Math Final

orthaB

b-projaB

How to determine range

determine the minimum and maximum values the function can take on (I think!!)

Double angle formulas

sin^2(theta)=(1-cos(2theta))/2 cos^2(theta)=(1+cos(2theta))/2

New** Stoke's Theorem

specified orientation

If a limit does not depend on m then it may exist and we have to apply the

squeeze theorem if it does exist (if it does not exist show it does not exist with another path test)

To find critical point

take the partial derivative of the function with respect to x and y and then set it equal to zero

Ellipses

two squared terms (both positive)

Find an equation of the tangent plane to the surface

z-zo=Fx(x-xo)+Fy(y-yo)

law of sin

|axb|=|a||b|sin(t)

compaB

|b|cos(t)

Directional derivative

(gradient of f) dot (unit vector which has a length of one)

Parabolas

(have only one squared term) y=x^2 x=y^2 (graph is of y=x^2) (opens on non-squared term)

Equation of a sphere

(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2

New** The Divergence Theorem

*can use the divergence theorem when the surface is closed to calculate flux

dy/dx

-Fx/Fy

Hyperboloid of two sheets

-x^2/a^2 - y^2/b^2 + z^2/c^2 = 1

volume of a triangle

1/2*b*h*z

Average value

1/Area and then the integral

Find an equation of the plane through the point (−1,3,−3) with normal vector n=⟨3,2,4⟩.

3(x+1)+2(y-3)+4(z+3)=0

Find the equation of the tangent plane to the surface 2𝑥𝑦𝑧^2=12 at the point (2,3,−1) Find the equation of the normal line to the surface 2𝑥𝑦𝑧^2=12 at the point (2,3,−1)

6(x-2)+4(y-3)-24(z+1)=0 L(t)=<2+6t,3+4t,-1-24t>

Curl(F)

=delxF

Surface area of a parametric surface

A(s)=double integral of |ru x rv|dA

Find a time in which you have traveled 9 units along the curve from r(−1) in the positive direction using the arc length function 𝐫(𝑡)=⟨cos(−2𝑡)−2,sin(−2𝑡)+2,2𝑡+4⟩ starting from r(-1)

Answer: (9-2sqrt(2))/(2sqrt(2))

Component test

Curl(f)=0 Qx=Py (for 2D)

NEW** Surface Integrals

Explicit surface (z=g(x,y)) double integral of f(x,y,g(x,y))*sqrt((Gx)^2+(Gy)^2+1)dA Parametric Surface (r(s,t)) double integral of f(r(Stewart))|rs x rt|dsdt

Curl(F)=0 then

F is conservative

Distance from one plane to another

Find a point on the first plane P Find a point on the second plane Q find the normal vector of the second plane d=|PQ*n|/|n|

Find a nonzero vector orthogonal to the plane through the points: A = (1,1,0),B = (2,−3,−3),C = (−2,5,3)

Find vectors AB and BC then cross product ABxBC which gives you the n <8,3,-12>

How to calculate the determinant

Fxx+Fyy-(Fxy)^2

NEW** Surfaces and Area

How to calculate flux across a surface double integral of F*dS then its the double integral of F*ndS then double integral over D of F *+ or - <gx,gy,-1>dA when dealing with an explicit surface and when its a parametric surface its the double integral of F + or - (rs x rt) dA for a parametric surface

If a curve is parametrized by r(t)=<x(t),y(t)> then the length is given by

L(c)=integral from b to a |r'(t)| (two dimensional) L(c)=integral from b to a sqrt(x'(t)^2+y'(t)^2+z'(t)^2 (three dimensional)

Find the linearization of a function

L(x,y)=f(a.b)+Fx(a.b)(x-a)+Fy(a.b)(y-b)

Limits and Continuity

Limit exists: use algebra to simplify and then plug in variables Limit does not exist: choose two paths and see that the limits are different variables (common path y=mx) Limit exists: use the squeeze theorem to show the limit exists

Find the maximum rate of change of 𝑓(𝑥,𝑦)=−3𝑥𝑒^(2𝑥𝑦) at the point (2,0)

