formula $($below$),$ save result in the variable fprime $.$

$f^{\text{'}}(x_i)$~~$\frac{f(x_{i+1})-f(x_{i-1})}{2x}$

# YOUR CODE HERE

raise NotImplementedError$()$

We add to $y$ a scaled noise as follows

# RUN THIS CELL. DO NOT MODIFY.

numpy. random. seed $(42)$

numpy. random. rand $(y*size)**2-1$

fprime$\_$noise $=$ numpy.gradient $(yn,0\mathrm{.}001)$

c$)$ Next step is to compute smoothed values of the noisy data $($yn$)$ by using the average of a point with its two nearby neighbors.

Use slicing to compute the following, save your result in a variable called y$\_$smooth $.$

$y_{\text{s}\text{m}\text{o}\text{o}\text{t}\text{h}}=\frac{y_{i-1}+y_i+y_{i+1}}{3}$

# YOUR CODE HERE

raise NotImplementedError$()$

d$)$ Using an average of $5$ points as follows: