Flash cards for the final!

Question 1

What is a vector v?

Answer 1

A two dimensional one is determined by two points in the place: initial and a terminal point.
v= –> PQ

Question 2

what is //v//?

Answer 2

Length of the vector, is the distance from P to Q. to calculate use the distance formula. Find components of the vectors, square and add them all under a square root.

Question 3

When are v and w of nonzero lengths parallel?

Answer 3

if lines through v and w are also parallel. The parallel vectors point either in the same or in opposite directions.

Question 4

When does vector v undergo a translation?

Answer 4

when it is moved parallel to itself without changing its length or direction. Translates have the same length and direction, BUT different BASEPOINTS.

Question 5

when are v and w equivalent?

Answer 5

is w is a translate of v and if they have the same components.

Question 6

how to calculate the components of a vector?

Answer 6

given the coordinates of two points, you subtract a2-a1, b2-b1 and receive the vector components.

Question 7

why are components important?

Answer 7

they determine the length and direction of v, but don’t have the basepoint.

Question 8

What is the parallelogram law?

Answer 8

v+w is the vector pointing from the basepoint to the opposite vertex of the parallelogram formed by v and w.

Question 9

how to calculate v-w?

Answer 9

translate of the vector pointing from the tip of w to the tip of v.

Question 10

Vector operations using components: addition, subtraction, scalar, and adding

Answer 10

v+w=
v-w=
scalar v=
v+0=v

Question 11

linear combination of vectors v and w

Answer 11

rv+sw=u

Question 12

what is a unit vector?

Answer 12

a vector of length 1. often used to indicate direction. the head of the unit vector e based at the origin lies on the unit circle and has components
e=

Question 13

What equation is used to scale a nonzero vector v= to obtain a unit vector pointing in the same direction?

Answer 13

ev=(1/ llvll)(v) and if v= makes an angle with the positive x axis, =llvll

Question 14

What are standard basis vectors?

Answer 14

a way to introduce special notation for the unit vectors in the direction of positive x and y axes.
i= j=

Question 15

Linear combination of i and j?

Answer 15

v=ai+bj

Question 16

what is the triangle inequality theorem?

Answer 16

llv+wll< llvll+llwll

Question 17

What is optimization?

Answer 17

the process of finding the extreme values of a function. this amounts to finding the highest and lowest points on the graph over a given domain. IMPORTANT to distinguish between local and global extreme values.

Question 18

Local extreme values definition?

Answer 18

a function f(x,y_ has a local extremum at P=(a,b) if there exists an open disk D(P,r) such that…
local maximum: f(x,y)< or equal to f(a,b) for all (x,y) in the domain of D(p,r)
Local minimum same except > or equal to.

Question 19

Fermat’s Theorem?

Answer 19

If f(a) is a local extreme value, then a is a critical point and thus there is a tangent plane that must be horizontal.
a=f(a,b)+partial derivative of x(a,b)(x-a)+partial deriv. y(a,b)(y-b)

*if f(x,y) has a local min or max, at p=(a,b) then (a,b) is a critical point of f(x,y)

Question 20

What happens if z=f(a,b)

Answer 20

if the partial derivatives do not exist.

Question 21

Definition of a critical point?

Answer 21

A point P=(a,b) in the domain of f(x,y) is called a critical point if:
partial derivative of x(a,b) =0 or does not exist.
same applies to y

Question 22

What is a discriminant?

Answer 22

determines the type of critical point (a,b) of a function f(x,y)
D=D(a,b)=2nd partial derivative of fx multiplied by fy 2nd derivative - fxy (a,b) squared

Question 23

What is the 2nd derivative test?

Answer 23

P=(a,b) be a critical point of f(x,y)
if D>0, fxx(a,b)> 0, then f(a,b) is a local min.
if D>0 and fxx<0, then f has a saddle point at (a,b)
If D=0, test is inconclusive.

Question 24

What are global values?

Answer 24

the min or max or value of a function on a given domain. does not always exist.

Question 25

what is interior point of D

Answer 25

if D contains some open disk D(P,r) centered at P.

Question 26

What is boundary point of D?

Answer 26

if every disk centered at P contains points in D and points not in D.

Question 27

What is the interior of D?

Answer 27

Is the set of all interior points in the domain not lying on the boundary curve

Question 28

what is the boundary of D?

