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by: Nikki Bee

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# MA 262 Study guide! Get an A+ w/ this MA 261

Marketplace > Purdue University > Mathematics (M) > MA 261 > MA 262 Study guide Get an A w this
Nikki Bee

GPA 4.0
Calculus 3

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This will help with your second exam for Calc 3.
COURSE
Calculus 3
PROF.
TYPE
Study Guide
PAGES
3
WORDS
KARMA
50 ?

## Popular in Mathematics (M)

This 3 page Study Guide was uploaded by Nikki Bee on Wednesday June 10, 2015. The Study Guide belongs to MA 261 at Purdue University taught by in Spring 2015. Since its upload, it has received 28 views. For similar materials see Calculus 3 in Mathematics (M) at Purdue University.

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Date Created: 06/10/15
MA 261 Spring 2011 Study Guide 2 3 3 f f properties of gradients gradient points in 81 8y direction of maximum rate of increase of f The maximum value of the directional derivative is equal to Vfa0 yo J level curve fa y C and7 in the case of3 variables Vfa0 yo 20 I level surface f1 y z C Gradient vector for fay Vf337y nVfxoyo zo nVfxoyo x fxyc 39lt x0 Vo sz oy Relativelocal extrema critical points V f 6 or V f does not exist 2nd Derivatives Test A critical points is a local min if D fmfyy 3y gt 0 and fm gt 07 local max if D gt 0 and fm lt 07 saddle if D lt 0 absolute extrema Max Min Problems Lagrange Multipliers Extremize f subject to a constraint 92 2 C solve the system V f AVg and 92 2 C m n Double integrals Midpoint Rule for rectangle f f 3334 dA Z Z ay i AA R 21 31 h1y S 33 S h2y 9132 33139200 Type I region D Type II region D C S y S d 7 agxgb 92 11 b iterated integrals over Type I and II regions f 13 y dA f 13 y dy d3 and D 1 9130 d h2y f 51334 dA f 3334 d3 dy respectively Reversing Order of Integration regions D c h1y that are both Type I and Type II properties of double integrals Integral inequalities mA g dA 3 MA Where A 2 area of D and m S g M D on D Change of Variables Formula in Polar Coordinates if D hlw S T S h26 then ozlt6 6 h209 fxydA f7 cos67 rsin6rdrd6 D a 1110 T 6 Applications of double integrals a Area of region D is AD dA D b Volume of solid under graph of z f1 y Where fa y 2 0 is V fa y dA D c Mass of D is m p1y dA Where p1 y 2 density per unit area sometimes write D m dm Where dm 2 p1 y dA D d Moment about the az aXis M9 y p1 y dA moment about the y aXis My 2 1 p1 y dA D D MyDap33ydA Mm f ypx7ydA e Center of mass 5 y Where E 7 g E pct y M D m fDpw7ydf1 Remark centroid 2 center of mass When density is constant this is useful 7 Elementary solids E C R3 of Type 1 Type 2 Type 3 triple integrals over solids E fayzdv fxyzdz dA for E 33731 6 D May S 2 3 W731 VE volume of solid E is dV applications of triple integrals mass of a solid moments E about the coordinate planes Mmy Mm Myz center of mass of a solid 3amp2 8 Cylindrical Coordinates 7362 1 rcos6 From CC to RC y rsin6 22 X Going from RC to CC use 32 y2 r2 and tan6 y make sure 6 is in correct quadrant a 9 Spherical Coordinates p 6 gb Where 0 3 gb 3 7r 1 psin gb cos6 From SC to RC y psin gb sin6 z pcosgb Going from RC to SC use 32 y2 Z2 2 p2 tan6 g and cosgb E a p 10 11 12 1 rcos6 y rsin6 22 ffE xvvadVffEfrcos6rsin6zrdzdrd6 T de39rdzd39rdH Triple integrals in Cylindrical Coordinates 1 psin gb cos6 y psingb sin6 z pcosgb ffEfxyz dVEfpsingbcos6 psingbsinQ pcos p2sjn dpd d9 T dV 2 p2 singb dp dqb d0 Triple integrals in Spherical Coordinates Vector elds 011 R2 and R3 x 3 135 3 62013731 and FOE y Z 135 3 62613731 R013 31 F is a conservative vector eld if F V f for some real valued function f

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