Use differentials to approximate the value of the expression. Compare your answer with that of a calculator. (Round your
answers to four decimal places.)

using differentials
using a calculator

Question

Use differentials to approximate the value of the expression. Compare your answer with that of a calculator. (Round your
answers to four decimal places.)

using differentials
using a calculator

UpStudy AI · Accepted Answer

1. Define the function  
$$
f(x)=x^{\frac{1}{3}}.
$$

2. Choose a nearby value $$ a $$ where the cube root is known exactly. Let  
$$
a=27,\quad f(27)=27^{\frac{1}{3}}=3.
$$

3. Compute the derivative of $$ f(x) $$:  
$$
f'(x)=\frac{1}{3}x^{-\frac{2}{3}}.
$$

4. Evaluate $$ f'(x) $$ at $$ x=27 $$:  
$$
f'(27)=\frac{1}{3}(27)^{-\frac{2}{3}}.
$$
Since  
$$
27^{\frac{1}{3}}=3\quad\text{implies}\quad 27^{-\frac{2}{3}}=3^{-2}=\frac{1}{9},
$$
we have  
$$
f'(27)=\frac{1}{3}\cdot\frac{1}{9}=\frac{1}{27}.
$$

5. Using differentials, approximate $$ f(26) $$ by  
$$
f(26) \approx f(27)+f'(27)(26-27).
$$
Substitute the known values:  
$$
f(26) \approx 3 + \frac{1}{27}(-1)=3-\frac{1}{27}.
$$
Thus,  
$$
f(26) \approx 3-0.0370\approx2.9630,
$$
when rounded to four decimal places.

6. Using a calculator,  
$$
\sqrt[3]{26}\approx 2.9625
$$
(rounded to four decimal places).

Final answers:
- Using differentials: $$2.9630$$
- Using a calculator: $$2.9625$$
Using differentials, the approximate value of $$ \sqrt[3]{26} $$ is 2.9630. Using a calculator, the value is 2.9625.

Use differentials to approximate the value of the expression. Compare your answer with that of a calculator. (Round your
answers to four decimal places.)

using differentials
using a calculator

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Use differentials to approximate the value of the expression. Compare your answer with that of a calculator. (Round your answers to four decimal places.) using differentials using a calculator

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Latest Calculus Questions

Use differentials to approximate the value of the expression. Compare your answer with that of a calculator. (Round your
answers to four decimal places.)

using differentials
using a calculator