2A: Functions of Two Variables, Tangent Approximation and Optimization

We start this unit by learning to visualize functions of several variables using graphs and level curves. Following this we will study partial derivatives. These will be used in the tangent approximation formula, which is one of the keys to multivariable calculus. It ties together the geometric and algebraic sides of the subject and is the higher dimensional analog of the equation for the tangent line found in single variable calculus. We will use it in part B to develop the chain rule. We will apply our understanding of partial derivatives to solving unconstrained optimization problems. (In part C we will solve constrained optimization problems.) Last Modification: 2020-02-07 Fri 16:44 ...

Captured On [2020-02-05 Wed 22:39] Source Session 24: Functions of Two Variables: Graphs | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Function of 1 variable Figure 2: Example of multivariable functions Figure 3: For simplicity, functions based on 2 or 3 variables Figure 4: Visualize a function of 2 variables...

Captured On [2020-02-05 Wed 22:40] Source Session 25: Level Curves and Contour Plots | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Contour plot definition Figure 2: Level of contour plot Figure 3: Contour plot examples Figure 4: Contour plot guide 2 What is a contour plot? 2.1 Front What is a contour plot?...

Captured On [2020-02-05 Wed 22:40] Source Session 26: Partial Derivatives | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Derivative of a function of 1 variable Figure 2: Approximation formula Figure 3: Partial derivative definition Figure 4: Partial derivative geometrically Figure 5: Example of partial derivative 2 What is a partial function of a multivariable function?...

Captured On [2020-02-06 Thu 10:31] Source Session 27: Approximation Formula | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Review partial derivatives Figure 2: Approximation formula Figure 3: Justify approximation formula Figure 4: Tangent line to the same point Figure 5: Tagent plane through 2 tangent lines 2 What is a tangent plane?...

Captured On [2020-02-06 Thu 10:32] Source Session 28: Optimization Problems | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Applications of Partial Derivatives Figure 2: At local min or max Figure 3: Definition of critical point Figure 4: Solve system of equations of partial derivatives Figure 5: What kind of critical point is it?...

Captured On [2020-02-06 Thu 10:32] Source Session 29: Least Squares | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Least squares interpolation Figure 2: Find best “a” and “b” Figure 3: Partial derivatives “a” and “b” Figure 4: Simply minimization of partial derivatives Figure 5: 2x2 Linear system Figure 6: Least square for exponential data...

Captured On [2020-02-06 Thu 10:32] Source Session 30: Second Derivative Test | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall critical points Figure 2: How do we find global min/man? Figure 3: Second derivative test Figure 4: Squared terms Figure 5: Studying cases Figure 6: When \(4ac - b^{2} > 0\)...