Question

Use differential, i.e., linear approximation, to approximate (125.4^(1/3)) as follows:
Let f(x)=x^(1/3) . The linear approximation to f(x) at x=125 can be written in the form y=mx+b where m is:
m=1/75
b=?
How do I find b?

Answer 1

f(x) = x^(1/3) = 125^(1/3) = 5
f'(x) = (1/3) x^(-2/3) = 1/3 / x^(2/3)
for x = 125, this is (1/3)/ 25 = 1/75 by the way
f(x+delta x) = f(x) + delta x * f'(x)
f(125+.4) = 5 + .4 *(1/75)
so b = 5 and m = (1/75)