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Review for Midterm II

                          Math 20


                     December 4, 2007



Announcements
   Midte...
Outline



                                       Graphing/Contour Plots
   Rank and other Linear Algebra
                ...
Rank and other Linear Algebra
Learning Objectives




          Determine whether a set of vectors is linearly independent...
Linear Independence



   Definition
   Let {a1 , a2 , . . . , an } be a set of vectors in Rm . We say they are
   linearly...
Deciding linear dependence


   We showed
   a1 , . . . , an LD ⇐⇒ c1 a1 + · · · + cn an = 0 has a nonzero sol’n
         ...
Example
Determine if the vectors
                        
                         1    3      0
                   ...
Example
Determine if the vectors
                        
                         1    3      0
                   ...
Example
Determine if the vectors
                        
                         1    3      0
                   ...
Deciding linear independence




   So
   a1 , . . . , an LI ⇐⇒ every column of rref(A) has a leading entry to it
        ...
Example
Determine if the vectors
                        
                         1  3     0
                      ...
Solution

           1    3   0            1   0     3
                                            
                  ...
Rank




  Definition
  The rank of a matrix A, written r (A) is the maximum number of
  linearly independent column vector...
Rank




  Definition
  The rank of a matrix A, written r (A) is the maximum number of
  linearly independent column vector...
Example
Since                              
                     130        10 3
               rref 0 −2 2 = 0 1 −...
Example
Since                                
                      1  30    1   0   0
                     0 −2 2 ...
Another way to compute rank




   Theorem
   Book Theorem 14.1 The rank of A is the size of the largest
   nonvanishing m...
Rank and consistency




   Fact
   Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A
   augmented by b.
  ...
Rank and redundancy




  Fact
  Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A
  augmented by b. Suppos...
Rank and freedom




  Fact
  Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A
  augmented by b. Suppose t...
Outline



                                       Graphing/Contour Plots
   Rank and other Linear Algebra
                ...
Eigenbusiness
Learning Objectives




          Determine if a vector is an eigenvalue of a matrix
          Determine if ...
Eigenbusiness




   Definition
   Let A be an n × n matrix. The number λ is called an eigenvalue
   of A if there exists a...
Example
Midterm II, Fall 2006, Problem 4



                 4 −2
    Let A =
                 11
    Problem
         2
 ...
Example
Midterm II, Fall 2006, Problem 4



                 4 −2
    Let A =
                 11
    Problem
         2
 ...
4 −2
Let A =
          11
Problem
Is 0 an eigenvalue for A?
4 −2
Let A =
           11
Problem
Is 0 an eigenvalue for A?

Solution
The number 0 is an eigenvalue for A if and only if ...
Methods




     To find the eigenvalues of a matrix A, find the determinant of
     A − λI. This will be a polynomial in λ ...
Diagonalization Procedure




      Find the eigenvalues and eigenvectors.
      Arrange the eigenvectors in a matrix P an...
Example


  Problem
            23
  Let A =      . Diagonalize.
            21
Example


  Problem
              23
  Let A =        . Diagonalize.
              21

  Solution
  To find the eigenvalues...
To find an eigenvector corresponding to the eigenvalue −1,

                               33      11
                  A+I...
To find an eigenvector corresponding to the eigenvalue 4,

                              −2 3       1 −3/2
               A...
Let
                          13
              P=
                         −1 2
so
                          2 −3
        ...
The Spectral Theorem


   Theorem (Baby Spectral Theorem)
   Suppose An×n has n distinct real eigenvalues. Then A is
   di...
The Spectral Theorem


   Theorem (Baby Spectral Theorem)
   Suppose An×n has n distinct real eigenvalues. Then A is
   di...
Outline



                                       Graphing/Contour Plots
   Rank and other Linear Algebra
                ...
Functions of several variables
Learning Objectives




          identify functions, graphs, and contour plots
          fi...
Types of functions




      linear
      polynomial
      rational
      Cobb-Douglas
      etc.
Examples


  Problem
  In each of the following, find the domain and range of the function.
  Is it linear? polynomial? rat...
Graphing/Contour Plots




   A function of two variables can be visualized by
       its graph: the surface (x, y , f (x,...
Example
Graph and contour plot of f (x, y ) = y − x
Example
Graph and contour plot of f (x, y ) = y − x
                                                 2




               ...
Example
                                        √
                                            y −x
Graph and contour plot ...
Example
                                                √
                                                        y −x
Gra...
Example
Graph and contour plot of f (x, y ) = 4x 2 + 9y 2
Example
Graph and contour plot of f (x, y ) = 4x 2 + 9y 2
                                              2




            ...
Example
Graph and contour plot of f (x, y ) = x 2 − y 2
Example
Graph and contour plot of f (x, y ) = x 2 − y 2
                                                 2




           ...
Example
Graph and contour plot of f (x, y ) = xy
Example
Graph and contour plot of f (x, y ) = xy
                                                 2




