Problem on lagrange interpolation polynomial

Question

Assignment:

Q1. Expand the following function into Maclaurin Series y =tan^-1x using properties of the power series.

Q2. The Lagrange interpolation polynomial may be compactly written as is a shape function. Sketch the shape function in a graphic form.

Q3. Write a forward and backward difference Newton's interpolation formulas based on the following data. Interpolate the values of y at x = 1.3, 2.5, and 2.7.
 
x       y
1   1.302
2   1.616
3   1.954

Q4. Find the root of the following transcendental equation by the numerical method of your choice 4x - cosx = 0 (1)

Q5. Find the following integral by the trapezoidal rule and by the Simpson's rule. Find the theoretically predicted errors. Compare to the exact value. Explain the results.

∫²₁(2x + x³)dx

Provide complete and step by step solution for the question and show calculations and use formulas.