the derivative of the function is given by , and . if the linear approximation to at is used to estimate , at what value of does the linear approximation estimate that ?
The linear approximation estimation value:
To find the value of where the linear approximation estimates , we need to use the linear approximation formula. The linear approximation to a function at a point is given by:
L(x) = f(a) + f'(a)(x-a)
Given that the derivative of the function is , the linear approximation to at can be written as:
L(x) = f(2) + f'(2)(x-2)
Now, we need to find the value of when the linear approximation estimates . Let’s represent this as:
L(1.8) = 13 + 5(1.8 - 2)
Solving this equation, we get:
L(1.8) = 13 + 5(-0.2)
L(1.8) = 13 - 1
L(1.8) = 12
Therefore, the linear approximation estimates at the value of .