                  ...
Example
                                        y
Graph and contour plot of f (x, y ) =   x2
Example
                                                    y
Graph and contour plot of f (x, y ) =               x2

    ...
Example
                                             1
Graph and contour plot of f (x, y ) =
                             ...
Example
                                                             1
Graph and contour plot of f (x, y ) =
             ...
Example
                                        9 − x2 − y2
Graph and contour plot of f (x, y ) =
Example
                                                9 − x2 − y2
Graph and contour plot of f (x, y ) =
                ...
Example
Graph and contour plot of f (x, y ) = ln(x 2 + y 2 )
Example
Graph and contour plot of f (x, y ) = ln(x 2 + y 2 )
                                       3




                ...
Example
                                              2 +y 2 )
Graph and contour plot of f (x, y ) = e −(x
Example
                                              2 +y 2 )
Graph and contour plot of f (x, y ) = e −(x
               ...
Example
Graph and contour plot of f (x, y ) = arcsin(y − x)
Example
Graph and contour plot of f (x, y ) = arcsin(y − x)
                                                2




        ...
Example
                                               y
Graph and contour plot of f (x, y ) = arctan   x
Example
                                                        y
Graph and contour plot of f (x, y ) = arctan            ...
The process




   To differentiate a function of several variables with respect to one
   of the variables, pretend that t...
Examples



  Example
  Let f (x, y ) = 3x + 2xy 2 − 2y 4 . Find both the partial derivatives of
  f.
Examples



  Example
  Let f (x, y ) = 3x + 2xy 2 − 2y 4 . Find both the partial derivatives of
  f.

  Solution
  We hav...
Example
Let w = sin α cos β. Find both the partial derivatives of w .
Example
Let w = sin α cos β. Find both the partial derivatives of w .

Solution
We have
           ∂w                     ...
Example
Let f (u, v ) = arctan(u/v ). Find both the partial derivatives of f .
Example
Let f (u, v ) = arctan(u/v ). Find both the partial derivatives of f .

Solution
For this it’s important to rememb...
Example
           2    2            2
          x1 + x2 + · · · + xn . Find all the derivatives of u.
Let u =
Example
            2    2            2
           x1 + x2 + · · · + xn . Find all the derivatives of u.
Let u =

Solution...
Example
Let f (x, y ) = 3x + 2xy 2 − 2y 4 . Find all the second derivatives.
Example
Let f (x, y ) = 3x + 2xy 2 − 2y 4 . Find all the second derivatives.

Solution

                ∂2f               ...
Tangent Planes


   Fact
   The tangent plane to z = f (x, y ) through (x0 , y0 , z0 = f (x0 , y0 ))
   has normal vector ...
Tangent Planes


   Fact
   The tangent plane to z = f (x, y ) through (x0 , y0 , z0 = f (x0 , y0 ))
   has normal vector ...
Tangent Planes


   Fact
   The tangent plane to z = f (x, y ) through (x0 , y0 , z0 = f (x0 , y0 ))
   has normal vector ...
Outline



                                       Graphing/Contour Plots
   Rank and other Linear Algebra
                ...
Fact (The Chain Rule, version I)
When z = F (x, y ) with x = f (t) and y = g (t), then

         z (t) = F1 (f (t), g (t))...
Fact (The Chain Rule, version I)
When z = F (x, y ) with x = f (t) and y = g (t), then

         z (t) = F1 (f (t), g (t))...
Tree Diagrams for the Chain Rule

                                      F
                                          ∂F
   ...
Fact (The Chain Rule, Version II)
When z = F (x, y ) with x = f (t, s) and y = g (t, s), then

                         ∂z...
Example
                                                   ∂z         ∂z
Suppose z = xy 2 , x = t + s and y = t − s. Find ...
Example
                                                        ∂z         ∂z
Suppose z = xy 2 , x = t + s and y = t − s. ...
Solution (ii)
We have
                z = xy 2                 y =t −s
                           x =t +s
So
       ∂z   ∂...
Theorem (The Chain Rule, General Version)
Suppose that u is a differentiable function of the n variables
x1 , x2 , . . . , ...
Implicit Differentiation
The Big Idea




    Fact
    Along the level curve F (x, y ) = c, the slope of the tangent line i...
Tree diagram



                         F


                                  y
                   x


                  ...
More than two variables
   The basic idea is to close your eyes and use the chain rule:
   Example
   Suppose a surface is...
More than two variables
   The basic idea is to close your eyes and use the chain rule:
   Example
   Suppose a surface is...
Tree diagram



                              F


                              y
                    x                 z
...
Example
Problem 16.8.4




    Problem
    Let D = f (r , P) denote the deman for an agricultural commodity
    when the p...
Solution

   We have
                   f (r , P) − g (w , P) ≡ 0

                                  f −g


              ...
Answer


  So
                                            ∂g
                        ∂P                  ∂w
              ...
Outline



