calculus

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 [2020-02-05 Wed 19:43] Source Session 3: Uses of the Dot Product: Lengths and Angles | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Computing lengths and angles Figure 2: Resolution of computing an angle Figure 3: Meaning of sign of a dot product Figure 4: Detect orthogonality Figure 5: Plane throught \(O\), perpendicular to \(\vec{A}\)

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 \(4ac - b^{2} > 0\)...

Captured On [2020-02-06 Thu 12:28] Source Session 32: Total Differentials and the Chain Rule | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Differentials and implicit differentiation Figure 2: Example of implicit differentiation Figure 3: Total differential Figure 4: What can do with differentials Figure 5: What can do with differentials (2) Figure 6: Chain Rule...

Captured On [2020-02-06 Thu 12:28] Source Session 34: The Chain Rule with More Variables | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Chain rule with more variables Figure 2: Expansion of chain rule Figure 3: Partial derivative with more variables Figure 4: Example with polar coordinates 2 How is the chain rule for partial derivative with more than one independent variable?...

Captured On [2020-02-06 Thu 12:28] Source Session 35: Gradient: Definition, Perpendicular to Level Curves | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall chain rule Figure 2: Definition of gradient Figure 3: Gradient is perpendicular to level surface Figure 4: Example of Gradient (1) Figure 5: Example of Gradient (2) 2 What is the definition of gradient?...

Captured On [2020-02-06 Thu 12:29] Source Session 36: Proof | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Curve that stays on the level \(w=c\) Figure 2: Chain rule Figure 3: Gradient perpendicular to tangent vector Figure 4: Any vector of tangent plane is perpendicular to gradient 2 How can we proof that gradient is perpendicular to level curves and surfaces?...

Captured On [2020-02-06 Thu 12:29] Source Session 37: Example | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Find tangent plane to surface Figure 2: Gradient is the normal vector of tangent plane Figure 3: Another way throught linear approximation Figure 4: Meaning of approximation 2 How can we proof that gradient is perpendicular to level curves and surfaces?...

Captured On [2020-02-06 Thu 12:29] Source Session 38: Directional Derivatives | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Directional derivatives Figure 2: Straight line trayectory Figure 3: Directional derivative in direction of \(\vu{u}\) Figure 4: Slope of a slice Figure 5: Directional derivative as a components of gradient Figure 6: Geometrically Figure 7: Fastest increase of \(w\)...

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 [2020-02-05 Wed 19:46] Source Session 4: Vector Components | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Review of dot product Figure 2: Components of \(\vec{A}\) along direction \(\vec{u}\) Figure 3: Pendulum problem with projections 2 Which is it the angle between 2 vectors? 2.1 Front Which is it the angle between 2 vectors?...

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)...

Captured On [2020-02-06 Thu 18:02] Source Session 45: Review of Topics | Exam 2 | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare Figure 1: Review 1 Figure 2: Review 2 Figure 3: Review 3 Figure 4: Review 4 Figure 5: Review 5 Figure 6: Review 6 Figure 7: Review 7 Figure 8: Review 8 Figure 9: Review 9 Figure 10: Review 10 Figure 11: Review 11...

Captured On [2020-02-06 Thu 13:14] Source Session 47: Definition of Double Integration | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall integrals of 1 variable function Figure 2: Meaning of double integral Figure 3: Small pieces of area Figure 4: Computing double integrals taking slices Figure 5: Iterated integrals 2 Can we compute a double integrals as a limit of Riemann sums?...

Captured On [2020-02-06 Thu 13:14] Source Session 48: Examples of Double Integration | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Example of iterated integral Figure 2: Inner integral Figure 3: Outer integral Figure 4: Same function, other region Figure 5: Inner function, more complex region Figure 6: Inner cont., outer function...