Flash cards for the final!
What is a vector v?
A two dimensional one is determined by two points in the place: initial and a terminal point.
v= –> PQ
what is //v//?
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.
When are v and w of nonzero lengths parallel?
if lines through v and w are also parallel. The parallel vectors point either in the same or in opposite directions.
When does vector v undergo a translation?
when it is moved parallel to itself without changing its length or direction. Translates have the same length and direction, BUT different BASEPOINTS.
when are v and w equivalent?
is w is a translate of v and if they have the same components.
how to calculate the components of a vector?
given the coordinates of two points, you subtract a2-a1, b2-b1 and receive the vector components.
why are components important?
they determine the length and direction of v, but don’t have the basepoint.
What is the parallelogram law?
v+w is the vector pointing from the basepoint to the opposite vertex of the parallelogram formed by v and w.
how to calculate v-w?
translate of the vector pointing from the tip of w to the tip of v.
Vector operations using components: addition, subtraction, scalar, and adding
v+w=
v-w=
scalar v=
v+0=v
linear combination of vectors v and w
rv+sw=u
what is a unit vector?
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=
What equation is used to scale a nonzero vector v= to obtain a unit vector pointing in the same direction?
ev=(1/ llvll)(v) and if v= makes an angle with the positive x axis, =llvll
What are standard basis vectors?
a way to introduce special notation for the unit vectors in the direction of positive x and y axes.
i= j=
Linear combination of i and j?
v=ai+bj
what is the triangle inequality theorem?
llv+wll< llvll+llwll
What is optimization?
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.
Local extreme values definition?
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.
Fermat’s Theorem?
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)
What happens if z=f(a,b)
if the partial derivatives do not exist.
Definition of a critical point?
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
What is a discriminant?
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
What is the 2nd derivative test?
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.
What are global values?
the min or max or value of a function on a given domain. does not always exist.
what is interior point of D
if D contains some open disk D(P,r) centered at P.
What is boundary point of D?
if every disk centered at P contains points in D and points not in D.
What is the interior of D?
Is the set of all interior points in the domain not lying on the boundary curve
what is the boundary of D?
is the set of all boundary curves. it is the curve surrounding the domain.
what is a closed domain?
if D contains all its boundary points.
what is an open domain?
is every point of D is an interior point. (does not have boundary points, dotted lines for the boundary)
Theorem: Existence and location of local extrema
Let f(x,y) be a continuous function on a closed, bounded domain D in R squared…
- f(x,y) takes on both a minimum and maximum value on D.
- The extreme values occur either a critical points in the interior of D or at point son the boundary of D.
Optimizing with a constraint
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
Lagrange Multipliers
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.
Langrange conditions and equations!
f of x (a,b)= scalar g of x (a,b)
same for y
Equations of Spheres
(x-a)squared+(y-b) squared+(z-c) squared=R squared
Equation of Cylinders
(x-a)squared+ (y-b) squared=R squared
Two nonzero vectors are parallel if…
v= lambda w for some scalar lambda
Equation of a Line: Vector Parametrization
r(t) arrow=+ t
Parametric Equations
x=x0+at y=y0+bt z=z0+ct
Line through Two Points: Vector Parametrization
r(t)=(1-t)+t
Line through two points: parametric equations
x=a1+(a2-a1)t
y=b1+(b2-b1)t
z=c1+(c2-c1)t
Determining if two lines intersect
two lines intersect if there exist parameter values t1 and t2 such that r1(t1)=r2(t2)
Dot Product
V*w=a1a2+b1b2+c1c2
Dot product and the Angel
Let theta be the angle between two nonzero vectors v and w
v*w= llvllllwllcos theta
two nonzero vectors v and w are called perpendicular or orthogonal when…
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
Testing Obtuseness
If V*W is <0 then obtuse
Projection Theorem/Formula
PROJECTION OF U ALONG V
ull=(uev)ev pr ull=((uv)/(v*v))v
V ALONG U
v//(vev)ev or vll=((vu)/(u*u))u
What does the sum of projection of u and vector u that is orthogonal?
it is equal to U.
Geometric Description of the Cross Product
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
w x v
= -v x w
What is the volume of a parallelepiped?
llv x wll * llull * lcos thetal
or we can use…
lu* (v x w)l
Area of parallelogram P spanned by v and w
A= llv x wll
What is a normal vector?
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.