                                       Graphing/Contour Plots
   Rank and other Linear Algebra
                ...
Optimization
Learning Objectives




          Find the critical points of a function defined on an open set
          (so ...
Theorem (Fermat’s Theorem)
Let f (x, y ) be a function of two variables. If f has a local
maximum or minimum at (a, b), an...
Theorem (The Second Derivative Test)
Let f (x, y ) be a function of two variables, and let (a, b) be a
critical point of f...
Example

  Problem
  Find and classify the critical points of

                        f (x, y ) = 4xy − x 4 − y 4
Example

  Problem
  Find and classify the critical points of

                        f (x, y ) = 4xy − x 4 − y 4


  Sol...
The second derivatives are
              ∂2f                        ∂2f
                   = −12x 2                   =4
 ...
Graph and contour plot of f (x, y ) = 4xy − x 4 − y 4
                                              2




                ...
Theorem (The Method of Lagrange Multipliers)
Let f (x1 , x2 , . . . , xn ) and g (x1 , x2 , . . . , xn ) be functions of s...
Problem
Find the critical points and values of

                    f (x, y ) = ax 2 + 2bxy + cy 2

subject to the constra...
Problem
Find the critical points and values of

                    f (x, y ) = ax 2 + 2bxy + cy 2

subject to the constra...
The critical values are
                                 ab      x
            f (x, y ) = x   y
                         ...
The critical values are
                                 ab         x
            f (x, y ) = x   y
                      ...
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Midterm II Review Session Slides

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Midterm II Review Session Slides

