Schaum's Outline of Calculus

Frank Ayres

Chapter 52

Directional Derivatives. Maximum and Minimum Values - all with Video Answers

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Chapter Questions

Problem 1

Derive formula (52.1).
In Fig. 52-1, let $P^{* *}(x+\Delta x, y+\Delta y)$ be a second point on $P^* L$ and denote by $\Delta s$ the distance $P^* P^{* *}$. Assuming that $z=f(x, y)$ possesses continuous first partial derivatives, we have, by Theorem 49.1,

Raj Bala

Raj Bala

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Problem 2

Find the directional derivative of $z=x^2-6 y^2$ at $P^*(7,2)$ in the direction: (a) $\theta=45^{\circ}$; (b) $\theta=135^{\circ}$. The directional derivative at any point $P^*(x, y)$ in the direction $\theta$ is

Lucas Finney

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Problem 3

Find the directional derivative of $z=y e^x$ at $P^*(0,3)$ in the direction (a) $\theta=30^{\circ}$; (b) $\theta=120^{\circ}$. Here, $d z / d s=y e^r \cos \theta+e^x \sin \theta$.

Lucas Finney

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Problem 4

The temperature $T$ of a heated circular plate at any of its points $(x, y)$ is given by $T=\frac{64}{x^2+y^2+2}$, the origin being at the center of the plate. At the point $(1,2)$, find the rate of change of $T$ in the direction $\theta=\pi / 3$.

Lucas Finney

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Problem 5

The electrical potential $V$ at any point $(x, y)$ is given by $V=\ln \sqrt{x^2+y^2}$. Find the rate of change of $V$ at the point $(3,4)$ in the direction toward the point $(2,6)$.

Lucas Finney

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Problem 6

Find the maximum directional derivative for the surface and point of Problem 2 .
At $P^*(7,2)$ in the direction $\theta, d z / d s=14 \cos \theta-24 \sin \theta$.

Lucas Finney

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Problem 7

Find the maximum directional derivative for the function and point of Problem 3 .
At $P^*(0,3)$ in the direction $\theta, d z / d s=3 \cos \theta+\sin \theta$.

William Semus

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Problem 8

In Problem 5, show that $V$ changes most rapidly along the set of radial lines through the origin.

Uma Kumari

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Problem 9

Find the directional derivative of $F(x, y, z)=x y+2 x z-y^2+z^2$ at the point $(1,-2,1)$ along the curve $x=t$, $y=t-3, z=t^2$ in the direction of increasing $z$

Lucas Finney

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Problem 10

Examine $f(x, y)=x^2+y^2-4 x+6 y+25$ for maximum and minimum values.

Darshan Maheshwari

Darshan Maheshwari

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Problem 11

Examine $f(x, y)=x^3+y^3+3 x y$ for maximum and minimum values.

Darshan Maheshwari

Darshan Maheshwari

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Problem 12

Divide 120 into three nonnegative parts such that the sum of their products taken two at a time is a maximum.

Adrian Co

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Problem 13

Find the point in the plane $2 x-y+2 z=16$ nearest the origin.

Cameron Bunney

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Problem 14

Show that a rectangular parallelepiped of maximum volume $V$ with constant surface area $S$ is a cube.

Lucas Finney

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Problem 15

Find the volume $V$ of the largest rectangular parallelepiped that can be inscribed in the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$.

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Problem 16

Find the directional derivatives of the given function at the given point in the indicated direction.
(a) $z=x^2+x y+y^2,(3,1), \theta=\frac{\pi}{3}$.
(b) $z=x^3-3 x y+y^3,(2,1), \theta=\tan ^{-1}\left(\frac{2}{3}\right)$.
(c) $z=y+x \cos x y,(0,0), \theta=\frac{\pi}{3}$.
(d) $z=2 x^2+3 x y-y^2,(1,-1)$, toward $(2,1)$.

Carson Merrill

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Problem 17

Find the maximum directional derivative for each of the functions of Problem 16 at the given point.

Lucas Finney

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Problem 18

Show that the maximal directional derivative of $V=\ln \sqrt{x^2+y^2}$ of Problem 8 is constant along any circle $x^2+y^2=r^2$.

Lucas Finney

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Problem 19

On a hill represented by $z=8-4 x^2-2 y^2$, find (a) the direction of the steepest grade at $(1,1,2)$ and (b) the direction of the contour line (the direction for which $z=$ constant). Note that the directions are mutually perpendicular.

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 20

Show that the sum of the squares of the directional derivatives of $z=f(x, y)$ at any of its points is constant for any two mutually perpendicular directions and is equal to the square of the maximum directional derivative.

Lucas Finney

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Problem 21

Given $z=f(x, y)$ and $w=g(x, y)$ such that $\partial z / \partial x=\partial w / \partial y$ and $\partial z / \partial y=-\partial w / \partial x$. If $\theta_1$ and $\theta_2$ are two mutually perpendicular directions, show that at any point $P(x, y), \partial z / \partial s_1=\partial w / \partial s_2$ and $\partial z / \partial s_2=-\partial w / \partial s_1$.

Harshita Goel

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Problem 22

Find the directional derivative of the given function at the given point in the indicated direction:
(a) $x y^2 z,(2,1,3),[1,-2,2]$.
(b) $x^2+y^2+z^2,(1,1,1)$, toward $(2,3,4)$.
(c) $x^2+y^2-2 x z,(1,3,2)$, along $x^2+y^2-2 x z=6,3 x^2-y^2+3 z=0$ in the direction of increasing $z$.

Subhadeepta Sahoo

Subhadeepta Sahoo

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Problem 23

Examine each of the following functions for relative maximum and minimum values.
(a) $z=2 x+4 y-x^2-y^2-3$
(b) $z=x^3+y^3-3 x y$
(c) $z=x^2+2 x y+2 y^2$
(d) $z=(x-y)(1-x y)$
(e) $z=2 x^2+y^2+6 x y+10 x-6 y+5$
(f) $z=3 x-3 y-2 x^3-x y^2+2 x^2 y+y^3$
(g) $z=x y(2 x+4 y+1)$

Lucas Finney

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Problem 24

Find positive numbers $x, y, z$ such that
(a) $x+y+z=18$ and $x y z$ is a maximum
(b) $x y z=27$ and $x+y+z$ is a minimum
(c) $x+y+z=20$ and $x y z^2$ is a maximum
(d) $x+y+z=12$ and $x y^2 z^3$ is a maximum

Sriram Soundarrajan

Sriram Soundarrajan

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Problem 25

Find the minimum value of the square of the distance from the origin to the plane $A x+B y+C z+D=0$.

Chris Trentman

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Problem 26

(a) The surface area of a rectangular box without a top is to be $108 \mathrm{ft}^2$. Find the greatest possible volume.
(b) The volume of a rectangular box without a top is to be $500 \mathrm{ft}^3$. Find the minimum surface area.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

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Problem 27

Find the point on $z=x y-1$ nearest the origin.

Lucas Finney

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Problem 28

Find the equation of the plane through $(1,1,2)$ that cuts off the least volume in the first octant.

Victor Salazar

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Problem 29

Determine the values of $p$ and $q$ so that the $\operatorname{sum} S$ of the squares of the vertical distances of the points $(0,2),(1,3)$, and $(2,5)$ from the line $y=p x+q$ is a minimum.

Angela Guo

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