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• Notes

Q1: $g(x, y)=\cos (x+2 y)$

(a) g$(2,-1)$
(b) Domain?
(c) Range?

(a) 9$(2,-1)$

$g(x, y)=\cos (x+2 y)$

(1) $g(2,-1)=\cos (2-2)$

(2) $g(2,-1)=\cos (0)$

(3) $g(2,-1)=1$

(b) Domain

$g(x, y)=\cos (x+2 y)$

$D=\left\{(x, y) \in R^{2}\right\}$

(c) Range

$y=\cos x \quad$ Range $y(-1,1)$

$g(x)$ Range

$1\leq \cos (x+2 y) \leq 1$

Range $= [-1 , 1]$

Q2: $g(x, y, z)=x^{3} y^{2} z \sqrt{10-x-y-z}$

(1) $g(1,2,3)$

(2) Domain

(1) $g(1,2,3)=(1)^{3} \times(2)^{2} \times 3 \times \sqrt{10-1-2-3}$

$=1 \times 4 \times 3 \times \sqrt{4}$

$= 12 \times 2$

$g(1,2,3)=24$

(2) Domain: $g(x , y , z)$

(1) $10-x-y-z \geq 0$

$-x-y-z \geq-10$

$x+y+z \leq 10$

$D=\{(x, y, z) | x+y+z \leq 10\}$

Q3: $f(x, y)=\ln \left(9-x^{2}-9 y^{2}\right)$

Req: (1) D??

(2) graph

$9-x^{2}-9 y^{2}>0$

$D_{f}=\left\{(x, y) \in R^{2} | 9-x^{2}-9 y^{2}>0\right\}$

(2) $9-x^{2}-9 y^{2}>0$

$\Rightarrow x^{2}+9 y^{2}<9$

$\Rightarrow \frac{x^{2}}{9}+\frac{y^{2}}{1}<1$