  1. 1. Review for Midterm II Math 20 December 4, 2007 Announcements Midterm I 12/6, Hall A 7–8:30pm ML Office Hours Wednesday 1–3 (SC 323) Old exams and solutions on website
  2. 2. Outline Graphing/Contour Plots Rank and other Linear Algebra Partial Derivatives Linear dependence Differentiation Rank The Chain Rule Eigenbusiness Implicit Differentiation Eigenvector and Eigenvalue Diagonalization Optimization The Spectral Theorem Unconstrained Optimization Functions of several variables Constrained Optimization
  3. 3. Rank and other Linear Algebra Learning Objectives Determine whether a set of vectors is linearly independent Find the rank of a matrix
  4. 4. Linear Independence Definition Let {a1 , a2 , . . . , an } be a set of vectors in Rm . We say they are linearly dependent if there exist constants c1 , c2 , . . . , ck ∈ R, not all zero, such that c1 a1 + c2 a2 + · · · + ck an = 0. If the equation only holds when all c1 = c2 = · · · = cn = 0, then the vectors are said to be linearly independent.
  5. 5. Deciding linear dependence We showed a1 , . . . , an LD ⇐⇒ c1 a1 + · · · + cn an = 0 has a nonzero sol’n  c1 . ⇐⇒ a1 . . . an  .  = 0 has a nonzero sol’n . cn A c ⇐⇒ system has some free variables ⇐⇒ rref(A) has a column with no leading entry to it
  6. 6. Example Determine if the vectors   1 3 0 0 , −2 , 2 1 2 1 are linearly dependent.
  7. 7. Example Determine if the vectors   1 3 0 0 , −2 , 2 1 2 1 are linearly dependent. Solution   1 3 0 −1   1 30 −2 2 − 2 2 0 0   −1 1 0 1 ←+ − 1 2     ←+ − 1 3 0 10 3 1 − 1 −3 − 1 0 0 1     00 0 00 0 So the vectors are linearly dependent.
  8. 8. Example Determine if the vectors   1 3 0 0 , −2 , 2 1 2 1 are linearly dependent. Solution   1 3 0 −1   1 30 −2 2 − 2 2 0 0   −1 1 0 1 ←+ − 1 2     ←+ − 1 3 0 10 3 1 − 1 −3 − 1 0 0 1     00 0 00 0 So the vectors are linearly dependent.
  9. 9. Deciding linear independence So a1 , . . . , an LI ⇐⇒ every column of rref(A) has a leading entry to it In ⇐⇒ A ∼ O
  10. 10. Example Determine if the vectors    1 3 0 0 −2  2   , ,  1  2   1  −1 1 0 are linearly dependent.
  11. 11. Solution 1 3 0 1 0 3     −1 − 2 2 1 −1 0 0         1 2 1 0 0 0 −1 −1 0 1 ←+ − 0 1 10 3 100     0 1 −1  0 1 0         0 0 0 0 0 1 −2 00 000 So the vectors are linearly independent.
  12. 12. Rank Definition The rank of a matrix A, written r (A) is the maximum number of linearly independent column vectors in A.
  13. 13. Rank Definition The rank of a matrix A, written r (A) is the maximum number of linearly independent column vectors in A. If A is a zero matrix, we say r (A) = 0.
  14. 14. Example Since    130 10 3 rref 0 −2 2 = 0 1 −1 121 00 0 this matrix has rank 2.
  15. 15. Example Since    1 30 1 0 0  0 −2 2 0 1 0 rref  =  1 2 1 0 0 1 −1 0 1 0 0 0 this matrix has rank 3.
  16. 16. Another way to compute rank Theorem Book Theorem 14.1 The rank of A is the size of the largest nonvanishing minor of A.
  17. 17. Rank and consistency Fact Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A augmented by b. Then the system of linear equations Ax = b has a solution (is consistent) if and only if r (A) = r (Ab ).
  18. 18. Rank and redundancy Fact Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A augmented by b. Suppose that r (A) = r (Ab ) = k < m (m is the number of equations in the system Ax = b). Then m − k of the equations are redundant; they can be removed and the system has the same solutions.
  19. 19. Rank and freedom Fact Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A augmented by b. Suppose that r (A) = r (Ab ) = k < n (n is the number of variables in the system Ax = b). Then n − k of the variables are free; they can be chosen at will and the rest of the variables depend on them, getting infinitely many solutions.
  20. 20. Outline Graphing/Contour Plots Rank and other Linear Algebra Partial Derivatives Linear dependence Differentiation Rank The Chain Rule Eigenbusiness Implicit Differentiation Eigenvector and Eigenvalue Diagonalization Optimization The Spectral Theorem Unconstrained Optimization Functions of several variables Constrained Optimization
