Lesson17: Functions Of  Several  Variables
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Lesson17: Functions Of Several Variables

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We introduce the concept of function of several variables and various types of functions

We introduce the concept of function of several variables and various types of functions

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Lesson17: Functions Of  Several  Variables Lesson17: Functions Of Several Variables Presentation Transcript

  • Lesson 17 (Section 15.1) Functions of Several Variables Math 20 October 29, 2007 Announcements Congratulations to the 2007 World Series Champion Boston Red Sox! Problem Set 6 on the website. Due October 31. OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323) Prob. Sess.: Sundays 6–7 (SC B-10), Tuesdays 1–2 (SC 116)
  • Outline Multivariable Functions Types of Functions Linear Functions Polynomials Rational functions Cobb-Doublas Functions Finding Domains
  • Multivariable Functions A function is a box which changes numbers to numbers, or vectors to vectors, or dogs to cats, or whatever. There are lots of functions which naturally have multiple inputs and a single output.
  • Multivariable Functions A function is a box which changes numbers to numbers, or vectors to vectors, or dogs to cats, or whatever. There are lots of functions which naturally have multiple inputs and a single output. The temperature in this room is a function of position and time.
  • Multivariable Functions A function is a box which changes numbers to numbers, or vectors to vectors, or dogs to cats, or whatever. There are lots of functions which naturally have multiple inputs and a single output. The temperature in this room is a function of position and time. The production of an economy is a function of capital (money and goods invested) and labor
  • Multivariable Functions A function is a box which changes numbers to numbers, or vectors to vectors, or dogs to cats, or whatever. There are lots of functions which naturally have multiple inputs and a single output. The temperature in this room is a function of position and time. The production of an economy is a function of capital (money and goods invested) and labor I derive happiness from eating bacon and eggs for breakfast.
  • Multivariable Functions A function is a box which changes numbers to numbers, or vectors to vectors, or dogs to cats, or whatever. There are lots of functions which naturally have multiple inputs and a single output. The temperature in this room is a function of position and time. The production of an economy is a function of capital (money and goods invested) and labor I derive happiness from eating bacon and eggs for breakfast. In Math 1a and its equivalents we studied functions of one variable. Now let’s look at functions of two variables.
  • Outline Multivariable Functions Types of Functions Linear Functions Polynomials Rational functions Cobb-Doublas Functions Finding Domains
  • Definition A linear function of the variables x1 , . . . , xn takes the form f (x1 , . . . , xn ) = a1 x1 + · · · + an xn where a1 , . . . , an are constants.
  • Example of a linear function Example Suppose a baker makes rolls and cookies. Rolls contain two ounces of flour and one ounce of sugar per each. Cookies contain two ounces of sugar and one ounce of flour per each. The profit on each roll sold is 8 cents, while the profit on each cookie sold is 10 cents. Express explicitly the profit, amount of flour needed, and amount of sugar needed to product x rolls and y cookies.
  • Example of a linear function Example Suppose a baker makes rolls and cookies. Rolls contain two ounces of flour and one ounce of sugar per each. Cookies contain two ounces of sugar and one ounce of flour per each. The profit on each roll sold is 8 cents, while the profit on each cookie sold is 10 cents. Express explicitly the profit, amount of flour needed, and amount of sugar needed to product x rolls and y cookies. Solution p(x, y ) = 8x + 10y
  • Example of a linear function Example Suppose a baker makes rolls and cookies. Rolls contain two ounces of flour and one ounce of sugar per each. Cookies contain two ounces of sugar and one ounce of flour per each. The profit on each roll sold is 8 cents, while the profit on each cookie sold is 10 cents. Express explicitly the profit, amount of flour needed, and amount of sugar needed to product x rolls and y cookies. Solution p(x, y ) = 8x + 10y bf (x, y ) = 2x + y
