Question

Use linear approximation, i.e. the tangent line, to approximate \sqrt[3] { 7.9 } as follows:
The equation of the tangent line to f(x) at x = 8 can be written in the form y =
Using this, we find our approximation for \sqrt[3] {7.9} is

Answer 1

It appears you are trying to approximate ∛7.9

So, if y = ∛x, then
dy = 1/(3∛(x^2)) dx
at x=8, y=2, so if dx = -0.1,

dy = 1/(3∛64) (-0.1)
= -1/240 = -.0042

So, ∛7.9 ≈ 1.9958

check: ∛7.9 is 1.9916
The estimate is expected to be a bit high, since ∛x is concave down.

Oh, yeah. The tangent line is

y-2 = 1/24 (x-8)