Max rate of change=sqrt(585) Which direction does it occur? (Your answer must be a unit vector). u=<-3/sqrt(585),-24/sqrt(585)>

Gradient

Partial derivative with respect to x and then with respect to y

Find the length of a curve

Using the given equation, taken the derivative then place the derivative under a square root and square each term (length equation)

Classify the equations phi=pi/3 p=2 p=6sin(theta)sin(phi)

a half cone sphere sphere

Two vectors are perpendicular (orthogonal) if and only if

a*b=0 cos(90)=0

law of cosine

a*b=|a||b|cos(t)

Find a decomposition of a=<5,5,-5> into a vector c parallel to b=<7,6,2> and a vector d perpendicular to b such that c+d=a

c=projbA=75/89<7,6,-2> d=a-c <5,5,-5>-75/89<7,6,-2>

Level Curves

curves w/ equations f(x,y)=k, where k is a constant

The distance between a point P and a plane: 𝑎(𝑥−𝑥0)+𝑏(𝑦−𝑦0)+𝑐(𝑧−𝑧0)=0 can be given by:

d=|PR*n|/|n|

The distance between a point p and a line r(t)=ro+tv

d=|PRxv|/|v|

The distance between a point 𝑃 and a line: 𝐫(𝑡)=𝐫0+𝑡𝐯 can be given by

d=|PRxv|/|v| R is any point on the line v is the normal of the r given

divF=

del*F (take the partial derivative of each component)

vectors are parallel if they

differ by a scalar

Find the distance between the plane x+3y−2z=6 and the line ⟨−6−2t,−2−2t,−2−4t⟩.

direction vector of the line: <-2,-2,4> normal vector of the plane: <1,3,-2> random point: <-6,-2,-2> R=<6,0,0> =14/sqrt(14) |PR*n|/|n|=| (distance between a point and a plane)

Surface area of a graph

double integral of the square root of the partial derivative with respect to x squared and the partial derivative with respect to y squared =1 multiplied by dA

Chain Rule Formula

dz/dt=partial with respect to x*dx/dt+partial with respect to y*dy/dt

Find the differential of the function 𝑧=−2cos(4πt)e^(−3x)

dz=(6e^(-3x)cos(4pit))dx+(8pie^(-3x)sin(4pit))dt dz=Fxdx+Ftdt

Polar integral

extra factor of r

Find unit vectors parallel to the tangent line of f(x)=-(2x+x^2) through the point (0,0)

f(x)=-2x-x^2 f'(x)=-2-2x f'(0)=-2 y=-2x+a 0=-2(0)+a a=0 y=-2x direction: <0-1,0+2> <-1,2> c=sqrt(1^2+2^2)=sqrt(5) b=+-c<-1,2> b=+-1/sqrt(5)<-1,2>

Setup and evaluate a double integral for the area of the loop of the rose r=cos(4theta) that crosses the positive x-axis

first integral from (-pi/8) to (pi/8) and second integral from 0 to cos(4theta) of rdrdtheta

cross product

given two non-parallel vectors and be able to find a vector perpendicular to them

Classify each of the equations theta=pi/4 r=2 z=2-7r^2

half plane cylinder elliptical paraboloid

vector definition

has both magnitude and direction

dot product

helps to find the angle between two vectors (results in a scalar)

Find the mass of a wire in the shape of a helix x=7t, y=2cos(t), z=2sin(t) 0<t<2pi given that the density of the wire is equal to the square of the distance from the origin

integral from 0 to 2pi of 49t^2+4*sqrt(53)

Work

integral of F*T then integral of F*dr then integral of F*r'(t) and finally integral of Pdx+Qdy and double integral of Qx-PydA

Flux

integral of F*nds=integral of Pdy-Qdx= double integral of Px+QydA

Arc Length Function

integral of |r'(u)| with no specified stopping point u is the unit vector

Length of the vector

magnitude, sqrt(each vector point squared)

Find the mass of the lamina that occupies the region D={(x,y) | 1≤x≤6, 4≤y≤8} and has the density function ρ(x,y)=2y.

