Question

How do you find the solutions for f'(3) f'(4) and f'(5), if you are given a graph with the curve `y=f(x)`?

Answer 1

See details and example.

Explanation:

Make another graph for `f' ( x )` and make lines x = 3, 4, 5 to read

f' values, respectively.

Exemplification:

`y = f ( x ) = xsqrtx - 1 and g ( x ) = f' ( x ) =3/2 sqrtx`

g-graph:reveals values near 2.5, 3 and 3.5.
graph{(y - 1.5sqrtx)(x-3+0y)(x-4+0y)(x-5+0y)=0}.

4-sd values, by trial and error y-bracketing method:

Graph for 4-sd `f' ( 3 ) = 2.598`
graph{(y - 1.5 sqrt x)(x-3)=0[2.95 3.05 2.597 2.599]}
By substitution, f ' ( 4 ) = 3, exactly.

Graph for 4-sd `f' ( 5 ) = 3.354`
graph{(y - 1.5 sqrt x)(x-5)=0[4.95 5.05 3.353 3.355]}