1) Calculate the following for the function f(x) = e^-3x - 2x.

a. Calculate the derivation of the function by hand. Write a MATLAB function that calculates the derivative of this function and calculate the derivative at x = 0.5.

b. Develop an M-to evaluate the centered finite-difference approximations, at x = 0.5. Assume that dx = 0.1.

c. Repeat part (b) for the second-order forward and backward differences. Again Assume that dx = 0.1.

d. Using the results obtained in parts b and C, calculate the percentage error between actual value and the results of (b) and (c).

Use the following formulas:

Centered diff. df/dx = (f(i+1) -  f(I - 1))/2dx

Fwd. diff. df/dx  = (-f(i+2) + 4f (i+1)- 3f(i))/2dx

Bkwd. diff. df/dx = (3f(i)-4f(i-1)+f(i-2))/2dx

2) a. Use the three-point Gauss-Legendre formula to evaluate the following function between 0 and 0.8

f(x) = 3x + 5x²

Remember that you have to make a change of variable to make the integration between -1 to 1. Do all your calculations in MATLAB.

b) Calculate the value of the integral using a MATLAB built in function of your choosing.

3) Use any Matlab built-in function to solve the following differential equation in the range of t = 0 to 10 and plot the solution.

y' = 4e^0.8t - 5y,   y(0) = 2.