# 9.1 - MATH 1081  Wednesday, March 16   ...

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Unformatted text preview: MATH 1081  Wednesday, March 16    Chapter 9 – Section 1    FUNCTIONS OF SEVERAL VARIABLES    Homework #9 (due 3/28):    Section 6.3 #10, 18, 22                    Section 9.1 #28, 46, 48    Clicker Check­in:  Choose any letter to check in now.  In Chapter 5 and 6, we used the first derivative to determine the relative extrema of a function of a single variable. In Chapter 9, we want to find extrema of a function of two variables. We will again use the derivative of the function to help us determine the location of extrema, but to classify an extrema as a maximum or a minimum, we will use a test involving the second derivative, rather than the first derivative. So, to begin, let’s look at the Second Derivative Test for a function of a single variable from Section 5.3. Notice that, unlike the First Derivative Test, this text can only be applied to critical numbers where f ' ( c ) = 0 , not where the derivative does not exist. Further, there is a case (3) where the test is inconclusive. So, it made more sense in the case of the function of a single variable to just use the First Derivative Test, since it always gives a result. The idea behind why the Second Derivative Test works for a function of a single variable can be seen in the following graph. So far this semester, we have considered only functions of a  single variable.  However, many (perhaps even most) models  of applications require more than one variable.    For example, if you want to compute the total interest earned  from a CD investment, you would need to know the amount  invested, the length of time of the investment, and the percent  rate of interest paid on that investment.    As another example, if you are a scuba diver, there are dangers  from too much nitrogen entering your system.  The amount  that enters your system depends both on the depth of your  dive and the length of time of your dive.    In such cases, we need to analyze functions of more than one  variable (or multi‐variable functions).  The notation that we  use for such functions is similar to that for single‐variable  functions.    We will write  f ( x1 , x2 , x3 , ..., xn )  for a function with  n  variables.   As with a function of a single variable, there is exactly ONE  output value from each  n ‐tuple input of real numbers in the  domain of  f .    So, for example, we might write the total amount of interest  earned from a CD investment as  I = f ( P, t , r )  where  P   represents the amount invested,  t  represents the length of  time, and  r  represents the percent rate of interest paid.  For the sake of being able to visualize a graph, we will  primarily restrict ourselves to analyzing only functions of two  variables, which we will write as  z = f ( x, y ) .    The domain of this function is the set of all ordered pairs ( x, y )   that are allowed as inputs in the function.    The range of this function is the set of all  z  that are outputs of  the function.    The graph can be drawn in 3‐dimensions with the 3 axes: one  for  x , one for  y , and one for  z .     Such a set of axes would look like        An example of the graph of a function of 2 variables is          For this function, if look at the  cross­section for any fixed  z  we  will see a circle,  k = x 2 + y 2 .    If we look at a cross‐section for  any fixed  x  we will see an  upward parabola,  z = k + y 2 .    If we look at a cross‐section for  any fixed  y  we will see an  upward parabola,  z = x 2 + k .            You will not be asked to graph any such functions in this class,  but we will be evaluating and examining the rate of change of  such functions at various points.    To evaluate such a function, we do much like we do with a  single variable.  We simply plug in the indicated value for the  appropriate variable and complete the arithmetic operations  shown.    For example,   if  f ( x, y ) = 3x 2 − xy + 8 y ,     2 then  f ( 2, −4 ) = 3( 2 ) − ( 2 ) ( −4 ) + 8 ( −4 ) = −12 .  In order to examine the rate of change of such a function, we  will focus on only one variable at a time.  In other words, we  compute the derivative with respect to one variable by  considering all the other variables in the function to be  constant at that moment.    Looking at the paraboloid,  z = x 2 + y 2 , this means that we will  actually have 2 derivatives to examine:     one when we consider  x  to be constant and have  z = k + y 2  and     one when we consider  y  to be constant and have  z = x 2 + k .    We call these partial derivatives. In the end, our analysis is much the same, in that a positive  derivative indicates the values of the function are increasing  (as that particular variable increases and the others stay  constant) and a negative derivative indicates the values of the  function are decreasing (as that particular variable increases  and others stay constant).    At points where the direction of the function changes (from  increasing to decreasing or vice versa), we will have potential  maxima and minima.      Look at the point at ( 0, 0 )  on both the paraboloid and  hyperbolic paraboloid and think about where the function is  increasing and where it is decreasing.  Clicker Check­out:  Choose any letter to check out now.      Tomorrow in recitation: Workshop #5    Next time – Section 9.2      HAVE A FUN AND SAFE SPRING BREAK!!  ...
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## This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

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