Find the linear approximation of the function f(x) = 4 − x at a = 0. l(x) = use l(x) to approximate the numbers 3.9 and 3.99 . (round your answers to four decimal places.) 3.9 ≈ 3.99 ≈ illustrate by graphing f and the tangent line.

The linear approximate value of f(x) ≈ 20

Step-by-step explanation:

The linear approximation of a function  f(x) is given by  

f(x + Δx) = f(x) + f'(x) × Δx

It leverages the tangent line to approximate values of a function close to a particular point.

Using the Taylor form:

f(x) = f(a) + (x - a) × f'(a)

Now,

From the given question:

f(x) = 4 - x

On differentiate, we get

f'(x) = 4

And

At a = 0

f(0) = 4 - 0 = 4

f'(0) = 4

Therefore,

f(x) = f(a) + (x - a) × f'(a)

f(x) = f(0) + (x - 0) × f'(0)

f(x) = 4 + (x) × 4 = 4 + 4x

∴ at x = 3.9

f(x) = 4 + 4 × 3.9 = 19.6

At x = 3.99

f(x) = 4 + 4 × 3.99 = 19.96

Hence, the linear approximation of the function f(x) is

f(x) ≈ 20