  21. 21. Eigenbusiness Learning Objectives Determine if a vector is an eigenvalue of a matrix Determine if a scalar is an eigenvalue of a matrix Find all the eigenvalues of a matrix Find all the eigenvectors of a matrix for a given eigenvalue Diagonalize a matrix Know when a matrix is diagonalizable
  22. 22. Eigenbusiness Definition Let A be an n × n matrix. The number λ is called an eigenvalue of A if there exists a nonzero vector x ∈ Rn such that Ax = λx. (1) Every nonzero vector satisfying (1) is called an eigenvector of A associated with the eigenvalue λ.
  23. 23. Example Midterm II, Fall 2006, Problem 4 4 −2 Let A = 11 Problem 2 Is an eigenvector for A? 1
  24. 24. Example Midterm II, Fall 2006, Problem 4 4 −2 Let A = 11 Problem 2 Is an eigenvector for A? 1 Solution Use the definition of eigenvector: 4 −2 2 6 2 = =3 11 1 3 1 So the vector is an eigenvector corresponding to the eigenvalue 3.
  25. 25. 4 −2 Let A = 11 Problem Is 0 an eigenvalue for A?
  26. 26. 4 −2 Let A = 11 Problem Is 0 an eigenvalue for A? Solution The number 0 is an eigenvalue for A if and only if the determinant of A − 0I = A is zero. But det A = 4 · 1 − 1 · (−2) = 6. So it’s not.
  27. 27. Methods To find the eigenvalues of a matrix A, find the determinant of A − λI. This will be a polynomial in λ (called the characteristic polynomial of A, and its roots are the eigenvalues. To find the eigenvector(s) of a matrix corresponding to an eigenvalue λ, do Gaussian Elimination on A − λI.
  28. 28. Diagonalization Procedure Find the eigenvalues and eigenvectors. Arrange the eigenvectors in a matrix P and the corresponding eigenvalues in a diagonal matrix D. If you have “enough” eigenvectors so that the matrix P is square and invertible, the original matrix is diagonalizable and equal to PDP−1 .
  29. 29. Example Problem 23 Let A = . Diagonalize. 21
  30. 30. Example Problem 23 Let A = . Diagonalize. 21 Solution To find the eigenvalues, find the characteristic polynomial and its roots: 2−λ 3 |A − λI| = = (2 − λ)(1 − λ) − 6 1−λ 2 = λ2 − 3λ − 4 = (λ + 1)(λ − 4) So the eigenvalues are −1 and 4.
  31. 31. To find an eigenvector corresponding to the eigenvalue −1, 33 11 A+I= 22 00 1 So is an eigenvector. −1
  32. 32. To find an eigenvector corresponding to the eigenvalue 4, −2 3 1 −3/2 A − 4I = 2 −3 0 0 3 So is an eigenvector. 2
  33. 33. Let 13 P= −1 2 so 2 −3 1 P−1 = 11 5 Then −1 0 2 −3 1 13 A= −1 2 04 11 5
  34. 34. The Spectral Theorem Theorem (Baby Spectral Theorem) Suppose An×n has n distinct real eigenvalues. Then A is diagonalizable.
  35. 35. The Spectral Theorem Theorem (Baby Spectral Theorem) Suppose An×n has n distinct real eigenvalues. Then A is diagonalizable. Theorem (Spectral Theorem for Symmetric Matrices) Suppose An×n is symmetric, that is, A = A. Then A is diagonalizable. In fact, the eigenvectors can be chosen to be pairwise orthogonal with length one, which means that P−1 = P . Thus a symmetric matrix can be diagonalized as A = PDP ,
  36. 36. Outline Graphing/Contour Plots Rank and other Linear Algebra Partial Derivatives Linear dependence Differentiation Rank The Chain Rule Eigenbusiness Implicit Differentiation Eigenvector and Eigenvalue Diagonalization Optimization The Spectral Theorem Unconstrained Optimization Functions of several variables Constrained Optimization
  37. 37. Functions of several variables Learning Objectives identify functions, graphs, and contour plots find partial derivatives of functions of several variables
  38. 38. Types of functions linear polynomial rational Cobb-Douglas etc.
  39. 39. Examples Problem In each of the following, find the domain and range of the function. Is it linear? polynomial? rational? algebraic? Cobb-Douglas? 9 − x2 − y2 f (x, y ) = y − x (a) (h) f (x, y ) = √ f (x, y ) = y − x (b) (i) f (x, y ) = ln(x 2 + y 2 ) f (x, y ) = 4x 2 + 9y 2 (c) f (x, y ) = x 2 − y 2 (d) 2 +y 2 ) (j) f (x, y ) = e −(x (e) f (x, y ) = xy f (x, y ) = y /x 2 (f) (k) f (x, y ) = arcsin(y − x) 1 (g) f (x, y ) = y (l) f (x, y ) = arctan 16 − x 2 − y 2 x
  40. 40. Graphing/Contour Plots A function of two variables can be visualized by its graph: the surface (x, y , f (x, y ) in R3 a contour plot: a collection of level curves
  41. 41. Example Graph and contour plot of f (x, y ) = y − x
