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The Calculus with Analytic Geometry

Louis Leithold

Chapter 19

Differential Calculus Of Functions Of Severai Variables - all with Video Answers

Educators

Section 5

Differentiability And The Total Differential

Problem 1

If $f(x, y)=3 x^{2}+2 x y-y^{2}, \Delta x=0.03$, and $\Delta y=-0.02$, find (a) the increment of $f$ at $(1,4)$ and (b) the total differential of $f$ at $(1,4)$.

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Problem 2

If $f(x, y)=x y e^{x y}, \Delta x=-0.1$, and $\Delta y=0.2$, find (a) the increment of $f$ at $(2,-4)$ and (b) the total differential of $f$ at $(2,-4)$.

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Problem 3

If $f(x, y, z)=x y+\ln (y z), \Delta x=0.02, \Delta y=0.04$, and $\Delta z=-0.03$, find (a) the increment of $f$ at $(4,1,5)$ and (b) the total differential of $f$ at $(4,1,5)$.

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Problem 4

If $f(x, y, z)=x^{2} y+2 x y z-z^{3}, \Delta x=0.01, \Delta y=0.03$, and $\Delta z=-0.01$, find (a) the increment of $f$ at $(-3,0,2)$ and (b) the total differential of $f$ at $(-3,0,2)$.

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Problem 5

In Exercises 5 through 8 , prove that $f$ is differentiable at all points in its domain by doing each of the following: (a) Find $\Delta f\left(x_{0}, y_{0}\right)$ for the given function; (b) find an $\epsilon_{1}$ and an $\epsilon_{2}$ so that Eq. (3) holds; (c) show that the $\epsilon_{1}$ and the $\epsilon_{2}$ found in part (b) both approach zero as $(\Delta x, \Delta y) \rightarrow(0,0)$.
$$
f(x, y)=x^{2} y-2 x y
$$

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Problem 6

In Exercises 5 through 8 , prove that $f$ is differentiable at all points in its domain by doing each of the following: (a) Find $\Delta f\left(x_{0}, y_{0}\right)$ for the given function; (b) find an $\epsilon_{1}$ and an $\epsilon_{2}$ so that Eq. (3) holds; (c) show that the $\epsilon_{1}$ and the $\epsilon_{2}$ found in part (b) both approach zero as $(\Delta x, \Delta y) \rightarrow(0,0)$.
$$
f(x, y)=2 x^{2}+3 y^{2}
$$

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Problem 7

In Exercises 5 through 8 , prove that $f$ is differentiable at all points in its domain by doing each of the following: (a) Find $\Delta f\left(x_{0}, y_{0}\right)$ for the given function; (b) find an $\epsilon_{1}$ and an $\epsilon_{2}$ so that Eq. (3) holds; (c) show that the $\epsilon_{1}$ and the $\epsilon_{2}$ found in part (b) both approach zero as $(\Delta x, \Delta y) \rightarrow(0,0)$.
$$
f(x, y)=\frac{x^{2}}{y}
$$

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Problem 8

In Exercises 5 through 8 , prove that $f$ is differentiable at all points in its domain by doing each of the following: (a) Find $\Delta f\left(x_{0}, y_{0}\right)$ for the given function; (b) find an $\epsilon_{1}$ and an $\epsilon_{2}$ so that Eq. (3) holds; (c) show that the $\epsilon_{1}$ and the $\epsilon_{2}$ found in part (b) both approach zero as $(\Delta x, \Delta y) \rightarrow(0,0)$.
$$
f(x, y)=\frac{y}{x}
$$

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Problem 9

Given $f(x, y)= \begin{cases}x+y-2 & \text { if } x=1 \text { or } y=1 \\ 2 & \text { if } x \neq 1 \text { and } y \neq 1\end{cases}$
Prove that $D_{1} f(1,1)$ and $D_{2} f(1,1)$ exist, but $f$ is not differentiable at $(1,1)$.

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Problem 10

Given $f(x, y)= \begin{cases}\frac{3 x^{2} y^{2}}{x^{4}+y^{4}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$
Prove that $D_{1} f(0,0)$ and $D_{2} f(0,0)$ exist, but $f$ is not differentiable at $(0,0)$.

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Problem 11

Prove Theorem 19.5.4(ii).

