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12.1 Functions Notes

# 12.1 Functions Notes - 1 2.1  F u n c t io n s  o f S...

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Unformatted text preview: 1 2 .1  F u n c t io n s  o f S e v e r a l  V a r ia b l e s   O n e  v a r ia b l e   The statement y = f(x) or “y is a function of x” means that y  depends on x.    x = independent variable  y= dependent variable  Domain = {x | f(x) is defined}  Range = {f(x) | x is a member of the domain}                            T w o  v a r ia b l e s   z = f(x,y) or “ z is a function of x and y” means that z depends  on both x and y.  x and y = independent variables  (Values are assigned  independently to x and y.)  z = dependent variable  Domain = {(x, y) | f(x, y) is defined}  Range = {f(x, y) | (x, y) is an element in the domain}                         Three or more variables can’t be graphed, but the concepts  stay the same.   For example w = f(x, y, z):  w is the dependent  variable while x, y, and z are the independent variables.  Ex.  f ( x, y ) = y 2 ! x 2   Domain =  Range =      f ( x, y ) = x 2 + y 2   Ex.  Domain =  Range =    f ( x, y ) = x 2 + y   Ex.  Domain =  Range =      Ex.  ! y\$ f ( x, y) = arctan # & " x% Domain =  Range =      Ex.  Below is a chart of elevation above sea level at any point  (x, y) in feet.     x               y   ‐6   ‐4   ‐2   0  2  4  6  ‐4   152  186  209  218  209  186  152  ‐2   193  236  266  277  266  236  193  0  209  256  288  300  288  256  209  2  193  236  266  277  266  236  193  4  152  186  209  218  209  186  152    f(4,2) =  The value of x when y = 0 so that the elevation is 300 is  The value of x when y = 0 so that the elevation is 256 is      f ( x , y ) = x 2 + y 2 .  Ex.  Sketch the graph of                   Ex.  Sketch the graph of  f ( x , y ) = y 2 ! x 2 .                  Do:  Find the domain and range of problems 1 and 2 and  sketch the domain.  1. x!y g ( x, y ) = x+y       2.   h( x, y) = ln ( x + y ! 1)       3.  Graph      f ( x, y) = cos y   L e v e l  C u rv e s  A level curve (or contour) for a function z = f(x, y) is the curve  of intersection of the surface z = f(x, y) and the horizontal  plane z = k.    f ( x, y ) = x 2 + y 2   Ex.                y g ( x, y ) = 2 x Ex.                Note:  1. The equation of a contour (level curve) is f(x, y) = k,  plotted in the x‐y coordinate system.  2.  All points on a particular contour have the same z value.   That is, f is constant on a level curve.  3. Contours cannot intersect.    Do: 1.  Draw and describe the level curves for k = 0, 1, 2 of  f ( x, y) = x .  What is the domain and range of this  function?    2.  Draw and describe the level curves for k = 0, 2, 5 for  f ( x , y ) = 4 x 2 + y 2 + 1 .                                          ...
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