Question

Let f(x) = `root(3)(x)`. Find the linear approximation at x = 27. Use the linear approximation to approximate `root(3)(27.2)`

Answer 1

x = 27

f(x) = `root(3)(27)` = 3

We need to find the value of `root(3)(27.2)`

We know that

f(x₀ + Δx) = f(x₀) + f'(x₀) Δx

f(27.2) = `3 + 1/(3x^(2/3)) xx 0.2`

= `3 + 1/(3(27)^(2/3)) xx 0.2`

= `3 + 1/(3 xx 9) x 0.2`

= `3 + 0.2/27`

= `3 + 2/270`

= 3 + 0.0074

= 3.0074

∴ Approximate value of `root(3)(27.2)` = 3.0074