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CAL / Math / MATH 53 / What are the limits of multivariable functions?

What are the limits of multivariable functions?

What are the limits of multivariable functions?

Description

School: University of California Berkeley
Department: Math
Course: Multivariable Calculus
Professor: Edward frenkel
Term: Fall 2016
Tags: multivariable calculus
Cost: 50
Name: Math 53 Midterm 2 Study Guide
Description: Study Guide for Stankova's Spring 2018 Math 53 Multivariable Calculus Class.
Uploaded: 03/30/2018
13 Pages 15 Views 4 Unlocks
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Midterm 2 Study Guide


What is the limits of multivariable functions?



Tuesday, March 27, 2018 1:23 PM

Limits of Multivariable Functions

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I

the function is continuous:ConstantsSine CosinePolynomials+ ‐ * of these functions

the domain of f.Partial Derivatives

It is denoted, for a function f(x,y), fClairaut's Theorem says that if fcontinuous, then they are equal.

Linear Approximation

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Partial derivatives refers to what?



Approach a limit from different paths

f the result for the limit varies, then there exists no limit You can just use the value of a function as the limit if you know 

 

 

Division and square root (of these functions) are 

continuous on their domains

Composition of functions: f(g(x)), assuming range(g(x)) is in   

A partial derivative is a derivative of an expression or function  of multiple variables. We treat everything but our desired  partial variable as constants and take the derivative as normal.  x or fy.

xy (taking (fx)y) and fyx are 


Linear approximation refers to what?



 

- f(x,y) must have continuous partials fx and fy near (x0, y0). - The linear approximation is: We also discuss several other topics like What is the name of telemachus’ mother?

 Cheat Sheets Page 1

○ L = f(x0,y0) + fx(x0,y0)(x‐x0) + fy(x0,y0)(y‐y0)

The Chain Rule 

- The basic chain rule learned in Calculus 1:

- We extend this to the multivariable case with partials:

- We can represent this using a tree diagram:  Cheat Sheets Page 2

 Cheat Sheets Page 3

- Verification.

To verify a solution to a given differential  equation, we attempt to find the quantities in the  differential equation and see if those are indeed  equal, as defined by the equation.

Directional Derivatives: Gradient 

- Definition of the directional derivative:

 Cheat Sheets Page 4

- Gradients are tangent to level curves of a function. Tangent lines and planes 

- Don't forget about the age old question of What are the assumptions, in monopolistic competition?

Tangent line to a curve can be thought of as a tangent  line to an equivalent level curve of a surface:

 Cheat Sheets Page 5

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Find the gradient, and the point you want to be  tangent at.

- Since gradient is tangent to the level curve:

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The general equation for a tangent line/plane at point  P is:

Local min/max of a function of two variables. - Combine the first and second derivative tests. - Second derivative of a function:

- If:

○ D > 0, fxx > 0: minimum

○ D > 0, fxx < 0: maximum We also discuss several other topics like What do we mean by self-esteem or addiction?

○ D < 0: saddle point

○ D = 0: unsure. Try a different test.

 Cheat Sheets Page 6

Finding Global Extrema

Very similar to global extrema in single variable. 

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Check for critical points and then check the  endpoints. However, with more dimensions, our  endpoints may be actual equations or even surfaces. -

A bounded region has a boundary, or it can be  contained within a shape of finite size. Don't forget about the age old question of What is the difference between essential and non-essential of amino acids?
If you want to learn more check out Who uses water?

- A close region is a region that contains its boundary. - Finding the global extrema

Find the critical points of the function, where the  gradient of f = <0,0> or <0,0,0> If you want to learn more check out What does the law of superposition?

Calculate equations for the boundaries of f, or use  Lagrange multipliers.

With equations, find their min/max with critical  points, and find their endpoints as well. ▪

With Lagrange multipliers, solve this system of  equations:

 Cheat Sheets Page 7

○ Then, compare all your points

Note. Lagrange multipliers doesn't always solve  for all the min/max on a region. You have to  checks spots where the gradients or the  derivatives are discontinuous.

Multiple Integrals 

 Cheat Sheets Page 8

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Approximations are the same with these, but we  simply test with more points because there are extra  dimensions.

Iterated Integrals 

 Cheat Sheets Page 9

Polar coordinate integrals 

- You can do the same integral, substituting properly:  Cheat Sheets Page 10

But, you have to multiply your f(r,theta) by r after 

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substituting. This is the Jacobian, a factor based on  how you actually changed the area when you  changed to polar coordinates.

Spherical coordinates integrals 

- Spherical coordinates:

 Cheat Sheets Page 11

Change of Variables, change of basis 

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When changing from x, y to u, v to calculate some  integral, we need to find the Jacobian for that new  space. This is what we've done for polar and for  spherical coordinates.

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Find x = f(u,v)

        y = g(u,v)

- Compute:

 Cheat Sheets Page 12

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The Jacobian for more variables extends to the  determinant of a larger matrix (3 by 3 for three, 4 by 4  for four, etc.)

 Cheat Sheets Page 13

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