mass=240

Find the distance between the planes y−3x=−19 and y−3x=1.

n1=<-3,1,0> point (0,-19,0) n2=<-3,1,0> point (0,1,0) d=|PQ*n|/|n| PQ=<0,20,0> PQ*n=20 |n|=sqrt(9+1)=sqrt(10) 20/sqrt(10)

Find parametric equations for the tangent line of the curve r(t)=⟨5+2sqrt(t),4t^3−5t^2+5,t^3+5t^2−4⟩ through the point (7,4,2)

r'(t)=<1/sqrt(t),12t^2-10t,3t^2+10t> r'(1)=<1,2,13> x=7+t y=4+2t z=2+13t

T (unit tangent vector)

r'(theta)/|r'(theta)|

Line integrals

r(t)=(1-t)ro+tr1 0<t<1 ds=sqrt((dx/dt)^2+(dy/dt)^2) helps to find the length of a curve

Find a line segment from (0,−4,0) to (2,−8,1)

r(t)=<0,-4,0>+t<2,-4,1>

Find the work done by the force field F(x,y)=7x^2i-2xyj on a particle that moves once around the circle x^2+y^2=9 oriented in the ccw direction

r(t)=<3cos(t), 3sin(t)> t [0,2pi] integral of 7x^2i-2xyj then integral from 0 to 2pi of 2(3cos(t))^2(-3cos(t))-2(3cos(t))(3sin(t))(3cos(t))dt

Vector equation

r(t)=r0+tv

Triple integrals in Spherical coordinates

r=psin(phi) x=psin(phi)cos(theta) y=psin(phi)sin(theta) z=pcos(phi) X^2+y^2+z^2=p^2

Green's Theorem

relationship between double integrals and line integrals around simple closed curves positive orientation: ccw negative orientation: cw

Find an equation of the tangent plane to the parametric surface x=4u+4v y=3u^2 z=6u-4v at the point (28,12,-8)

ru=<4,6u,6> rv=<4,0-4> ru x rv= <-24u,40,-24u> solve for u and v using (28,12,-8) u=2 v=5 -48(x-28)+40(y-12)-48(z+8)=0

max value of directional derivative

the gradient's length and it occurs when U has the same direction as the gradient vector

When evaluating limits if they depend on m

they do not exist

Hyperbolas

two squared terms (one positive and one negative) (opens to the positive squared term)

The position function of a particle is given by: r(t)=⟨2+4t^2,5+3t,5t^2−2t⟩ When is the speed a minimum?

v(t)=<8t,3,10t-2> s(t)=sqrt((8t^2)+3^2+(10t-2)^2) s(t)^2=64t^2+9+(10t-2)^2 s'(t)^2=128t+0+2(10t-2)*10 s'(t)^2=128t+200t-40 328t=40 t=40/328

Find a vector function to represent the cylinder x^2+y^2=1 and plane x+z=5 intersection

x=rcos(t) y=rsin(t) x=cos(t) y=sin(t) z=5-cos(t)

Polar equations

x=rcos(theta) y=rsin(theta) x^2+y^2=r^2 tan(theta) = y/x

Find parametric equations of the curve given by the intersection of the surfaces: The cone: z=sqrt(x^2+y^2) The plane: z=1+y.

x=t y=(t^2-1)/2 z=1+(t^2-1)/2

Ellipsoid

x^2/a^2 + y^2/b^2 + z^2/c^2 = 1

Hyperboloid of One Sheet

x^2/a^2 + y^2/b^2 - z^2/c^2 = 1

Elliptical Paraboloid

x^2/a^2 + y^2/b^2 = z/c

Elliptical Cone

x^2/a^2 + y^2/b^2 = z^2/c^2

Hyperbolic Paraboloid

z/c = x^2/a^2 - y^2/b^2

Cylinder

z=y^2+1 consists of all lines that are parallel to a given line and pass through a given plane curve

Two vectors are parallel if the magnitude of the cross product equals

zero a(axb)=0 b(axb)=0