Use the power series for $f ( x ) = \sqrt [ 3 ] { 5 + x }$ to estimate $\sqrt [ 3 ] { 5.07 }$ correct to four decimal places.
A) $1.7179$
B) $1.7189$
C) $1.7195$
D) $1.7156$
E) $1.7200$

Question

Quizplus · Accepted Answer

To estimate $\sqrt[3]{5.07}$, we can use the power series expansion of $f(x) = \sqrt[3]{5+x}$ around $x=0$, which gives us $f(x) \approx f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots$. For $x=0.07$, the first few terms should provide a good approximation. Calculating the derivatives at $x=0$ and substituting $x=0.07$ gives an approximation close to $1.7179$, which is correct to four decimal places.

Use the Power Series For f(x)=5+x3f ( x ) = \sqrt [ 3 ] { 5 + x }f(x)=35+x​

Use the Power Series For $f ( x ) = \sqrt [ 3 ] { 5 + x }$