using linear approximation, estimate δf for a change in x from x = a to x = b. use the estimate to…

using linear approximation, estimate δf for a change in x from x = a to x = b. use the estimate to approximate f(b), and find the error using the calculator. f(x) = √x, a = 36, b = 44. (give your answer to four decimal places.) δf = 0.6667 f(b) = 6.6667 (give your answer to six decimal places.) error = note: you can earn partial credit on this problem.

using linear approximation, estimate δf for a change in x from x = a to x = b. use the estimate to approximate f(b), and find the error using the calculator. f(x) = √x, a = 36, b = 44. (give your answer to four decimal places.) δf = 0.6667 f(b) = 6.6667 (give your answer to six decimal places.) error = note: you can earn partial credit on this problem.

Answer

Explanation:

Step1: Find the derivative of $f(x)$

The function is $f(x)=\sqrt{x}=x^{\frac{1}{2}}$. Using the power - rule $(x^n)' = nx^{n - 1}$, we have $f'(x)=\frac{1}{2\sqrt{x}}$.

Step2: Evaluate $f'(a)$

Given $a = 36$, then $f'(36)=\frac{1}{2\sqrt{36}}=\frac{1}{2\times6}=\frac{1}{12}$.

Step3: Calculate $\Delta x$

$\Delta x=b - a$, with $a = 36$ and $b = 44$, so $\Delta x=44 - 36 = 8$.

Step4: Estimate $\Delta f$ using linear approximation

The linear approximation formula is $\Delta f\approx f'(a)\Delta x$. Substituting $f'(36)=\frac{1}{12}$ and $\Delta x = 8$ into the formula, we get $\Delta f\approx\frac{1}{12}\times8=\frac{2}{3}\approx0.6667$.

Step5: Approximate $f(b)$

We know that $f(b)\approx f(a)+\Delta f$. Since $f(a)=f(36)=\sqrt{36}=6$, then $f(b)\approx6 + 0.6667=6.6667$.

Step6: Calculate the actual value of $f(b)$

The actual value of $f(b)$ where $b = 44$ is $f(44)=\sqrt{44}\approx6.633249$.

Step7: Calculate the error

The error is $|f(b){approx}-f(b){actual}|=|6.6667 - 6.633249|=0.033451\approx0.0335$.

Answer:

$0.0335$