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by: Sarah Ma

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# Midterm Review Guide MA1024

Sarah Ma
WPI
GPA 3.8
Calculus 4
Rostami

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This is my basic skills review guide on the strategies from chapters 14.1 through 14.7. There are no examples of problems being solved, but this includes the basic strategies of how, say, direction...
COURSE
Calculus 4
PROF.
Rostami
TYPE
Study Guide
PAGES
3
WORDS
KARMA
50 ?

## 7

1 review
"Loved these! I'm a horrible notetaker so I'll be your #1 fan this semester"
Haylie Schamberger

## Popular in Math

This 3 page Study Guide was uploaded by Sarah Ma on Saturday February 7, 2015. The Study Guide belongs to MA1024 at Worcester Polytechnic Institute taught by Rostami in Winter2015. Since its upload, it has received 107 views. For similar materials see Calculus 4 in Math at Worcester Polytechnic Institute.

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## Reviews for Midterm Review Guide

Loved these! I'm a horrible notetaker so I'll be your #1 fan this semester

-Haylie Schamberger

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Date Created: 02/07/15
Calculus 4 Midterm Basics 141 Functions of Several Variables Level curves to draw a level curve 2D 0 If fxyxy1 choose different c values 0 c 1 O 1 9 set equation equal to c value 0 Graph equation and label it with its corresponding c value The closer together the lines on the level curve the steeper the 3D object is Level surfaces 3D objects 0 If fxy x2y2 9 z x2y2 0 Fix xxo yyo and 220 in which xoyo and 20 are all constants 0 Bring constants to one side then figure out its 2D shape 2D shapes 0 yZzxo2 9 parabola o x2y2 z0 9 circle 0 9y222 16 9 ellipse o 22169yo2 9 two lines positive amp negative 0 22y2 x0 9 hyperbola 142 Limits and Continuity 0 Limits limxyoo Jag 7 to figure out if this has a limit use these different methods 0 Substitute ymx line in the equation If m slope is left after simplifying then it does not have a limit because it differs when you substitute other equations in with different slopes ex Y2x 0 Substitute xaxis in which y0 or yaxis in which x0 0 Substitute a line ymx and a parabola yx2 to yield different results 0 Use sandwich theorem 39 limtxyo0 ysin 1X I 1 S sin1XS 1 I thus y S ysin1x S y I If the xy approach 00 then limit of ysin1x approaches 0 143 Partial Derivatives Partial derivative Differentiating one variable while the other is held constant 1 if 62f 6yax ayax 9 fxy a a f 62 f 9f 0 6x 6x 6x242 XX 6 6f 6 f o 9f 6x6y axay yx 11f 62f 6y6y ayAz 9f 146 Tangent Planes To find a tangent plane to a function use equation axxo by yo cz 20 O at point Pxo yo 20 0 Vector lta b cgt s normal vector of plane a a i X0 yo 20 a b X0 yo 20 C 3 x0 yo 20 Linearization You can do this because Lxy is a good approximation of fxy at the given point 0 LOW f W fxXo yoX Xo f Xo yo yyo 145 Gradients and Directional Derivatives Gradient of f vector 2 ltfX x0 yo fyxo y0gt Directional Derivative How fast a function is changing o 66 Duf fu gradient f I u 9 dot product gives numerical value not vector u direction gradient f magnitude W 0 Increase most rapidly is u o Decrease most rapidly is u vector value O 147 Extreme Values and Saddle Points Critical point is an interior point when 0 f x 0 o f x DNE Theorem If fx has a local min or max value at an interior point x of its domain then x is a critical point 0 However every critical point is not a min max value Theorem If fxy is a local min or max value at an interior point xy then xy is a critical point 0 If x is a critical point I fquot x gt O 9 local min I fquot x lt O 9 local max I fquot x O 9 inconclusive Let fxxo yo fy x0 yo 0 o A fXX x0 yo 0 B fxytxm yo 0 C fyy X0 yo 0 D AC B2 9 discriminant 1 IfDgt OandAgt O 9localmin DgtO and A lt0 9 local max 2 If Dlt 0 saddle point if you look at function from different angles the point can be either a maximum or minimum 3 If D O inconclusive Global Min Max To find a global minmax you must test critical points and boundaries 0 Find critical points and plug into original equation 0 Find boundaries and critical points from all of the function s segments then plug into original equation I Critical points are found by taking partial derivatives and setting them equal to O 0 Compare the points and see which value is highest and at which points

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