$\text { Use differentials to approximate the value of } \sqrt [ 3 ] { 7.5 } \text {. Round your answer to four }$ decimal places.
A)1.9683
B)1.9483
C)1.9383
D)1.9583
E)1.9783

Question

Quizplus · Accepted Answer

We start by using differentials. Let $f(x)=\sqrt[3]{x}$. Then, we have
\begin{align*}
f'(x) &= \frac{1}{3\sqrt[3]{x^2}} \
\Delta y &\approx f'(7)\cdot \Delta x \
\Delta y &\approx \frac{1}{3\sqrt[3]{7^2}}\cdot 0.5 \
\sqrt[3]{7.5} &\approx 7 + \Delta y \
\sqrt[3]{7.5} &\approx 7 + \frac{1}{3\sqrt[3]{7^2}}\cdot 0.5 \
\sqrt[3]{7.5} &\approx 1.9583\ldots \
\end{align*}
Therefore, the answer is D, rounded to four decimal places.

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$\text { Use differentials to approximate the value of } \sqrt [ 3 ] { 7.5 } \text {. Round your answer to four }$