Answer 28

is the set of all boundary curves. it is the curve surrounding the domain.

Question 29

what is a closed domain?

Answer 29

if D contains all its boundary points.

Question 30

what is an open domain?

Answer 30

is every point of D is an interior point. (does not have boundary points, dotted lines for the boundary)

Question 31

Theorem: Existence and location of local extrema

Answer 31

Let f(x,y) be a continuous function on a closed, bounded domain D in R squared... 1. f(x,y) takes on both a minimum and maximum value on D. 2. The extreme values occur either a critical points in the interior of D or at point son the boundary of D.

Question 32

Optimizing with a constraint

Answer 32

some optimization problems involve finding extreme values of a function f(x,y) subject to a constraint g(x,y)=o. e.g. minimize f(x,y)=sqrt x squared + y squared subject to g(x,y) =2x+3y-6=0

Question 33

Lagrange Multipliers

Answer 33

is a general procedure for solving optimization problems with a constraint. Assume that f(x,y) and g(x,y) are differentiable functions. If f(x,y) has a local min. or local max. pm the constraint curve g(x,y)=0 at P=(a,b) and if gradient of G of P doesnt equal to 0, there is a scalar such that gradient of f of p=scalar of gradient of g of p.

Question 34

Langrange conditions and equations!

Answer 34

f of x (a,b)= scalar g of x (a,b) | same for y

Question 35

Equations of Spheres

Answer 35

(x-a)squared+(y-b) squared+(z-c) squared=R squared

Question 36

Equation of Cylinders

Answer 36

(x-a)squared+ (y-b) squared=R squared

Question 37

Two nonzero vectors are parallel if...

Answer 37

v= lambda w for some scalar lambda

Question 38

Equation of a Line: Vector Parametrization

Answer 38

r(t) arrow=+ t

Question 39

Parametric Equations

Answer 39

x=x0+at y=y0+bt z=z0+ct

Question 40

Line through Two Points: Vector Parametrization

Answer 40

r(t)=(1-t)+t

Question 41

Line through two points: parametric equations

Answer 41

x=a1+(a2-a1)t y=b1+(b2-b1)t z=c1+(c2-c1)t

Question 42

Determining if two lines intersect

Answer 42

two lines intersect if there exist parameter values t1 and t2 such that r1(t1)=r2(t2)

Question 43

Dot Product

Answer 43

V*w=a1a2+b1b2+c1c2

Question 44

Dot product and the Angel

Answer 44

Let theta be the angle between two nonzero vectors v and w v*w= llvllllwllcos theta

Question 45

two nonzero vectors v and w are called perpendicular or orthogonal when...

Answer 45

the angle between them is 90 degrees. we can use the dot product to test if they are orthogonal. v is orthogonal to w only if v*w=0

Question 46

Testing Obtuseness

Answer 46

If V*W is <0 then obtuse

Question 47

Projection Theorem/Formula

Answer 47

PROJECTION OF U ALONG V ull=(u*ev)ev pr ull=((u*v)/(v*v))v V ALONG U v//(v*ev)ev or vll=((v*u)/(u*u))u

Question 48

What does the sum of projection of u and vector u that is orthogonal?

Answer 48

it is equal to U.

Question 49

Geometric Description of the Cross Product

Answer 49

cross product of v and w: v x w is orthogonal to v and w v x w has length llvll llwll sin theta v, w, v x w forms a right hand system

Question 50

w x v

Answer 50

= -v x w

Question 51

What is the volume of a parallelepiped?

Answer 51

llv x wll * llull * lcos thetal or we can use... lu* (v x w)l

Question 52

Area of parallelogram P spanned by v and w

Answer 52

A= llv x wll

Question 53

What is a normal vector?

Answer 53

when a plane passes through a point, we can determine the plane P by specifying it with a normal vector. n= that is orthogonal to plane P. We base the normal vector a the point on the plane.

Question 54

Equation of a plane: plane through point P initial = x0, y0, z0) with normal vector n=

Answer 54

Vector form: n* =d Scalar forms: a(x-a0)+b(y-b0)+c(z-c0)=0 or ax+by+cz=d 1. find value of d, d=n* = d

Question 55

If n is normal to a plane...

Answer 55

then any scalar factor multiplied by n is also normal to the same plane. e.g. x+y+z=1 4x+4y+4z=4

Question 56

Two planes are parallel if they have a common normal normal vectors...