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Problem 12

Show that Eq. (5) may be written in the form (7) where $\xi_{1}$ is between $x_{0}$ and $x_{0}+h$.

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Problem 13

Show that Eq. (6) may be written in the form $(8)$ where $\xi_{2}$ is between $y_{0}$ and $y_{0}+k$.

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Problem 14

In Exercises 14 through 17 , use Theorem $19.5 .4$ to find either a $\xi_{1}$ or a $\xi_{2}$, whichever applies.
$$
f(x, y)=x^{2}+3 x y-y^{2} ; x \text { is in }[1,3] ; y=4
$$

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Problem 15

In Exercises 14 through 17 , use Theorem $19.5 .4$ to find either a $\xi_{1}$ or a $\xi_{2}$, whichever applies.
$$
f(x, y)=x^{3}-y^{2} ; x \text { is in }[2,6] ; y=3
$$

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Problem 16

In Exercises 14 through 17 , use Theorem $19.5 .4$ to find either a $\xi_{1}$ or a $\xi_{2}$, whichever applies.
$$
f(x, y)=\frac{4 x}{x+y^{\prime}} y \text { is in }[-2,2] ; x=4
$$

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Problem 17

In Exercises 14 through 17 , use Theorem $19.5 .4$ to find either a $\xi_{1}$ or a $\xi_{2}$, whichever applies.
$$
f(x, y)=\frac{2 x-y}{2 y+x} ; y \text { is in }[0,4] ; x=2
$$

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Problem 18

Given $f(x, y)= \begin{cases}\frac{3 x^{2} y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$
This function is continuous at $(0,0)$ (see Example 3, Sec. 19.2, and Illustration 1, Sec. 19.3). Prove that $D_{1} f(0,0)$ and $D_{2} f(0,0)$ exist but $D_{1} f$ and $D_{2} f$ are not continuous at $(0,0)$.

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Problem 19

Given $f(x, y)= \begin{cases}\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$
Prove that $f$ is differentiable at $(0,0)$ by using Theorem 19.5.5.

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Problem 20

In Exercises 20 and 21, prove that $f$ is differentiable at all points in $R^{3}$ by doing each of the following: (a) Find $\Delta f\left(x_{0} ; y_{0}, z_{0}\right)$ (b) find an $\epsilon_{1}, \epsilon_{2}$, and $\epsilon_{3}$, so that Eq. (30) holds; (c) show that the $\epsilon_{1}, \epsilon_{2}$, and $\epsilon_{3}$ found in (b) all approach zero as $(\Delta x, \Delta y, \Delta z)$ approaches $(0,0,0)$.
$$
f(x, y, z)=2 x^{2} z-3 y z^{2}
$$

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Problem 21

In Exercises 20 and 21, prove that $f$ is differentiable at all points in $R^{3}$ by doing each of the following: (a) Find $\Delta f\left(x_{0} ; y_{0}, z_{0}\right)$ (b) find an $\epsilon_{1}, \epsilon_{2}$, and $\epsilon_{3}$, so that Eq. (30) holds; (c) show that the $\epsilon_{1}, \epsilon_{2}$, and $\epsilon_{3}$ found in (b) all approach zero as $(\Delta x, \Delta y, \Delta z)$ approaches $(0,0,0)$.
$$
f(x, y, z)=x y-x z+z^{2}
$$

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Problem 22

Given $f(x, y, z)= \begin{cases}\frac{x y^{2} z}{x^{4}+y^{4}+z^{4}} & \text { if }(x, y, z) \neq(0,0,0) \\ 0 & \text { if }(x, y, z)=(0,0,0)\end{cases}$
(a) Show that $D_{1} f(0,0,0), D_{2} f(0,0,0)$, and $D_{3} f(0,0,0)$ exist; (b) make use of the fact that differentiability implies continuity to prove that $f$ is not differentiable at $(0,0,0)$.

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Problem 23

Given $f(x, y, z)= \begin{cases}\frac{x y z^{2}}{x^{2}+y^{2}+z^{2}} & \text { if }(x, y, z) \neq(0,0,0) \\ 0 & \text { if }(x, y, z)=(0,0,0)\end{cases}$
Prove that $f$ is differentiable at $(0,0,0)$.

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Problem 24

Use the total differential to find approximately the greatest error in calculating the area of a right triangle from the lengths of the legs if they are measured to be 6 in. and 8 in., respectively, with a possible error of $0.1$ in. for each measurement. Also find the approximate percent error.

