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MATH 200 Midterm Study Guide

by: spencer.kociba

MATH 200 Midterm Study Guide Math 200

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Chapter 11.1-11.6, 12.1-12.2 concepts, formulas and properties to review for the first midterm on 10/19
Multivariate Calculus
Study Guide
Calculus, vectors, vector calculus, multivariable calculus, parametric equations, planes, skew lines, Derivatives
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This 6 page Study Guide was uploaded by spencer.kociba on Sunday October 16, 2016. The Study Guide belongs to Math 200 at Drexel University taught by in Summer 2016. Since its upload, it has received 40 views. For similar materials see Multivariate Calculus in Math at Drexel University.


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Date Created: 10/16/16
Spencer Kociba MATH 200­005  MIDTERM 1 EXAM REVIEW GUIDE ***Things that WILL be on the test: ○ 11.1: Rectangular coordinates in 3­space, spheres, cylindrical  surfaces ○ 11.2: Vectors ○ 11.3: Dot Products and Projections ○ 11.4: Cross Products ○ 11.5: Parametric Equations for lines ○ 11.6: Planes in 3­space ○ 12.1: Intro to Vector Valued Functions ○ 12.2: Calculus of Vector Valued Functions ○ ***All homework worksheets THROUGH Vector Valued  Functions EXCEPT Quadratic Surfaces ● Section 11.1 ○   = 3­space (or R3)  ■ “Floor” = xy­plane ■ “Right wall” = yz­plane ■ “Left wall” = xz­plane ■ “Inside the room” = the first octant ○ distance=  d= ❑√ ○ ○ Formula for a sphere= r =(x❑ −❑)❑ +(y❑ −b)❑ ❑(z❑ −c)❑ ❑ 2 where the center is C(a,b,c) and the distance from any point on the surface to the center=r ○ Cylindrical surface: a graph (in 3 space) of an equation with 2  variables projected along a 3rd variable’s axis ■ Ex.  ○ Midpoint between  A(x❑ ,1❑ ,z❑1) 1  and B(x❑ ,y❑ , z❑ ) 2 2 2  is  ● Section 11.2 ○ Vector= direction and magnitude ■ Ex. angle and displacement ■ ● Name: vector “v” or  vector AB  (which is not equal to vector BA) ● A=initial point ● B=terminal point ● Scalar= any real number ○ v=u if direction and magnitude are the same ■ Position is NOT important ○ (scalar)(v) multiplies the elements of v by the scalar value ■ (stretches or shrinks the vector or changes its  direction by 180 degrees) ○ Notation ■ ||v||= the magnitude of v (aka the length/norm of v) ■ ( ) = a point ■ < > = a vector ○ Unit vector: ||v||=1 ■ u= 1 ∗v   (reciprocal of the norm of  ¿∨v∨¿ v (results in scalar) multiplied by v) ■ 2 space unit vectors: i=<1,0> and j=<0,1> ■ 3 space unit vectors: i=<1,0,0> ; j=<0,1,0> and  k=<0,0,1> ○ v=¿∨v∨¿⋅<cosθ,sinθ>¿ ○ Resultant Force: 2 forces applied on a same point on an object ■ If F1 and F2 have the same initial point, F1+F2= resultant force ■ ● F 1=¿∨F 1∨¿⋅<cosA,sinA>¿ ● F 2=¿∨F 2∨¿⋅<cosB,sinB ● F3 is usually given= <0, ­Fg> ● F1+F2+F3=0 (static equilibrium) ○ Solve for forces of tension (||F1||, ||F2||) ● Section 11.3 ○ Direction cosines ■ cos(α)= v❑ 1 ¿∨v∨¿ ■ cos(β)= v❑ 2 ¿∨v∨¿ v❑ 3 ■ cos(Ɣ)= ¿∨v∨¿   ○ Angle between vectors u∙v ■ cos(θ)= ¿∨u∨¿∨¿v∨¿ ○ Law of cosines (for vectors): 2 2 2 ¿∨u−v∨¿ =¿∨u∨¿ +¿∨v∨¿ −2∨¿u∨¿∨¿v∨¿cos(θ) ■ ○ Dot product in 2 space=  u❑ 1❑ +1❑ v❑2 2 ■ If  u∙v=0 , u and v are perpendicular v∙b ○ proj❑b(v)=¿ 2⋅b ¿∨b∨¿ ■ ■ v=proj❑ (b)+(v−proj❑ (b)) ● Components ○ ■ F ❑ 1k❑ e1 1 where k is a scalar ■ F ❑ =k❑ e❑ where k is a scalar 2 2 2 ■ Fg=k❑ e1 +1❑ e❑2 2 ■ Fg∙e❑ 1k❑ 1 ■ Fg∙e❑ 2k❑ 2 ■ Fg=(Fg∙e❑ )1❑ +1Fg∙e❑ )e2 2 ○ To find orthogonal vectors to two given vectors v and u, ■ u∙w=0 ■ v∙ w=0 ■ This sets up an undetermined system  where 1 variable can be any non­zero and real number ● Section 11.4 e f d f d e |g h| | | |g i g h ○ = a ­b +c ○ Properties of cross products ■ (i) u x v = ­(v x u) ■ (ii) u x (v+w) = u x v + u x w ■ (iii) (u+v) x w = u x w + v x w ■ (iv) k(u x v) = (ku) x v = u x (kv) ■ (v) u x 0 = 0 ■ (vi) u x u = 0 ■ Unit vectors: i x j=k, j x k=i, k x i=j ○ ■ (i)  u∙(uxv)=0 ■ (ii) v∙(uxv)=0 ○ Use dot products to PROVE two vectors are orthogonal,  find projections and angles between vectors ○ Use cross products to find an orthogonal vector ○ Parallelepiped ■ ¿∨uxv∨¿=¿∨u∨¿⋅∨¿v∨¿⋅sin(θ) ■ A=¿∨u∨¿⋅∨¿v∨¿⋅sin(θ) ■ V= |u∙(vxw)| ● V=0 if u,v, and w lie in the  same plane ● Absolute value of Scalar Triple Product ( ) is the Volume of the  parallelepiped ● Section 11.5 ○ Theorem: the line (in 2­space) through the point P(x,y) parallel  to v=<a,b> has parametric equations x=x+at, y=y+bt ■ In 3­space with P(x,y,z) and v=<a,b,c> the  parametric equations are  x=x+at, y=y+bt and z=z+ct ○ Two lines that do not intersect and are not parallel are skew  lines ○ r(t)= ¿x❑ +ot ,y❑ +bo ,z❑ +co>¿ ● Section 11.6 ○ n=the normal line (perpendicular to the plane) ○ Acute Angle between planes ¿ ¿n❑ ∙1❑ ∨ 2 ■ ¿∨n❑ ∨1⋅∨¿n❑ ∨¿ 2 cosθ=¿ ○ Line of intersection between planes ■ n❑ 1n❑ 2 ○ To make an equation for a plane you need ■ A point and a vector or ■ 2 points on the plane or ■ 2 vectors on a plane or ■ 3 points on a plane or ■ A point and a vector/line on a plane  ○ Equation for a plane ■ ax+by+cz+d=0 ● Where d= −ax❑ −oy❑ −cz❑o o ■ a(x−x❑ )+o(y−y❑ )+c(zoz❑ )=0 o ○ Planes are parallel if they are scalar multiples of eachother or v x n =0 ○ Planes are perpendicular if  v∙n=0 ○ If planes are NOT parallel, they MUST intersect at some line ● Section 12.1 ○ Vector valued function: a function where the input is a number  and the output is a vector ○ Domains of VVF are all possible values of t ■ Keep in mind any pre­existing restrictions  before finding the domain of the derivative of the function ■ r’(t) domain should be within r(t) domain ○ Be aware of under and overdetermined systems ○ VVF form ■ r(t)=<f(t), g(t), h(t)> where f, g and h are real  valued functions ● Section 12.2 ○ Continuity definition ■ We say a VVF r(t) is continuous at “a” if lim r(t)=r(a) t→a ○ Derivative definition ■ If r(t) is a VVF, we define the derivative of r with  lim r(t+h)−r(h) respect to t to be the VVF r’ given br'(t)=h→ 0 h ● The domain of r’ is all values of t  for which this limit exists d dr ■ r'(t)=dt [r(t)]=dt ● r’(t)=vector ■ If r(t)=<x(t),y(t),z(t)> then r’(t)=<x’(t),y’(t),z’(t)> ■ Results in the “tangent vector” ○ Properties ■ All the same as regular derivatives except the  multiplication rule ● d [ f (t)r(t)]= [ f (t)]⋅r(t)+ [r(t)]⋅ f (t) dt dt dt ○ Theorem: if r and s are VVfs  ■ d [r(t)∙s(t)]= d [r(t)]∙s(t)+r(t)∙d [s(t)] dt dt dt ● r∙s turns into a real valued  function ■ d d d [r(t)xs(t)]= [r(t)]xs(t)+r(t)∙ [s(t)] dt dt dt ● Derivatives go on the outside,  VVFs go on the inside ○ Integrate each element of r(t) separately and add C= ¿C❑ ,1❑ ,C❑2>¿ 3


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