Please help me with no. b!!

Question

Let .

(a) Find the linearization of valid near .

  ..1/3z+1..

(b) Use the linearization to approximate and .

  ..........   .........

Answer 1

The function in question is the cuberoot function f(x) = ³√x also known as f(x) = x^1/3

At x = 1.00, the function value is also 1.00 [f(1) = 1^1/3 = 1] so point (1, 1) is a point on the function graph.

To get a linear approximation, you would determine what the line tangent to the function is that passes through (1,1) and use that to estimate f(0.99) and f(1.06).

The derivative of f(x) = x^1/3 is f '(x) = 1/3x^-2/3 so f '(1) = 1/3(1)^-2/3 = 1/3 which means the slope of the tangent line is 1/3. We can use the point-slope form [y - y₁ = m(x - x₁)] to write the equation of the tangent line:

y - 1 = 1/3(x - 1) or y = 1/3x + 2/3

So to estimate f(0.99) use 0.99 for "x" in this tangent line equation:

y = 1/3(0.99) + 2/3 ≈ 0.9967

And to estimate f(1.06) use x = 1.06 for "x" in this tangent line equation:

y = 1/3(1.06) + 2/3 ≈ 1.02