Answer 56

are parallel to one another. choosing a common normal vector and changing the value of d creates parallel planes. e.g. x+y+z= 5 x+y+z=10 4x+4y+4z=48598c their normal vector is

Question 57

How to determine n, given 3 points on a plane

Answer 57

n= PQ x PR , find d, then plug into the equation... | n*

Question 58

at time t, particle is at point..the parametric equations are..

Answer 58

c(t)=( f(t), g(t)) x=f(t) y=g(t)

Question 59

what is parametrization?

Answer 59

c(t) with parameter t

Question 60

what are parametric equations?

Answer 60

x=.... | y=...

Question 61

what is a parametric curve?

Answer 61

curve C

Question 62

parametrization of a line...

Answer 62

the line through P=(a,b) of slope m is parametrized by x=a+rt y=b+st the line through P=(a,b,) and Q=(c,d) has parametrization... x=a+t(c-a) and y=b+t(d-b)

Question 63

Circle of Radius R centered at origin has the parametrization...

Answer 63

x=a+ Rcos theta | y=b+Rsin theta

Question 64

Parametrization of an ellipse

Answer 64

c(t)=(acost, b sint)

Question 65

Parametrization of a Cycloid

Answer 65

x(t)=t-sint | y(t)=1-cost

Question 66

Parametrization of cycloid generating a circle

Answer 66

x=Rt=Rsint | y=R-Rcost

Question 67

Slope of the tangent Line

Answer 67

Let c(t) = (x(t), y(t)), where x(t) and y(t) are differentiable. =y'(t)/ x'(t)

Question 68

What is vector valued function?

Answer 68

a particle moving in R^3. who's coordinates at time t are (x(t), y(t0, z(t)). any function r(t) of the form equation ... represented with equation: r(t)==x(t)i+ y(t)j+z(t)k whose domain D is a set of real numbers and whose range is a set of position vectors r(t)= + tv

Question 69

Variable t in vector valued function...

Answer 69

is called a parameter...

Question 70

functions x(t), y(t), z(t) are...

Answer 70

components or coordinate functions

Question 71

When asked to parametrize...what is the answer in the form of?

Answer 71

points. sometimes you must place knowns into r(t) equation then get the points after multiplying it all out.

Question 72

Parametrizing a circle with a known radius and given center P=(a,b,c)

Answer 72

r(t)=(a,b,c) ( (s parallel to, to be more exact)

Question 73

Limit of a Vector-Valued Function

Answer 73

A vector valued function r(t) approaches the limit u (a vector) as t approaches t initial if limit llr(t)-ull=0 lim r(t)=u

Question 74

Vector Valued Limits are Computed Componentwise

Answer 74

A vector valued function r(t)= approaches a limit as t goes to t initial if and only if each component approaches a limit in order to take a limit of a component, you must take the limit of each individual one... e.g. : lim x(t), lim y(t), lim z(t)

Question 75

Vector valued function r(t)= is continuous...

Answer 75

if the limit as t approaches t0 is equal to r(t0)

Question 76

Vector Valued Derivative are Computed Componentwise

Answer 76

to take the derivative of a component, you must take the derivative of each individual one.. e.g. x'(t), y'(t), z'(t)

Question 77

Chain Rule for any differentiable scalar-valued function

Answer 77

d/dt r(g(t))= g'(t) r'(g(t))

Question 78

Product Rule for Dot Product

Answer 78

d/dt(r1(t)*r2(t)=r1(t)* r'2(t)+r'1(t) * r2(t)

Question 79

Product Rule for Cross Product

Answer 79

d/dt (r1(t) x r2(t) = [r1(t) x r'2(t) ] + [r'1(t) x r2(t)]

Question 80

what does r'(t) x r'(t) =?

Answer 80

anything cross product with itself is ALWAYS 0

Question 81

Tangent Line and r(t0)

Answer 81

L(t)=r(t0)+tr'(t0)

Question 82

Find the parametrization of the tangent line...

