In this part we will study a new type of optimization problem: that of finding the maximum (or minimum) value of a function w = f(x, y, z) when we are only allowed to consider points (x, y, z) which are constrained to lie on a surface. The technique we will use to solve these problems is called Lagrange multipliers.
Last Modification: 2020-09-17 Thu 12:19
Captured On [2020-02-06 Thu 12:56] Source Session 39: Statement of Lagrange Multipliers and Example | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Lagrange multipliers Figure 2: Example of using lagrange multipliers Figure 3: Subject to the constraint Figure 4: Both gradient vectors are parallel at level curves Figure 5: System of equations Figure 6: 2 gradiant vector and constraint...
Captured On [2020-02-06 Thu 12:57] Source Session 40: Proof of Lagrange Multipliers | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Why this method is valid? Figure 2: Tangent to contrained level curve Figure 3: Both gradient vector are parallel Figure 4: Warning: Can’t use second derivatives Figure 5: Compare values at \(f\)
Captured On [2020-02-06 Thu 12:57] Source Session 41: Advanced Example | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Build pyramid Figure 2: First draft, can’t solve Figure 3: Another approximation to the solution Figure 4: Function to minimize and constrain Figure 5: Solution
Captured On [2020-02-06 Thu 12:57] Source Session 42: Constrained Differentials | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Non-independent variables Figure 2: \(z\) as a function of \(x\) and \(y\) Figure 3: Take differential Figure 4: Partial derivatives Figure 5: In general 2 What problem can we find when we are computing partial derivatives with non-independent variables?...
Captured On [2020-02-06 Thu 12:59] Source Session 43: Clearer Notation | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Introduction to confussion Figure 2: Equals variable but different partial derivatives Figure 3: Need clearer notation 2 What differences are there between formal and actual partial derivatives? 2.1 Front What differences are there between formal and actual partial derivatives?...
Captured On [2020-02-06 Thu 12:59] Source Session 44: Example | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Area of triangle Figure 2: Rate of change of area wrt \(\theta\) Figure 3: Keeping the right triangle Figure 4: Keeping the right triangle (2) Figure 5: Method 0: Substituting values Figure 6: Systematic methods: Differentials (1)...