Chapter 3: Functions of Several Variables

Section 3.2: Limits and Continuity

Example 3.2.20

$$ Prove the inequality $\left|x^3-y^3\right|\le {\left(x^2\+y^2\right)}^{3\/2}$. $$ Solution $$ The requisite estimates are shown below. $$  $\left| x^3- y^3\right|$ $\le \left|x^3\right|\+\left| y^3\right|$ Inequality 3 Table 3.2.1 $$ $\=\left|x\right| x^2$ $\+$ $\left|y\right| y^2$ $$ $$ $\le \left(x^2\+y^2\right)\sqrt{x^2\+y^2}$ Inequalities 4 and 5 Table 3.2.1$$$$ $$ $\={\left(x^2\+y^2\right)}^{3\/2}$ $$ $$ Figure 3.2.20(a) compares ${\left(x^2\+y^2\right)}^{3\/2}$ with $\left|x^3-y^3\right|$, the first in green, the second, in red. The green surface lies above the red surface, indicating that ${\left(x^2\+y^2\right)}^{3\/2}$ is greater than $\left|x^3-y^3\right|$.   Figure 3.2.20(a)   $\left|x^3-y^3\right|$ in red, ${\left(x^2\+y^2\right)}^{3\/2}$ in green $$$$

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