  42. 42. Example Graph and contour plot of f (x, y ) = y − x 2 1 0 4 2 2 0 1 2 4 1 2 0 1 1 0 1 2 2 2 1 0 1 2 2
  43. 43. Example √ y −x Graph and contour plot of f (x, y ) =
  44. 44. Example √ y −x Graph and contour plot of f (x, y ) = 2 1 0 2.0 1.5 2 1.0 1 0.5 0.0 1 0 2 1 1 0 1 2 2 2 1 0 1 2 2
  45. 45. Example Graph and contour plot of f (x, y ) = 4x 2 + 9y 2
  46. 46. Example Graph and contour plot of f (x, y ) = 4x 2 + 9y 2 2 1 0 40 2 20 1 1 0 2 0 1 1 0 1 2 2 2 1 0 1 2 2
  47. 47. Example Graph and contour plot of f (x, y ) = x 2 − y 2
  48. 48. Example Graph and contour plot of f (x, y ) = x 2 − y 2 2 1 0 4 2 2 0 1 2 4 1 2 0 1 1 0 1 2 2 2 1 0 1 2 2
  49. 49. Example Graph and contour plot of f (x, y ) = xy
  50. 50. Example Graph and contour plot of f (x, y ) = xy 2 1 0 4 2 2 0 1 2 4 1 2 0 1 1 0 1 2 2 2 1 0 1 2 2
  51. 51. Example y Graph and contour plot of f (x, y ) = x2
  52. 52. Example y Graph and contour plot of f (x, y ) = x2 2 1 0 5 2 0 1 5 1 2 0 1 1 0 1 2 2 2 1 0 1 2 2
  53. 53. Example 1 Graph and contour plot of f (x, y ) = 16 − x 2 − y 2
  54. 54. Example 1 Graph and contour plot of f (x, y ) = 16 − x 2 − y 2 4 2 0 1.0 4 0.5 2 0.0 2 0 4 2 2 0 2 4 4 4 2 0 2 4 4
  55. 55. Example 9 − x2 − y2 Graph and contour plot of f (x, y ) =
  56. 56. Example 9 − x2 − y2 Graph and contour plot of f (x, y ) = 3 2 1 0 3 2 1 2 1 0 0 2 2 0 2 3 2 3 2 1 0 1 2 3
  57. 57. Example Graph and contour plot of f (x, y ) = ln(x 2 + y 2 )
  58. 58. Example Graph and contour plot of f (x, y ) = ln(x 2 + y 2 ) 3 2 1 0 2 1 1 2 0 1 0 2 2 0 2 3 2 3 2 1 0 1 2 3
  59. 59. Example 2 +y 2 ) Graph and contour plot of f (x, y ) = e −(x
  60. 60. Example 2 +y 2 ) Graph and contour plot of f (x, y ) = e −(x 3 2 1 0 1.0 1 0.5 2 0.0 0 2 2 0 2 3 2 3 2 1 0 1 2 3
  61. 61. Example Graph and contour plot of f (x, y ) = arcsin(y − x)
  62. 62. Example Graph and contour plot of f (x, y ) = arcsin(y − x) 2 1 0 1 2 0 1 1 1 0 2 1 1 0 1 2 2 2 1 0 1 2 2
  63. 63. Example y Graph and contour plot of f (x, y ) = arctan x
  64. 64. Example y Graph and contour plot of f (x, y ) = arctan x 2 1 0 1 2 0 1 1 1 0 2 1 1 0 1 2 2 2 1 0 1 2 2
  65. 65. The process To differentiate a function of several variables with respect to one of the variables, pretend that the others are constant.
  66. 66. Examples Example Let f (x, y ) = 3x + 2xy 2 − 2y 4 . Find both the partial derivatives of f.
  67. 67. Examples Example Let f (x, y ) = 3x + 2xy 2 − 2y 4 . Find both the partial derivatives of f. Solution We have ∂f ∂f = 3 + 2y 2 = 4xy − 8y 3 ∂x ∂y
  68. 68. Example Let w = sin α cos β. Find both the partial derivatives of w .
  69. 69. Example Let w = sin α cos β. Find both the partial derivatives of w . Solution We have ∂w ∂w = − sin α sin β = cos α cos β ∂α ∂β
  70. 70. Example Let f (u, v ) = arctan(u/v ). Find both the partial derivatives of f .
  71. 71. Example Let f (u, v ) = arctan(u/v ). Find both the partial derivatives of f . Solution For this it’s important to remember the chain rule! ∂f 1 ∂u 1 1 = = 2 ∂u v 2v ∂u 1 + (u/v ) 1 + (u/v ) −u ∂f 1 ∂u 1 = = 1 + (u/v )2 ∂v v 1 + (u/v )2 v 2 ∂u Another way to write this is −u ∂f v ∂f =2 =2 u + v2 u + v2 ∂u ∂v
  72. 72. Example 2 2 2 x1 + x2 + · · · + xn . Find all the derivatives of u. Let u =
  73. 73. Example 2 2 2 x1 + x2 + · · · + xn . Find all the derivatives of u. Let u = Solution We have a partial derivative for each index i, but luckily they’re symmetric. So each derivative is represented by: ∂u 1 ∂2 2 2 (x1 + x2 + · · · + xn ) = ∂xi 2 + x 2 + · · · + x 2 ∂xi 2 x n 1 2 xi = 2 2 2 x1 + x2 + · · · + xn
  74. 74. Example Let f (x, y ) = 3x + 2xy 2 − 2y 4 . Find all the second derivatives.
  75. 75. Example Let f (x, y ) = 3x + 2xy 2 − 2y 4 . Find all the second derivatives. Solution ∂2f ∂2f =0 = 4y ∂x 2 ∂x ∂y ∂2f ∂2f = −24y 2 = 4y ∂y 2 ∂y ∂x
  76. 76. Tangent Planes Fact The tangent plane to z = f (x, y ) through (x0 , y0 , z0 = f (x0 , y0 )) has normal vector (f1 (x0 , y0 ), f2 (x0 , y0 ), −1) and equation f1 (x0 , y0 )(x − x0 ) + f2 (x0 , y0 )(y − y0 ) − (z − z0 ) = 0
  77. 77. Tangent Planes Fact The tangent plane to z = f (x, y ) through (x0 , y0 , z0 = f (x0 , y0 )) has normal vector (f1 (x0 , y0 ), f2 (x0 , y0 ), −1) and equation f1 (x0 , y0 )(x − x0 ) + f2 (x0 , y0 )(y − y0 ) − (z − z0 ) = 0 or z = f (x0 , y0 ) + f1 (x0 , y0 )(x − x0 ) + f2 (x0 , y0 )(y − y0 )