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Problem 25

Find approximately, by using the total differential, the greatest error in calculating the length of the hypotenuse of the right triangle from the measurements of Exercise 24 . Also find the approximate percent error.

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Problem 26

If the ideal gas law (see Example 5, Sec. 19.4) is used to find $P$ when $T$ and $V$ are given, but there is an error of $0.3 \%$ in measuring $T$ and an error of $0.8 \%$ in measuring $V$, find approximately the greatest percent error in $P$.

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Problem 27

The specific gravity $s$ of an object is given by the formula
$$
s=\frac{A}{A-W}
$$
where $A$ is the number of pounds in the weight of the object in air and $W$ is the number of pounds in the weight of the object in water. If the weight of an object in air is read as $20 \mathrm{lb}$ with a possible error of $0.01 \mathrm{lb}$ and its weight in water is read as $12 \mathrm{lb}$ with a possible error of $0.02 \mathrm{lb}$, find approximately the largest possible error in calculating $s$ from these measurements. Also find the largest possible relative error.

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Problem 28

A wooden box is to be made of lumber that is $\frac{2}{3}$ in. thick. The inside length is to be $6 \mathrm{ft}$, the inside width is to be $3 \mathrm{ft}$, the inside depth is to be $4 \mathrm{ft}$, and the box is to have no top. Use the total differential to find the approximate amount of lumber to be used in the box.

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Problem 29

A company has contracted to manufacture 10,000 closed wooden crates having dimensions $3 \mathrm{ft}, 4 \mathrm{ft}$, and $5 \mathrm{ft}$. The cost of the wood to be used is $5 \notin$ per square foot. If the machines that are used to cut the pieces of wood have a possible error of $0.05 \mathrm{ft}$ in each dimension, find approximately, by using the total differential, the greatest possible error in the estimate of the cost of the wood.

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Problem 30

In Exercises 30 through 33 , we show that a function may be differentiable at a point even though it is not continuously differentiable there. Hence, the conditions of Theorem $19.5 .5$ are sufficient but not necessary for differentiability. The function $f$ in these exercises is defined by
In Exercises 30 through 33 , we show that a function may be differentiable at a point even though it is not continuously differentiable there. Hence, the conditions of Theorem $19.5 .5$ are sufficient but not necessary for differentiability. The function $f$ in these exercises is defined by
$$
f(x, y)= \begin{cases}\left(x^{2}+y^{2}\right) \sin \frac{1}{\sqrt{x^{2}+y^{2}}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}
$$
$$
\text { Find } \Delta f(0,0) \text {. }
$$

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Problem 31

In Exercises 30 through 33 , we show that a function may be differentiable at a point even though it is not continuously differentiable there. Hence, the conditions of Theorem $19.5 .5$ are sufficient but not necessary for differentiability. The function $f$ in these exercises is defined by
$$
f(x, y)= \begin{cases}\left(x^{2}+y^{2}\right) \sin \frac{1}{\sqrt{x^{2}+y^{2}}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}
$$
$$
\text { Find } D_{1} f(x, y) \text { and } D_{2} f(x, y) \text {. }
$$

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Problem 32

In Exercises 30 through 33 , we show that a function may be differentiable at a point even though it is not continuously differentiable there. Hence, the conditions of Theorem $19.5 .5$ are sufficient but not necessary for differentiability. The function $f$ in these exercises is defined by
$$
f(x, y)= \begin{cases}\left(x^{2}+y^{2}\right) \sin \frac{1}{\sqrt{x^{2}+y^{2}}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}
$$
Prove that $f$ is differentiable at $(0,0)$ by using Definition 19.5.2 and the results of Exercises 30 and 31 .

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Problem 33

In Exercises 30 through 33 , we show that a function may be differentiable at a point even though it is not continuously differentiable there. Hence, the conditions of Theorem $19.5 .5$ are sufficient but not necessary for differentiability. The function $f$ in these exercises is defined by
$$
f(x, y)= \begin{cases}\left(x^{2}+y^{2}\right) \sin \frac{1}{\sqrt{x^{2}+y^{2}}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}
$$
Prove that $D_{1} f$ and $D_{2} f$ are not continuous at $(0,0)$.

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