4. Error estimate for quadratic interpolation. Estimate the error for \( p_{2}(9.2) \) in Example 2 trom (5)
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Estimate the error if $P_{4}(x)=1+x+\left(x^{2} / 2\right)+\left(x^{3} / 6\right)+\left(x^{4} / 24\right)$ is used to estimate the value of $e^{x}$ at $x=1 / 2$
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Consider a closed rectangular box with a square base, as shown in the figure. Assume $x$ is measured with an error of at most 0.5$\%$ and $y$ is measured with an error of at most $0.75 \%,$ so we have $|d x| / x < 0.005$ and $|d y| / y < 0.0075 .$ a. Use a differential to estimate the relative error $|d V| / V$ in computing the box's volume $V .$ b. Use a differential to estimate the relative error $|d S| / S$ in computing the box's surface area $S$ . $$ \begin{aligned} \text {Hint for b: } & \frac{4 x^{2}+4 x y}{2 x^{2}+4 x y} \leq \frac{4 x^{2}+8 x y}{2 x^{2}+4 x y}=2 \text { and } \\ & \frac{4 x y}{2 x^{2}+4 x y} \leq \frac{2 x^{2}+4 x y}{2 x^{2}+4 x y}=1 \end{aligned} $$
Partial Derivatives
Tangent Planes and Differential
Consider a closed rectangular box with a square base, as shown in the figure. Assume $x$ is measured with an error of at most $0.5 \%$ and $y$ is measured with an error of at most $0.75 \%,$ so we have $|d x| / x<0.005$ and $|d y| / y<0.0075$ a. Use a differential to estimate the relative error $|d V| / V$ in computing the box's volume $V$. b. Use a differential to estimate the relative error $|d S| / S$ in computing the box's surface area $S$ Hint for b: $\frac{4 x^{2}+4 x y}{2 x^{2}+4 x y} \leq \frac{4 x^{2}+8 x y}{2 x^{2}+4 x y}=2 \quad$ and $$ \frac{4 x y}{2 x^{2}+4 x y} \leq \frac{2 x^{2}+4 x y}{2 x^{2}+4 x y}=1 $$
Tangent Planes and Differentials
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