  78. 78. Tangent Planes Fact The tangent plane to z = f (x, y ) through (x0 , y0 , z0 = f (x0 , y0 )) has normal vector (f1 (x0 , y0 ), f2 (x0 , y0 ), −1) and equation f1 (x0 , y0 )(x − x0 ) + f2 (x0 , y0 )(y − y0 ) − (z − z0 ) = 0 or z = f (x0 , y0 ) + f1 (x0 , y0 )(x − x0 ) + f2 (x0 , y0 )(y − y0 ) This is the best linear approximation to f near (x0 , y0 ). is is the first-degree Taylor polynomial (in two variables) for f .
  79. 79. Outline Graphing/Contour Plots Rank and other Linear Algebra Partial Derivatives Linear dependence Differentiation Rank The Chain Rule Eigenbusiness Implicit Differentiation Eigenvector and Eigenvalue Diagonalization Optimization The Spectral Theorem Unconstrained Optimization Functions of several variables Constrained Optimization
  80. 80. Fact (The Chain Rule, version I) When z = F (x, y ) with x = f (t) and y = g (t), then z (t) = F1 (f (t), g (t))f (t) + F2 (f (t), g (t))g (t) or dz ∂F dx ∂F dy = + dt ∂x dt ∂y dt
  81. 81. Fact (The Chain Rule, version I) When z = F (x, y ) with x = f (t) and y = g (t), then z (t) = F1 (f (t), g (t))f (t) + F2 (f (t), g (t))g (t) or dz ∂F dx ∂F dy = + dt ∂x dt ∂y dt We can generalize to more variables, too. If F is a function of x1 , x2 , . . . , xn , and each xi is a function of t, then dz ∂F dx1 ∂F dx2 ∂F dxn + ··· + = + dt ∂x1 dt ∂x2 dt ∂xn dt
  82. 82. Tree Diagrams for the Chain Rule F ∂F ∂F ∂y ∂x y x dx dy dt dt t t To differentiate with respect to t, find all “leaves” marked t. Going down each branch, chain (multiply) all the derivatives together. Then add up the result from each branch. dz dF ∂F dx ∂F dy = = + dt dt ∂x dt ∂y dt
  83. 83. Fact (The Chain Rule, Version II) When z = F (x, y ) with x = f (t, s) and y = g (t, s), then ∂z ∂F ∂x ∂F ∂y = + ∂t ∂x ∂t ∂y ∂t ∂z ∂F ∂x ∂F ∂y = + ∂s ∂x ∂s ∂y ∂s F y x s s t t
  84. 84. Example ∂z ∂z Suppose z = xy 2 , x = t + s and y = t − s. Find and at ∂t ∂s (t, z) = (1/2, 1) in two ways: (i) By expressing z directly in terms of t and s before differentiating. (ii) By using the chain rule.
  85. 85. Example ∂z ∂z Suppose z = xy 2 , x = t + s and y = t − s. Find and at ∂t ∂s (t, z) = (1/2, 1) in two ways: (i) By expressing z directly in terms of t and s before differentiating. (ii) By using the chain rule. Solution (i) We have z = (t + s)(t − s)2 = s 3 − ts 2 − t 2 s + t 3 So ∂z = −s 2 − 2ts + 3t 2 ∂t ∂z = 3s 2 − 2ts − t 2 ∂s
  86. 86. Solution (ii) We have z = xy 2 y =t −s x =t +s So ∂z ∂z ∂x ∂z ∂y = + ∂t ∂x ∂t ∂y ∂t = y · 1 + 2xy · 1 = (t − s)2 + 2(t + s)(t − s) 2 ∂z ∂z ∂x ∂z ∂y = + ∂s ∂x ∂s ∂y ∂s = y 2 · 1 + 2xy (−1) = (t − s)2 − 2(t + s)(t − s) These should be the same as in the previous calculation.
  87. 87. Theorem (The Chain Rule, General Version) Suppose that u is a differentiable function of the n variables x1 , x2 , . . . , xn , and each xi is a differentiable function of the m variables t1 , t2 , . . . , tm . Then u is a function of t1 , t2 , . . . , tm and ∂u ∂u ∂x1 ∂u ∂x2 ∂u ∂xn + ··· + = + ∂ti ∂x1 ∂ti ∂x2 ∂ti ∂xn ∂ti In summation notation n ∂u ∂xj ∂u = ∂ti ∂xj ∂ti j=1
  88. 88. Implicit Differentiation The Big Idea Fact Along the level curve F (x, y ) = c, the slope of the tangent line is given by dy dy ∂F /∂x F (x, y ) =− 1 =− = dx dx F ∂F /∂x F2 (x, y )
  89. 89. Tree diagram F y x x ∂F ∂F dy + =0 ∂x ∂y dx F
  90. 90. More than two variables The basic idea is to close your eyes and use the chain rule: Example Suppose a surface is given by F (x, y , z) = c. If this defines z as a function of x and y , find zx and zy .
  91. 91. More than two variables The basic idea is to close your eyes and use the chain rule: Example Suppose a surface is given by F (x, y , z) = c. If this defines z as a function of x and y , find zx and zy . Solution Setting F (x, y , z) = c and remembering z is implicitly a function of x and y , we get ∂F ∂F ∂z ∂z Fx =− + = 0 =⇒ ∂x ∂z ∂x ∂x Fz F F Fy ∂F ∂F ∂z ∂z =− + = 0 =⇒ ∂y ∂z ∂y ∂y Fz F F
  92. 92. Tree diagram F y x z x ∂F ∂F ∂z ∂z Fx =− + = 0 =⇒ ∂x ∂z ∂x ∂x Fz F F
  93. 93. Example Problem 16.8.4 Problem Let D = f (r , P) denote the deman for an agricultural commodity when the price is P and r is the producers’ total advertising expenditure. Let supply be given by S = g (w , P), where w is an index for how favorable the weather has been. Assume gw (w , P) > 0. Equilibrium now requires f (r , P) = g (w , P). Assume that this equation defines P implicitly as a differentiable function of r and w . Compute Pw and comment on its sign.
  94. 94. Solution We have f (r , P) − g (w , P) ≡ 0 f −g r w P w ∂f ∂P ∂g ∂g ∂P − − ∂P ∂w ∂w ∂P ∂w f =g f =g
  95. 95. Answer So ∂g ∂P ∂w = ∂f ∂g ∂w f =g − ∂P ∂P ∂f ∂g ∂g < 0 and > 0. We assumed that > 0. So in this case, ∂P ∂P ∂w ∂P < 0, ∂w f =g meaning the price decreasing with improving weather.
  96. 96. Outline Graphing/Contour Plots Rank and other Linear Algebra Partial Derivatives Linear dependence Differentiation Rank The Chain Rule Eigenbusiness Implicit Differentiation Eigenvector and Eigenvalue Diagonalization Optimization The Spectral Theorem Unconstrained Optimization Functions of several variables Constrained Optimization
  97. 97. Optimization Learning Objectives Find the critical points of a function defined on an open set (so unconstrained) Classify the critical points of a function Find the critical points of a function restricted to a surface (constrained)
  98. 98. Theorem (Fermat’s Theorem) Let f (x, y ) be a function of two variables. If f has a local maximum or minimum at (a, b), and is differentiable at (a,b), then ∂f ∂f (a, b) = 0 (a, b) = 0 ∂x ∂y As in one variable, we’ll call these points critical points.
  99. 99. Theorem (The Second Derivative Test) Let f (x, y ) be a function of two variables, and let (a, b) be a critical point of f . Then 2 2 2 2 ∂2f If ∂xf2 ∂yf2 − ∂x∂y ∂∂ ∂f > 0 and > 0, the critical point is a ∂x 2 local minimum. 2 2 2 2 ∂2f If ∂xf2 ∂yf2 − ∂x∂y ∂∂ ∂f > 0 and < 0, the critical point is a ∂x 2 local maximum. 2 ∂2f ∂2f ∂2f − If < 0, the critical point is a saddle point. ∂x 2 ∂y 2 ∂x∂y All derivatives are evaluated at the critical point (a, b).
  100. 100. Example Problem Find and classify the critical points of f (x, y ) = 4xy − x 4 − y 4
  101. 101. Example Problem Find and classify the critical points of f (x, y ) = 4xy − x 4 − y 4 Solution ∂f ∂f We have ∂x = 4y − 4x 3 and ∂y = 4x − 4y 3 . Both of these are zero when y = x 3 and x = y 3 So x 9 = x. Since x 9 − x = x(x 8 − 1) = x(x 4 + 1)(x 2 + 1)(x + 1)(x − 1) the real solutions are x = 0, x = 1, and x = −1. The corresponding y values are 0, 1, and −1. So the critical points are (0, 0), (1, 1), (−1, −1)
  102. 102. The second derivatives are ∂2f ∂2f = −12x 2 =4 ∂x 2 ∂y ∂x ∂2f ∂2f = −12y 2 =4 ∂x ∂y ∂y ∂x So −3x 2 1 H(x, y ) = 4 −3y 2 1 01 At (0, 0), the matrix is , which has determinant < 0. So 10 it’s a saddle point. At the other two points, the matrix is −3 1 , which has positive determinant. So those points are 1 −3 local maxima.
  103. 103. Graph and contour plot of f (x, y ) = 4xy − x 4 − y 4 2 1 0 0 2 10 20 1 30 1 2 0 1 1 0 1 2 2 2 1 0 1 2 2
  104. 104. Theorem (The Method of Lagrange Multipliers) Let f (x1 , x2 , . . . , xn ) and g (x1 , x2 , . . . , xn ) be functions of several variables. The critical points of the function f restricted to the set g = 0 are solutions to the equations: ∂f ∂g (x1 , x2 , . . . , xn ) = λ (x1 , x2 , . . . , xn ) for each i = 1, . . . , n ∂xi ∂xi g (x1 , x2 , . . . , xn ) = 0. Note that this is n + 1 equations in the n + 1 variables. x1 , . . . , xn , λ.
  105. 105. Problem Find the critical points and values of f (x, y ) = ax 2 + 2bxy + cy 2 subject to the constraint that x 2 + y 2 = 1.
  106. 106. Problem Find the critical points and values of f (x, y ) = ax 2 + 2bxy + cy 2 subject to the constraint that x 2 + y 2 = 1. Solution We have fx = λgx =⇒ 2ax + 2by = λ(2x) fy = λgy =⇒ 2bx + 2cy = λ(2y ) So the critical points happen when ab x x =λ bc y y
  107. 107. The critical values are ab x f (x, y ) = x y bc y x = λ(x 2 + y 2 ) = λ =x yλ y
  108. 108. The critical values are ab x f (x, y ) = x y bc y x = λ(x 2 + y 2 ) = λ =x yλ y So The critical points are eigenvectors! The critical values are eigenvalues!

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