Use linear approximation, i.e. the tangent line, to approximate 3^√(8.1) as follows:

Question

Let f(x)=³√(x). The equation of the tangent line to f(x) at x=8 can be written in the form y=mx+b

Answer 1

Let f(x) = (x)^1/3; f(8) = 8^1/3 =2. We have tangent point (8, 2). Now equation of tangent line: f'(x) = 1/3x^-2/3;

f'(8) = 1/3(8)^-2/3 = 1/12; So y = 2 + 1/12(x - 8); y = 1/12x + 4/3

Linear approximation L(x) = x/12 + 4/3.

Now x = 8.1; L(12) = 8.1/12 + 4/3 = 2.0083.

So 8^1/3 ≈ 2.0083