Multivariate Calculus - Bruce E. Shapiro
Multivariate Calculus - Bruce E. Shapiro Multivariate Calculus - Bruce E. Shapiro
100 LECTURE 13. TANGENT PLANES Revised December 6, 2006. Math 250, Fall 2006
Lecture 14 Unconstrained Optimization Every continuous function of two variables f(x, y) that is defined on a closed, bounded set attains both a minimum and a maximum. The process of finding these points is called unconstrained optimization. The process of finding a maximum or a minimum subject to a constraint will be discussed in the following section. Definition 14.1 Let f(x, y) be a function defined on some set S, and let (a, b) ∈ S be a point in S. Then we say that 1. f(a, b) is a global maximum of f on S if f(a, b) ≥ f(x, y) for all (x, y) ∈ S. 2. f(a, b) is a global minimum of f on S if f(a, b) ≤ f(x, y) for all (x, y) ∈ S. 3. f(a, b) is a global extremum if it is either a global maximum or a global minimum on S Theorem 14.1 Let f(x, y) be a continuous function on some closed, bounded set S ⊂ R 2 . Then f(x, y) has both a global maximum value and a global minimum value on S. The procedure for finding the extrema (maxima and minima) of functions of two variables is similar to the procedure for functions of a single variable: 1. Find the critical points of the function to determine candidate locations for the extrema (in the single-dimensional case, these were points where the derivative is zero or undefined, and the boundary points of the interval); 2. Examine the second derivative at the candidate points that do not lie on the border (in one-dimension, we had f ′′ (a) < 0 at local maxima and f ′′ (a) > 0 at local minima). 3. Compare internal extrema with the value of the function on the boundary points and at any points where the derivative or second derivative is undefined to determine absolute extrema. Definition 14.2 Let f(x, y) be defined on some set J ⊂ R 2 . Then the candidate extrema points occur at 101
- Page 61 and 62: LECTURE 7. CYLINDRICAL AND SPHERICA
- Page 63 and 64: Lecture 8 Functions of Two Variable
- Page 65 and 66: LECTURE 8. FUNCTIONS OF TWO VARIABL
- Page 67 and 68: LECTURE 8. FUNCTIONS OF TWO VARIABL
- Page 69 and 70: LECTURE 8. FUNCTIONS OF TWO VARIABL
- Page 71 and 72: Lecture 9 The Partial Derivative De
- Page 73 and 74: LECTURE 9. THE PARTIAL DERIVATIVE 6
- Page 75 and 76: LECTURE 9. THE PARTIAL DERIVATIVE 6
- Page 77 and 78: LECTURE 9. THE PARTIAL DERIVATIVE 6
- Page 79 and 80: Lecture 10 Limits and Continuity In
- Page 81 and 82: LECTURE 10. LIMITS AND CONTINUITY 6
- Page 83 and 84: LECTURE 10. LIMITS AND CONTINUITY 7
- Page 85 and 86: Lecture 11 Gradients and the Direct
- Page 87 and 88: LECTURE 11. GRADIENTS AND THE DIREC
- Page 89 and 90: LECTURE 11. GRADIENTS AND THE DIREC
- Page 91 and 92: LECTURE 11. GRADIENTS AND THE DIREC
- Page 93 and 94: Lecture 12 The Chain Rule Recall th
- Page 95 and 96: LECTURE 12. THE CHAIN RULE 83 The p
- Page 97 and 98: LECTURE 12. THE CHAIN RULE 85 Examp
- Page 99 and 100: LECTURE 12. THE CHAIN RULE 87 becau
- Page 101 and 102: LECTURE 12. THE CHAIN RULE 89 Solut
- Page 103 and 104: LECTURE 12. THE CHAIN RULE 91 Examp
- Page 105 and 106: Lecture 13 Tangent Planes Since the
- Page 107 and 108: LECTURE 13. TANGENT PLANES 95 Solut
- Page 109 and 110: LECTURE 13. TANGENT PLANES 97 Multi
- Page 111: LECTURE 13. TANGENT PLANES 99 Accor
- Page 115 and 116: LECTURE 14. UNCONSTRAINED OPTIMIZAT
- Page 117 and 118: LECTURE 14. UNCONSTRAINED OPTIMIZAT
- Page 119 and 120: LECTURE 14. UNCONSTRAINED OPTIMIZAT
- Page 121 and 122: LECTURE 14. UNCONSTRAINED OPTIMIZAT
- Page 123 and 124: LECTURE 14. UNCONSTRAINED OPTIMIZAT
- Page 125 and 126: LECTURE 14. UNCONSTRAINED OPTIMIZAT
- Page 127 and 128: LECTURE 14. UNCONSTRAINED OPTIMIZAT
- Page 129 and 130: Lecture 15 Constrained Optimization
- Page 131 and 132: LECTURE 15. CONSTRAINED OPTIMIZATIO
- Page 133 and 134: LECTURE 15. CONSTRAINED OPTIMIZATIO
- Page 135 and 136: LECTURE 15. CONSTRAINED OPTIMIZATIO
- Page 137 and 138: LECTURE 15. CONSTRAINED OPTIMIZATIO
- Page 139 and 140: Lecture 16 Double Integrals over Re
- Page 141 and 142: LECTURE 16. DOUBLE INTEGRALS OVER R
- Page 143 and 144: LECTURE 16. DOUBLE INTEGRALS OVER R
- Page 145 and 146: LECTURE 16. DOUBLE INTEGRALS OVER R
- Page 147 and 148: Lecture 17 Double Integrals over Ge
- Page 149 and 150: LECTURE 17. DOUBLE INTEGRALS OVER G
- Page 151 and 152: LECTURE 17. DOUBLE INTEGRALS OVER G
- Page 153 and 154: LECTURE 17. DOUBLE INTEGRALS OVER G
- Page 155 and 156: LECTURE 17. DOUBLE INTEGRALS OVER G
- Page 157 and 158: Lecture 18 Double Integrals in Pola
- Page 159 and 160: LECTURE 18. DOUBLE INTEGRALS IN POL
- Page 161 and 162: LECTURE 18. DOUBLE INTEGRALS IN POL
100 LECTURE 13. TANGENT PLANES<br />
Revised December 6, 2006. Math 250, Fall 2006
Change language