1. ** Question 1 - Plotting the Legendre polynomial **

The Mathematica code I used to plot the Legendre polynomial P_{10}(x) is the following:

And the received graph:

2.** Question 2 - Calculating the Legendre polynomials for x=0.66 **

The Mathematica code I used to calculate and plot the values of the Legendre polynomial and its derivative for x=.66 and n=1,2,3,....10 is the following:

And the received graph:

Where the blue dots represent the Legendre polynomials values at x=0.66 and the orange dots represent their derivatives at the same point.

3.** Question 3 - Calculating an integral **

This integral can be easily calculated using Mathematica code:

*Integrate[(1 + t)^0.5,{t,0,3}]*

And the received answer is 4.66667.

We shall note that this integral can be also solved analytically which is always more accurate:

The Mathematica code I used to plot the Legendre polynomial P

And the received graph:

2.

The Mathematica code I used to calculate and plot the values of the Legendre polynomial and its derivative for x=.66 and n=1,2,3,....10 is the following:

And the received graph:

Where the blue dots represent the Legendre polynomials values at x=0.66 and the orange dots represent their derivatives at the same point.

3.

This integral can be easily calculated using Mathematica code:

And the received answer is 4.66667.

We shall note that this integral can be also solved analytically which is always more accurate: