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