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# Exam 1 Study Guide MATH 265

ISU

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This 5 page Study Guide was uploaded by Elena Camp on Monday September 14, 2015. The Study Guide belongs to MATH 265 at Iowa State University taught by Butler, Steve in Fall 2015. Since its upload, it has received 122 views. For similar materials see Calculus 3 in Mathematics (M) at Iowa State University.

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Date Created: 09/14/15

Math 265 Study Guide for Midterm 1 Professor Steve Butler Iowa State University Fall 2015 This midterm consists of 5 questions and 10 points each Important ideas that will be covered in this exam are 0 Arc length 0 Vectors derivatives integrals component by component 0 Dot product projection angle testing perpendicular 0 Cross product area of shapes creating perpendicular 0 Motion rt vt at aTT aNN o Curvature 0 Various coordinate systems Cartesian Cylindrical Spherical Arc Length b b L f Ilr tdt f vx39t2o39t 2z39t2 dt a m ScaHng ka b c 2 ka kb kc Addition a b c d ef a d b ec f Magnitude a b cII m Special Vectors 171 I vector with length 1 pointing in their respective directions x y and z 0 000 the zero vector which is a vector that has no direction or magnitude Derivatives f Ft ft gt ht is a vector to find the derivative of the vector take the derivative of each component So F t f tg th t Rules for derivatives Sum Rule ver Gm M Gm Chain Rule d E F pt F ptp39t Product Rule d E ptFt p tFt ptF t d E Ft Gti F t 60 F t 39 0 0 gm x em M x Ga no x 63903 Integrals f Ft ft gt ht is a vector to find the integral of the vector take the interal of each component So Fm fft dtfgt dtfht dt This is similar for definite integrals pre bro dt bgt dt bawdt Dot Product The dot product takes two vectors and creates a scalar product The dot product can be used to find the angle between two vectors fi 13 fii3c06 6 cos391 W Il llll ll Proiection r 1 pro j 17 The projection of Ti onto 17 ie the shadow that ii casts onto 1 1 D1rect10n of pro 3 u m 12 Length of pT OJB u uc039 gt u 39 93 a v v 39 9 proj U 2 Cross Product The cross product helps determine a vector parallel to two other vectors i J x abcgtltd f a b C a Ill 93961 jlcd lell d e f l 9 g 17 X 17 Ill illll llsmg Area of Shages 12 Area of a parallelogram 2 base height Area Ill lllll llsin l gt v 1 Area of a triangle 2 E fl13sin6 1 Volume of a Parallelepiped baseheight Volume of a Parallelepiped fl x llllWlIcosO tl x 13 WI Motion T39t Tt IIquot 15 Unit vector pointing in the direction of the motion T39t N t HT 15 quotunit normal vector d d act Em EIlr tllT t 6105 aTtTt aN tNt I TI TI X Ti Tt Nt T39 a 7quot 7quot Curvature The curvature or quotbendinessquot is used to measure the how quickly the curve is turning To do this the the curvature equation is used K IIquot x 7quot II 7quot 3 Coordinate Systems Cartesian x y z Cartesian uses distance from the axis s to define a point Cylindrical r 62 Cylindrical is polar coordinates but in 3 dimensions Spherical p 6 go Spherical uses distance and direction to define a point Conversion factors between the different coordinate systems x rcosQ y rsin l x2 y2 r2 tanH Z x2 y2 z2 p2 z pcosgo r psingo x psingocosgo y psingosingo

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