Question

suppose that $f(x)$ is a function with $f(85)=85$ and $f(85)=7$. estimate $f(89)$. $f(89)=$

Explanation:

Step1: Recall linear - approximation formula

The linear - approximation formula is $L(x)=f(a)+f^{\prime}(a)(x - a)$, where $a$ is a known point and $x$ is the point we want to estimate the function value at. Here, $a = 85$ and $x=89$.

Step2: Identify values of $a$, $x$, $f(a)$ and $f^{\prime}(a)$

We know that $a = 85$, $x = 89$, $f(85)=85$ and $f^{\prime}(85)=7$.

Step3: Calculate $x - a$

$x - a=89 - 85=4$.

Step4: Apply the linear - approximation formula

$f(89)\approx f(85)+f^{\prime}(85)(89 - 85)$. Substitute $f(85)=85$, $f^{\prime}(85)=7$ and $89 - 85 = 4$ into the formula: $f(89)\approx85+7\times4$.

Step5: Perform the arithmetic

$85+7\times4=85 + 28=113$.

Answer:

$113$