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Calculus: An Intuitive and Physical Approach

Morris Kline

Chapter 22

Partial Differentiation - all with Video Answers

Educators

Section 1

Functions of Two or More Variables

11:01

Problem 1

If $z=x^{3}+3 x y-y^{2}$, what is the value of $z$ when $x$ and $y$ have the following values?
(a) $x=3, y=5$.
(b) $x=3, y=0 \quad$ Ans. 63 .
(b) $x=3, y=0$.
(c) $x=0, y=3$. Ans. $-9$.
(d) $x=2, y=-2$.
(e) $x-2, y=2$. Ans. $-24$.
(f) $x=-3, y=-4$.

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
00:46

Problem 2

Show that the values of $\mathrm{z}$ in the function $z=x^{2}-2 x y+y^{2}$ can never be negative.

Lucas Finney
Lucas Finney
Numerade Educator
02:32

Problem 3

If $z=\sin x \cos y$, what is the value of $z$ when $x$ and $y$ are the following?
(a) $x=\frac{\pi}{4}, y=\frac{\pi}{2} \quad$ Ans. 0 .
(b) $x=0, y=\frac{\pi}{4}$
(C) $x=\frac{\pi}{4}, y=\frac{\pi}{4} \quad$ Ans. $\frac{1}{2}$
(d) $x=\frac{3 \pi}{2}, y=\pi$.
(e) $x=2 \pi, y=\frac{3 \pi}{2}$
Ans. 0 .

Ruchika Sangwan
Ruchika Sangwan
Numerade Educator
01:03

Problem 4

If $f(x, y)=\frac{x^{2} y}{x^{2}+3}$, what is the value of the following?
(a) $f(1,5)$. Ans. $\frac{5}{4}$
(b) $f(0,5)$.
(c) $f(5,-3)$. Ans. $\frac{75}{28}$
(d) $f(0,0)$.

Thomas Emment
Thomas Emment
Numerade Educator
01:54

Problem 5

If $f(x, y, z)=x^{2}+3 x y z+z^{2}$, what is the value of the following?
(a) $f(1,-1,1)$. Ans. $-1$.
(b) $f(0,0,1)$.
(c) $f(2,-2,-3)$. Ans. 49 .
(d) $f\left(\frac{3}{4}, \frac{1}{2}, \frac{2}{3}\right)$.

Bobby Barnes
Bobby Barnes
University of North Texas
00:52

Problem 6

What is the domain of $x$ and $y$ for which the function $z=\sqrt{1-x^{2}-y^{2}}$ has real values?

Vikash Ranjan
Vikash Ranjan
Numerade Educator
01:09

Problem 7

Consider the law of gravitation $F=G M m / r^{2} . G$ is, of course, a constant, but the masses $M$ and $m$ and the distance $r$ between them can take on different values; that is, $M, m$, and $r$ are independent variables. To indicate that we wish to consider $\mathrm{F}$ as a function of all three variables, we can write $F=f(M, m, r)$. What is the value of the following?
(a) $f(2,3,4)$. Ans. $3 G / 8$.
(b) $f(2,3,5)$.
(c) $f(3,4,0)$. Ans. Undefined.

Carson Merrill
Carson Merrill
Numerade Educator
03:14

Problem 8

If $f(x, y)=\sin (x+y)$, what is $f(0,0), f(0, \pi), f\left(-\frac{\pi}{2}, 0\right), f\left(\frac{\pi}{2}, \frac{\pi}{4}\right)$ Ans. $0,0,-1,1 / \sqrt{2}$

Gregory Higby
Gregory Higby
Numerade Educator