Question

Suppose f(5) = 2 and f′(5) = −0.5. Estimate f(6).

Answer 1

Given , f(5) = 2 and f′(5) = −0.5

To estimate f(6)

We can estimate f(6) by finding linearization of function about the point x=5

Linearization of a function f(x) means " the linear function that best approximate f(x)

Linearization of f(x) about point x=a is given by-

L(x) = f(a) + f'(a)(x-a)

Put a=5

=> L(x) = f(5)+ f'(5)(x-5)

Put f(5) = 2 and f′(5) = −0.5

=> L(x) = 2+(-0.5)(x-5)

=> L(x) = 2 - 0.5 x +2.5

=> L(x) = 4.5 -0.5 x------(1)

To find estimate of f(6), put x= 6 in equation (1)

=> L(6) = 4.5 - (0.5)(6)

=> L(6) = 4.5 -3.0

=> L(6) = 1.5

So the estimated value of f(6) = 1.5

Answer.