Equation of a plane: plane through point P initial = x0, y0, z0) with normal vector n=
Vector form: n* =d
Scalar forms: a(x-a0)+b(y-b0)+c(z-c0)=0
or ax+by+cz=d
- find value of d,
d=n* = d
If n is normal to a plane…
then any scalar factor multiplied by n is also normal to the same plane.
e.g.
x+y+z=1
4x+4y+4z=4
Two planes are parallel if they have a common normal normal vectors…
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
How to determine n, given 3 points on a plane
n= PQ x PR , find d, then plug into the equation…
n*
at time t, particle is at point..the parametric equations are..
c(t)=( f(t), g(t))
x=f(t)
y=g(t)
what is parametrization?
c(t) with parameter t
what are parametric equations?
x=….
y=…
what is a parametric curve?
curve C
parametrization of a line…
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)
Circle of Radius R centered at origin has the parametrization…
x=a+ Rcos theta
y=b+Rsin theta
Parametrization of an ellipse
c(t)=(acost, b sint)
Parametrization of a Cycloid
x(t)=t-sint
y(t)=1-cost
Parametrization of cycloid generating a circle
x=Rt=Rsint
y=R-Rcost
Slope of the tangent Line
Let c(t) = (x(t), y(t)), where x(t) and y(t) are differentiable.
=y’(t)/ x’(t)
What is vector valued function?
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
Variable t in vector valued function…
is called a parameter…
functions x(t), y(t), z(t) are…
components or coordinate functions
When asked to parametrize…what is the answer in the form of?
points. sometimes you must place knowns into r(t) equation then get the points after multiplying it all out.
Parametrizing a circle with a known radius and given center P=(a,b,c)
r(t)=(a,b,c) ( (s parallel to, to be more exact)
Limit of a Vector-Valued Function
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
Vector Valued Limits are Computed Componentwise
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)
Vector valued function r(t)= is continuous…
if the limit as t approaches t0 is equal to r(t0)
Vector Valued Derivative are Computed Componentwise
to take the derivative of a component, you must take the derivative of each individual one..
e.g. x’(t), y’(t), z’(t)
Chain Rule for any differentiable scalar-valued function
d/dt r(g(t))= g’(t) r’(g(t))
Product Rule for Dot Product
d/dt(r1(t)r2(t)=r1(t) r’2(t)+r’1(t) * r2(t)
Product Rule for Cross Product
d/dt (r1(t) x r2(t) = [r1(t) x r’2(t) ] + [r’1(t) x r2(t)]
what does r’(t) x r’(t) =?
anything cross product with itself is ALWAYS 0
Tangent Line and r(t0)
L(t)=r(t0)+tr’(t0)
Find the parametrization of the tangent line…
use the L(t)=r(t0) + t r(t0) equation :D
calculating length of a path
s= integral from b to a llr’(t)ll dt = sqrt x’(t) squared+ y’(t) squared + z’(t) squared
Speed at time t
ds/dt= llr’(t)ll
Arc length parametrization, finding it given r(t) =
find the length of s.
integral of it from t to 0.
solve for s=#t
plug it into the original r(t)
parametrize the path r(t)
plug it in, substitute,
solve for s, and substitute it into the boundaries.
what is curvature measuring?
how much a curve bends
What is a unit tangent vector and how do you find it?
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
To find Arc Length Parametrization
we compute the arc length function
integral of llr’(t)ll
then plug in into the points
Formula for curvature
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
Curvature of Graph in the Plane
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
relationship between T’(t) and T(t)
they are orthogonal
What is a unit normal vector?
the unit vector in the direction of T’(t), assuming it’s nonzero.
Denoted by N(t) = T’(t)/ llT’(t)ll
Where is N pointing?
N is pointing in the direction that the curve is turning.
What are vertical traces?
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)
Vertical trace in the place x=a
and in the place y=b
intersection of the graph with the vertical place x=a, consisting of all points (a, y, f(a,y))
…(x, b, f(x,b))
What is a level curve?
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.
What is the horizontal trace at height c?
intersection of the graph of the horizontal place z=c, consisting of points (x,y, f(x,y)) such that f(x,y)=c
Calculate average rate of change from P to Q on a contour map,
change of altitude over change of horizontal distances.
13.6 equations!!!!!!!
look at physical cards in bp & memorize
Linearization equation
L(x,y)=f(a,b)+fx(a,b)(x-a)+fy(a,b)(y-b)
Tangent plane equation to the graph at (a,b, f(a,b)
z=f(a,b)(x-a)+fx(a,b)(x-a)+fy(a,b)(y-b)
Linear Approximation
f(a+h, b+k)=f(a,b)+fx(a,b)h+ fy(a,b)k
x=a+h
y=b+k
Gradient
gradient of a function f(x.y) at a point P=(a,b) is the vector
gradient fp= < fx(a,b), fy(a,b)>
Chain Rule for Gradient
gradient of F(f(x,y,z))= F’(f(x,y,z) *gradient of f
Chain Rule for Paths
d/dt= f(c(t))= gradient fc(t) * c’(t)
Directional Derivative
the directional derivative in the direction of a unit vector u= is the formula…..
respect to V
Dvf(a,b)= gradient f(a,b) *
Computing Directional Derivative in direction of V
Duf(p)= 1/llvll *gradient fp *
Interpretation fo Gradient
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.
Implicit Differentiation
partial z/ partial x= - Fx/Fz (same for y, plug in y for x)