  • Example of a linear function Example Suppose a baker makes rolls and cookies. Rolls contain two ounces of flour and one ounce of sugar per each. Cookies contain two ounces of sugar and one ounce of flour per each. The profit on each roll sold is 8 cents, while the profit on each cookie sold is 10 cents. Express explicitly the profit, amount of flour needed, and amount of sugar needed to product x rolls and y cookies. Solution p(x, y ) = 8x + 10y bf (x, y ) = 2x + y bs (x, y ) = x + 2y
  • Polynomials A polynomial in variables x1 , . . . , xn is a linear combination of products of the variables x1 , . . . , xn
  • able to form groups of different sizes and composition. For the defined boarding problem, we want to assign each ExampleN,(i,j j)!theaaircraft. Letgroupdecision kvariablewith G 1= if{1,seat 3, …,isg}, where to is the total number0of groups used all seat to boarding k, with ! G, 2, g in boarding the x= (i, j) assigned group k and x = otherwise, for ijk ijk i! M, k ! G. The cost of assigning passengers to board a plane in a certain The complete formulation of the aircraft -boarding problem is presented below. In this formulation, the equation numbers alongside the model serve to clarify the purpose of each set of expressions. In the objective function, we have different penalties for each type of interference. Seat interferences have penalties represented by ! s and aisle order can be modeled by interferences by ! a . The penalties associated with the different types of interferences, capture their relative contribution to the total delay of the boarding procedure. These penalties are explained in detail in the next section. (1a) s s quot; quot;x x iBk x iCk + !1 quot; quot;x Minimize: z = !1 x iEk x iDk + iAk iFk i! N k !G i! N k !G (1b) s s x x + !s quot; quot; quot;x quot; quot;x quot;x x x +! xx+ ! 2 iAk iBk iCl 3 iAk iBl iCk 4 iAl iBk iCk i! N k ,l !G :k <l i! N k ,l !G :k < l i! N k ,l !G :k <l !s ! x iEk x iDl + ! s ! x iEl x iDk + ! s ! !x !x !x x iEk x iDk + 2 iFk 3 iFk 4 iFl iquot; N k ,l quot;G :k <l iquot; N k , lquot;G : k < l iquot; N k ,l quot;G :k <l (1c) s s x x + !s ! ! !x ! !x !x x x +! xx + ! 5 iAk iBl iCl 6 iAl iBk iCl 7 iAl iBl iCk iquot; N k , lquot;G : k < l iquot; N k ,lquot; G :k < l iquot; N k ,lquot; G :k <l !s ! ! xiDk xiEl xiFl + ! s6 ! ! xiFl xiEk xiDl + ! s7 ! ! xiFl xiEl xiDk + 5 i quot;N k ,lquot; G :k <l iquot; N k , lquot;G : k < l iquot; N k ,lquot; G: k < l (1d) !s ! x iBm x iCk + ! s ! s !x !x x iBl x iCm + !10 ! !x x iBl x iCk + 8 iAl 9 iAk iAm i quot;N k ,l , mquot; G :k <l < m iquot; N k ,l ,m quot;G :k < l < m iquot; N k , l ,m quot;G : k < l < m s s ! !x ! !x x iBm x iCl + ! x iBk x iCl + ! 11 iAk 12 iAm i quot;N k ,l , mquot;G : k < l < m iquot; N k , l ,m quot;G : k < l < m !s ! x iEm x iDk + ! s ! s !x !x x iEl x iDm + !10 ! !x x iEl x iDk + 8 iFl 9 iFk iFm i quot;N k ,l , mquot; G :k <l < m iquot; N k ,l ,m quot;G :k < l < m iquot; N k , l ,m quot;G : k < l < m s s !11 ! !x x iEm x iDl + !12 ! !x x iEk x iDl + iFk iFm iquot; N k , l ,m quot;G :k < l < m iquot; N k ,l , mquot;G : k < l < m (2a) a a !1 ! ! !x xivk + !1 ! ! !x x ivk + iuk iuk iquot;N u ,vquot;L :v # u k quot;G iquot;N u ,vquot;R:u # v kquot;G (2b) 2!a ! ! !x xivk + 2 iuk iquot; N u,v quot;M :uquot;L ,vquot; R k quot;G (2c) !a ! ! !x xbvk + 3 auk a ,bquot; N :a <b u ,vquot;M k quot;G (2d) !a ! ! xiuk xivl + ! a4 ! ! ! ! xiuk xivl + 4 iquot; N u, vquot; M k , lquot;G :k < l iquot; N u, vquot; M k ,lquot; G :k <l (2e) !a ! ! xiuk xivl + ! a5 ! ! ! ! xivl xiuk + 5 iquot; N u , vquot; M :u quot;L , vquot; R k ,lquot; G :k <l iquot; N u , vquot; M :u quot;L , vquot; R k ,lquot; G :k <l (2f) a ! ! !x x bvl ! 6 auk a, bquot; N :a < b u , vquot; M k ,lquot; G :k <l Subject to: for all i quot; N , j quot; M (3) x =1
  • Outline Multivariable Functions Types of Functions Linear Functions Polynomials Rational functions Cobb-Doublas Functions Finding Domains
  • Cobb-Douglas Production functions In 1928 Charles Cobb and Paul Douglas published a study in which they modeled the growth of the American economy during the period 1899–1922. They considered a simplified view of the economy in which production output is determined by the amount of labor involved and the amount of capital invested. The function they used was of the form P(L, K ) = bLα K β ,
  • Cobb-Douglas Production functions In 1928 Charles Cobb and Paul Douglas published a study in which they modeled the growth of the American economy during the period 1899–1922. They considered a simplified view of the economy in which production output is determined by the amount of labor involved and the amount of capital invested. The function they used was of the form P(L, K ) = bLα K β , Functions of this form are known as Cobb-Douglas functions.
  • Outline Multivariable Functions Types of Functions Linear Functions Polynomials Rational functions Cobb-Doublas Functions Finding Domains