Question

Use linear approximation to estimate f (2.85 )given that f (3 )equals 2 and f prime (3 )equals 5.

Answer 1

Answer:

Step-by-step explanation:

Given that,

f(3) = 2

f'(3) = 5.

We want to estimate f(2.85)

The linear approximation of "f" at "a" is one way of writing the equation of the tangent line at "a".

At x = a, y = f(a) and the slope of the tangent line is f'(a).

So, in point slope form, the tangent line has equation

y − f(a) = f'(a)(x − a)

The linearization solves for y by adding f(a) to both sides

f(x) = f(a) + f'(a)(x − a).

Given that,

f(3) = 2,

f'(3) = 5

a = 3, we want to find f(2.85)

x = 2.85

Therefore,

f(x) = f(a) + f'(a)(x − a)

f(2.85) = 2 + 5(2.85 - 3)

f(2.85) = 2 + 5×-0.15

f(2.85) = 2 - 0.75

f(2.85) = 1.25