10. (a) Produce the linear Taylor polynomials to f(x) = In (x) on 1 < x < 2, expanding about x, = }. Graph the error.

If the sign combination of f' and f'' changes from ++ to +− at x = c, then (choose the correct answer):(a) f (c) is a local min. (b) f (c) is a local max.(c) f (c) is a point of inflection.

What is the residue of the function f(z) = (z^2 + 1) / (z^3 - 2z) at the point z = i?

22. Find f/(x) and simplify 1 -1 a) f(z) 3-52+1 3 b) f(x) 1.8Vc) f(x) = 32

A particular finite difference formula for the first derivative of a funtion is: f '(xi) = (-f(xi+3) + 9 f(xi+1) - 8 f(xi))/6h where the points xi, xi+1, xi+2, and xi+3 are all equally spaced with step size h. What is the order of the truncation or descretization error?

Determine the interpolating polynomial for f(x)=x^(2)+sin\pi x through (0,0);(1,1) (2,4). What is the error when x=(1)/(2) ?

For a positive integer n, prove that summation d/n mu^(2)(d) phi(d) = n / phi(n). [Hint: Both sides of the equation are multiplicative functions].

Let f(x) = x^4 + 3x^3 − 4x^2 + 3x − 2.Find f(1.6) and f'(1.6) by Horner’s method using a table form (with the help of a calculator).

Ex. 5) For R and Q functions and given that: Q(-1) = 5, R(-1) = -4, R(5) = 7; || Q'(-1) =-3, R'(-1) =2, R'(5) =-1 If f(x) = R(Q(x)), then evaluate f"(-1). f"(-1)=

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