Question

Given the values of the derivative f′(x) in the table and that f(0)=150, find or estimate f(x) for x=0,2,4,6.

Answer 1

Here's what i found online:
We have in general: f(b) - f(a) = Integral[x=a to b; f ' (x) dx] 
Since we don't have a lot of information, we're going to divide the interval [a, b] in half and use base the right rectangle on f(a), and the left rectangle on f(b): 

f(b) - f(a) ~ (1/2)(b-a) f '(a) + (1/2)(b-a) f ' (b) = (1/2)(b-a) ( f '(a) + f '(b) ) 

That's all we can use for the Riemann integral. 
In all cases here, we'll have b-a=2. 

Now: 
f(2) - f(0) ~ ( f '(0) + f '(2) ) = 40, so f(2) ~ 130 
f(4) - f(2) ~ (f '(2) + f '(4)) = 57, so f(4) ~ 57+130 = 187 
f(6) - f(4) ~ (f '(4) + f '(6)) = 69, so f(6) ~ 187+69 = 256


Hope this helps you! :-)