Question

Solve part a and b for the below question

f(1) = 1, f(2) = 2, f(4) = 1. (a) Use the idea outlined in Q2 of tutorial 5 to determine an approximate values of the first f'(2) and the second derivatives f"(2) at point x = 2. (b) Which additional information about the function should be given so that the third derivative f''(2) can be estimated?

Extra information for part A of the question.

Below is Question 2 from Tutorial 5

Below is answer for question 2 from tutorial 5

Answer 1

(a) We can frame equations using Taylor expansion in exactly the same way. For the point x = 4,

f(4) = 1 = f(2) + (4 - 2)f'(2) + (4-2)'f"(2) + ...

$2f'(2)+4f''(2)=-1~~...(1)~\textup{ (approximation)}$

For the point x = 1,

$f(1)=1=f(2)+(1-2)f'(2)+(1-2)^{2}f''(2)+...$

$f'(2)-f''(2)=1~~...(2)~\textup{ (approximation)}$

$\textup{Solving, we get }~f'(2)=0.5,~~f''(2)=-0.5$

(b) We were able to form a system of linear equations in two variables, when two points in vicinity of x = 2 were given. If the value of another point in vicinity such as f(3) , f(2.5) etc. could be given, we can expand the Taylor expansion till f'''(x-a) and solve them. Hence, what we require is another function value in the vicinity of x = 2