Respuesta :

Kazeemsodikisola

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

facundo3141592

Using the linear approximation, we will see that a good estimate is:

f(2.85) = 1.25

How to get an estimate?

For any function, the Taylor polinomial around the value a is given by:

f(x) = f(a) + f'(a)*(x - a) + ....

But it works better for values close to a.

If we use only the first two terms, we have linear approximation.

In this case we know:

f(3) = 2

f'(3) = 5

So we can replace a by 3 in the above equation, then we have:

f(x) = f(3) + f'(3)*(x - 3)

f(x) =   2 + 5*(x - 3)

Then we can estimate:

f(2.85) = 2 + 5*(2.85 - 3) = 1.25

If you want to learn more about estimations, you can read:

https://brainly.com/question/9211177

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