1/ For which surface below is z a function of x and y ? Explain why or why not b

Question

1/ For which surface below is z a function of x and y ? Explain why or why not by recalling the definition of a function "For every value of...., there is at most one value of..."

a/ z=sqrt(x+y)

b/ 2x-y+3z=6

c/ z=5

d/ x+y=2

e/ x^2+y^2+z^2=1

2/ Sketch the domain of the function f(x,y)=1/sqrt(x^2-y) on the axes on the right. What kind of (x,y) pairs can you put into this function and get out of value? (Be sure to shade the appropriate region in the xy-plane where these (x,y) pairs are provide the proper boundary

b/ State the range of f(x,y) using interval notation. What are the possible output values from this function ?

3/

a/ What conic section is formed by the intersection of the surface z=4x^2+y^2 and the plane y=1 ?

b/ what conic section is formed by the intersection of the surface z=4x^2 +y^2 and the plane z=1 ?

c/ What is formed by the intersection of the same surface and the xyplane ?

Explanation / Answer

1) a) function bcoz for a given number only one sqrt exists

b) this is an equation of a plane , for a certain values of x and y the number of values for z can only be one as this equation can be written as z = interms of x and y

c) z =5 is a function bcoz for certain values of x and y z can only be 5

d) not a function bcz z can be anything

e)not a function bcoz for certain value of x and y there are 2 values for z +ve and -ve

3) y = 1 so z =4x^2 + 1 so parabola

b) 4x^2 +y^2 =1 ellipse

c) for xy plane z = 0 so x = 0 and y =0 a point

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