Answer 82

use the L(t)=r(t0) + t r(t0) equation :D

Question 83

calculating length of a path

Answer 83

s= integral from b to a llr'(t)ll dt = sqrt x'(t) squared+ y'(t) squared + z'(t) squared

Question 84

Speed at time t

Answer 84

ds/dt= llr'(t)ll

Question 85

Arc length parametrization, finding it given r(t) =

Answer 85

find the length of s. integral of it from t to 0. solve for s=#t plug it into the original r(t)

Question 86

parametrize the path r(t)

Answer 86

plug it in, substitute, | solve for s, and substitute it into the boundaries.

Question 87

what is curvature measuring?

Answer 87

how much a curve bends

Question 88

What is a unit tangent vector and how do you find it?

Answer 88

at every point P along the path there is a unit tangent vector pointing int he direction of motion of the parametrization. T(t) = r'(t)/ llr'(t)ll

Question 89

To find Arc Length Parametrization

Answer 89

we compute the arc length function integral of llr'(t)ll then plug in into the points

Question 90

Formula for curvature

Answer 90

If r(t) is a regular parametrization, then the curvature at r(t) is THIS IS r(t) given points!!! k(t) = ll r'(t) x r''(t)ll / llr'(t) ll ^3

Question 91

Curvature of Graph in the Plane

Answer 91

The curvature at the point (x, f(x)) on the graoh of y=f(x) is equal tp... THIS ONE IS f(x)!!! given..it's an EQUATION k(x) = /f''(x)l / (1+ f'(x)^2) ^3/2

Question 92

relationship between T'(t) and T(t)

Answer 92

they are orthogonal

Question 93

What is a unit normal vector?

Answer 93

the unit vector in the direction of T'(t), assuming it's nonzero. Denoted by N(t) = T'(t)/ llT'(t)ll

Question 94

Where is N pointing?

Answer 94

N is pointing in the direction that the curve is turning.

Question 95

What are vertical traces?

Answer 95

a way of analyzing the graph of a function f(x,y) by freezing the x or y coordinate. e.g. f(a,y) or f(x,b)

Question 96

Vertical trace in the place x=a | and in the place y=b

Answer 96

intersection of the graph with the vertical place x=a, consisting of all points (a, y, f(a,y)) ...(x, b, f(x,b))

Question 97

What is a level curve?

Answer 97

The curve f(x,y) =c in the x-y plane. The level curve consists of all points (x,y) in the plane where the function takes the value c. Each level curve is the projection onto the xy place of the horizontal trace on the graph that lies above it.

Question 98

What is the horizontal trace at height c?

Answer 98

intersection of the graph of the horizontal place z=c, consisting of points (x,y, f(x,y)) such that f(x,y)=c

Question 99

Calculate average rate of change from P to Q on a contour map,

Answer 99

change of altitude over change of horizontal distances.

Question 100

13.6 equations!!!!!!!

Answer 100

look at physical cards in bp & memorize

Question 101

Linearization equation

Answer 101

L(x,y)=f(a,b)+fx(a,b)(x-a)+fy(a,b)(y-b)

Question 102

Tangent plane equation to the graph at (a,b, f(a,b)

Answer 102

z=f(a,b)(x-a)+fx(a,b)(x-a)+fy(a,b)(y-b)

Question 103

Linear Approximation

Answer 103

f(a+h, b+k)=f(a,b)+fx(a,b)h+ fy(a,b)k x=a+h y=b+k

Question 104

Gradient

Answer 104

gradient of a function f(x.y) at a point P=(a,b) is the vector gradient fp= < fx(a,b), fy(a,b)>

Question 105

Chain Rule for Gradient

Answer 105

gradient of F(f(x,y,z))= F'(f(x,y,z) *gradient of f

Question 106

Chain Rule for Paths

Answer 106

d/dt= f(c(t))= gradient fc(t) * c'(t)

Question 107

Directional Derivative

Answer 107

the directional derivative in the direction of a unit vector u= is the formula..... respect to V Dvf(a,b)= gradient f(a,b) *

Question 108

Computing Directional Derivative in direction of V

Answer 108

Duf(p)= 1/llvll *gradient fp *

Question 109

Interpretation fo Gradient

Answer 109

Duf(P)=llgradient fpll cos theta gradient Fp in the direction of maximum rate of increase of F at p. negative gradient Fp in the direction of maximum rate of decrease at P. Gradient Fp is normal to the level curve of f at P.

Question 110

Implicit Differentiation

Answer 110

partial z/ partial x= - Fx/Fz (same for y, plug in y for x)