Solve a polynomial (search the roots and factorize it) using the method of substitution.

Given the polynomial obtain the roots of it.

Calculating the solutions of an equation.

Show that there are exactly two solutions of the following equation, and find them (where ln(x) is the natural logarithm function) ln(2x squared - 1) - 2ln(2 + x) = 0

Working with iterative methods.

The sequence of value given by the iterative formula is X n+1=2/9-lnXn with initial value X=1,converges to alpha. 1) State an equation satisfied by alpha, and hence show that alpha is the x coordinate of a point on the curve where y=3. 2) Use this iterative formula to find alpha correct to 2 decimal places, showing each iteration.

Maclaurin Series

Given The Maclaurin series for the inverse hyperbolic tangent is of the form x+x^3/3+x^5/5...x^7/7. Show that this is true through the third derivative term.

Differentiation of Polynomials : Proofs

Please see the attached file for the fully formatted problems. Let g be a function which can be differentiated four times on the interval [-1,1]. Denote . 1) Show that when g is a polynomial of degree less than or equal to 3. 2) Let P be the interpolation polynomial of f at the points -1, , , 1. a) Show that . b) Show that , where and is a constant which you will evaluate. c) Deduce a number which is greater than or equal to the error . 3) Let f be a function which can be differentiated four times on an interval [a,b]. Let . Using x, show that the integral f can be calculated on [a,b] with the help of an integral on [-1,1]. 4) Deduce an approximation of . 5) U... click for more