VC.03 Gradient Vectors, Level Curves, Maximums/Minimums/Saddle Points.

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Presentation transcript:

VC.03 Gradient Vectors, Level Curves, Maximums/Minimums/Saddle Points

Example 1: The Gradient Vector

Example 2: The Gradient Vector

Definition: The Gradient Vector

Example 3: A Surface and Gradient Field

Example 4: A Surface and Contour Plot

Example 4: Contour Plots in Real Life

Example 4: A Surface and Contour Plot

Example 5: The Gradient Points in the Direction of Greatest Initial Increase

Example 6: A Path Along our Surface

Example 6: Tangent Vectors and Gradient Vectors on Our Path

The Derivative of f(x(t),y(t)) With Respect to t

The Chain Rule in n-Dimensions

Chain Rule Proves the Gradient is Perpendicular to the Level Curve

Example 7: Using the Chain Rule

Example 8: Identifying Local Extrema from the Gradient

Example 9: Another Visit to Our Surface

Example 10: Level Surfaces and Entering the Fourth Dimension

Example 11: A Different Analogy for the Fourth Dimension?

z = f(x,y)w = f(x,y,z) Graph3-D surface4-D (hyper)surface Can’t truly graph it! Level SetsLevel Curve k = f(x,y) Level Surface k = f(x,y,z) Gradient Vectors2D Vectors3D Vectors Final Thoughts: f(x,y) versus f(x,y,z)