Consider the interval I=[6,7.6]. Break I into four subintervals of length 0.4, namely the four subintervals

Question

[6,6.4],[6.4,6.8],[6.8,7.2],[7.2,7.6].
Suppose that f(6)=19, f'(6)=0, f'(6.4)=−0.5, f'(6.8)=−0.1, and f'(7.2)=−0.1. What is the approximate value of f(7.6)?

Answer 1

I bet it is (using the left side) is

first figure the values a the left side of each interval.

f(6)=given 19
f(6.4)=f(6)+.4*f'(6)=19
f(6.8)=f(6.4)+f'(6.4)*.4=19-.2= 18.8
f(7.2)=f(6.8)+f'(6.8)*.4=18.8-.04=18.76
f(7.6)=18.76-.4*.1=18.72

so check all those, it is easy to make an error when doing it on a keyboard.