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Chapter 14: Partial Derivatives (14.5 The Chain Rule (Implicit…
Chapter 14: Partial Derivatives
14.1 Functions of several variables
Definition
The format of domain
How to draw the graph given the equation
Level Curves
Definition
Problem types
Given the level curve, determine the value asked by the question
Sketch different level curves in a family
Level Surface
14.2 Limits and Continuity
Definition
Determine whether the limit exists
x-axis / y-axis approach
y = kx
y^2 = x
Spherical coordinate
Continuous Definition
All polynomials are continuous on R^2; any rational functions are continuous on its domain
Composition and the relation with continuity
How to determine if a function f(a, b) is continuous on (a, b)
f(a, b) exists
lim f(x, y) = f(a, b)
lim f(x, y) exists
14.6 Directional Derivatives and the Gradient Vector
Definition
Calculation/Theorem
Proof: still have questions about
Partial derivatives with respect to x and y are just special cases of directional derivatives
Gradient
Definition
In 3 var cases
Maximizing the directional derivatives
Tangent Plane
Definition
The normal line to S at P is the line passing through P and perpendicular to the tangent plane
Significance of Gradient Vector
Since normal line is the perpendicular line to the tangent plane, and gradient vector is the vector whose direction is vertically perpendicular to every point on the surface, the gradient vector determines the direction of the tangent plane
14.3 Partial Derivatives
Definition:
Notations
Interpretation
Higher derivatives
Definition
Case when fxy = fyx
Note: when doing higher derivative fxy, you should take the partial derivative of x first, and then take the partial derivative of y
14.5 The Chain Rule
Case 1
Case 2
General Case
Example 6
Example 7
Implicit Differentiation
3 var case
Implicit function theorem
14.4 Tangent Planes and Linear Approximation
Tangent Planes
Definition: let C1 and C2 be two curves obtained by intersecting x=x0, y=y0 planes with the surface S(z = f(x, y)). Then the point P lies on both C1 and C2. Let T1 and T2 be the tangent lines to the curves C1 and C2 at the point P. Then the tangent plane to the surface at the point P is defined to be the plane that contains both tangent lines T1 and T2
14.7 Maximum and Minimum Values
Definition
Theorem
Proof
Geometric meaning: but why horizontal tangent plane
Critical point
Definition
Critical points -> f'x = 0, f'y = 0 || one of these derivatives does not exist
Meaning: could have local maximum/minimum or neither
Example:
saddle point
At (x,y) it is both the maximum in x direction and minimum in y direction
Second Derivative Test
Application
Find the extreme value: can actually establish a function based on the setting and then use second derivative test
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Absolute maximum and minimum
Closed set
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Proof
Techniques to discern a graph given the functions
as x or y increases, the other remain fixes, how f(x,y) goes
limit
maximum / minimum
