Question

suppose that $f(t)$ is a function with $f(25)=3.2$ and $f(25)= - 0.3$. estimate $f(26)$ and $f(27)$. $f(26)approx$ $f(27)approx$

Explanation:

Step1: Recall linear - approximation formula

The linear - approximation formula is $f(x)\approx f(a)+f^{\prime}(a)(x - a)$. Here $a = 25$.

Step2: Estimate $f(26)$

For $x = 26$, we have $f(26)\approx f(25)+f^{\prime}(25)(26 - 25)$. Substitute $f(25)=3.2$ and $f^{\prime}(25)=- 0.3$ into the formula: $f(26)\approx3.2+( - 0.3)\times(26 - 25)=3.2-0.3 = 2.9$.

Step3: Estimate $f(27)$

For $x = 27$, we use the linear - approximation formula $f(27)\approx f(25)+f^{\prime}(25)(27 - 25)$. Substitute $f(25)=3.2$ and $f^{\prime}(25)=-0.3$ into the formula: $f(27)\approx3.2+( - 0.3)\times(27 - 25)=3.2-0.3\times2=3.2 - 0.6=2.6$.

Answer:

$f(26)\approx2.9$
$f(27)\approx2.6$