PLEASE SHOW WORK IM REALLY CONFUSED Let f be a differentiabl

PLEASE SHOW WORK! I\'M REALLY CONFUSED!

Let f be a differentiable function satisfying the following table of values: x 3.0 3.2 3.4 3.6 3.8 f(x) 8.2 9.5 10.5 11.0 13.2 If the graph of f is as shown, estimate f\'(3.2) and f\'(3.5). (In each case, there can be more than one answer that is acceptable.)

Solution

Well we know that the derivative at a point is the rate of change of the function at that point. We also know that when the function goes from 3.2 to 3.4 f(x) goes from 9.5 to 10.5. So an estimation of the derivative would be (10.5-9.5)/(3.4-3.2)= 1/.2=5. We don\'t have values at 3.5 but we do know that when the function goes from 3.4 to 3.6 f(x) goes from 10.5 to 11. So an estimation of the derivative at 3.5 would be (11-10.5)/(3.6-3.4)= .5/.2= 2.5.