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Applied Calculus For Business, Economics, and Finance

Warren B. Gordon, Walter O. Wang, April Allen Materowski

Chapter 6

An Introduction to Functions of Several Variables

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Problem 1

$f(x, y)=2 x y-3 x^{2} y^{3},$ determine (a) $f(2,1),(\mathrm{b}) f(1,2)$.

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Problem 2

$f(x, y)=x^{2}-y^{2}-3 x^{3} y,$ determine (a) $f(1,-1),(\mathrm{b}) f(2,3)$

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Problem 3

$f(x, y)=100 x^{1 / 4} y^{3 / 4},$ determine (a) $f(1,16),(\text { b) } f(16,81)$.

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Problem 4

$f(x, y)=75 x^{1 / 3} y^{2 / 3},$ determine (a) $f(8,1),(b) f(27,8)$

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Problem 5

$f(x, y, z)=100 x^{1 / 2} y^{1 / 3} z^{1 / 6},$ determine (a) $f(4,8,1),(\mathrm{b}) f(16,27,64)$

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Problem 6

$f(x, y)=2 x^{2}-3 x y^{2}-2 y^{3},$ determine (a) $f(2,-1),(\mathrm{b}) f(-1,2)$

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Problem 7

$f(x, y)=4 x^{3} y^{2}$ determine (a) $f(3,2),(b) f(2,5)$

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Problem 8

$f(x, y, z)=2 x^{2}-3 x y^{2}-2 y^{3} z^{2}+z^{2},$ determine (a) $f(1,-2,3)$, (b) $f(0,1,-2)$

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Problem 9

$f(x, y)=x / y,$ determine (a) $f(3,2),(\text { b) } f(2,3),(\text { c) } f(x+h, y)$

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Problem 10

$f(x, y)=10 e^{x^{2}-y^{2}},$ determine (a) $f(1,1),$ (b) $f(x, x),$ (c) $f(x, y+k)$

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Problem 11

$f(x, y)=25 \ln \left(x^{2}+y^{2}\right),$ determine (a) $f(1,0),$ (b) $f(0,1),(\text { c) } f(x, x)$

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Problem 12

$f(x, y, z)=100 \sqrt{x+2 y-3 z},$ determine (a) $f(5,4,0)$, (b) $f(42,-4,-3)$

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Problem 13

$S(R, r, n)=R\left(\frac{(1+r)^{n}-1}{r}\right)$ determine $S(10,000,0.0025,24)$

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Problem 14

In Exercise 1, determine (a) $\lim _{h \rightarrow 0} \frac{f(x+h, y)-f(x, y)}{h}$
(b) $\lim _{k \rightarrow 0} \frac{f(x, y+k)-f(x, y)}{k}$,
(c) $\lim _{h \rightarrow 0} \frac{f(2+h, y)-f(2, y)}{h}$,
(d) $\lim _{k \rightarrow 0} \frac{f(x, 1+k)-f(x, 1)}{k}$.

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Problem 15

In Exercise $2,$ determine (a) $\lim _{h \rightarrow 0} \frac{f(x+h, y)-f(x, y)}{h}$,
(b) $\lim _{k \rightarrow 0} \frac{f(x, y+k)-f(x, y)}{k},$
(c) $\lim _{h \rightarrow 0} \frac{f(2+h, y)-f(2, y)}{h},$
(d) $\lim _{k \rightarrow 0} \frac{f(x, 3+k)-f(x, 3)}{k}.$

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Problem 16

In Exercise $8,$ determine (a) $\lim _{h \rightarrow 0} \frac{f(x+h, y, z)-f(x, y, z)}{h}$,
(b) $\lim _{k \rightarrow 0} \frac{f(x, y+k, z)-f(x, y, z)}{k},$
(c) $\lim _{l \rightarrow 0} \frac{f(x, y, z+l)-f(x, y, z)}{l},$

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Problem 17

The cost of manufacturing a rectangular box is as follows: The base costs 4 dollar per square foot, each of the sides cost 2 dollar per square foot and the top costs 1 dollar per square foot. Determine the cost function for the manufacture of this box. (Let $x$ be the length, $y$ the width and $z$ the height of the box.)

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