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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) $$9-x^{2}-9 y^{2}>0$$

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

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

 

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