Calculus III Flashcards
if a=<x,y,z> and b=<i,j,k>, what is the dot product of a and b?
(x)(i)+(y)(j)+(z)(k)
is a vector plus a vector a scalar or vector?
Vector
Is the dot product of a vector and a scalar a scalar or a vector?
Vector
Is the dot product of two vectors a scalar or a vector?
Scalar
if a=<x,y,z> and b=<a,b,c> what is the cross product of a and b?
(yc-zb)i - (xc-za)j + (xb-ya)k
What is <x,y,z>=<xo+at, yo+bt, zo+ct>
The vector equation of a line thru a point <xo, yo, zo> and is parallel to vector <ai, bj, ck>
What are the parametric equations of the line that passes thru point <5, 1, 3> and is parallel to the vector i-4j+2k?
x=5+t
y=1-4t
z=3+2t
find 2 other points besides <5, 1, 3> on the line given the parametric equations
x=5+t
y=1-4t
z=3+2t
if t=1
(5, -3, 5)
if t=-1
(4, 5, 1)
if the vector v=<a, b, c> is used to describe the direction of a line L, what are the numbers a, b, and c called?
Direction numbers
Let P1=<x1, y1, z1> and P2=<x2, y2, z2>, what is the symmetric equation of L?
(x-x1)/(x2-x1) = (y-y1)/(y2-y1) = (z-z1)/(z2-z1)
How do you find the equation of a line thru a given point P that is orthogonal to a given vector V?
Vx(x-Px) + Vy(y-Py) + Vz(z-Pz) = 0
What are the parametric equations of line L thru point P0=(x0,y0,z0) and parallel to vector V=<a,b,c>.
r=r0+tV
r= point along line L
r0= based on P0
x=x0+at
y=y0+bt
z=z0+ct
What is the scalar equation of a plane thru point P0=(x0,y0,z0) with normal vector n=<a,b,c>
n(r-r0)=0
a(x-x0)+b(y-y0)+c(z-z0)=0
Two planes are parallel if what?
Their normal vectors are parallel.
How does one determine if 2 given planes (labeled a and b) with points in them are parallel or not.
(xa)/(xb)=(ya)/(yb)=(za)/(zb)
the planes are parallel if the statement is true.
What is the equation cos(theta)=(n1 DotProd n2)/(||n1|| DotProd ||n2||) used for?
Finding acute angle between non-parallel planes.
a) Find the derivative of
r(t)=<1+t^3, te^(-t), sin(2t)>
b) find the unit tangent vector at the point t=0
a) r’(t)=<3t^2, e^(-t)-te^(-t), 2cos(2t)>
b) <0, 1/(sqrt(5)), 2/(sqrt(5))>
What is the equation T(t)=(r’(t))/(|r’(t)|)
The unit tangent vector
If r(t)=<2cos(t), sin(t), 2t>, find the integral of r(t).
<2sin(t)+C1, -cos(t)+C2, t^(2)+C3>
what is the equation L=Integral(A-B)[|r’(t)|dt]
Arc Length
Find the arc length of the circular helix with the equation
r(t)=cos(t)i+sin(t)j+tk
from the point (1,0,0) to the point (1,0,2pi)
|r’(t)|=<-sin(t), cos(t), 1>
|r’(t)|=sqrt((-sin(t))^(2) + (cos(t))^(2) + 1^(2)) = sqrt(2)
r(t)=(1,0,0) => t=0
r(t)=(1,0,2pi) => t=2pi
L=integral(0-2pi)[|r’(t)|]
L=2sqrt(2)pi
A parametrization r(t) is called ________ on an interval I if r’(t) is continuous and r’(t) does not equal 0.
smooth
what is the curvature equation?
k=|(dT/ds)|
where T is the tangent vector, so
k(t)=|T’(t)|/|r’(t)|
this creates:
k(t)=|r’(t) X r”(t)|/|r’(t)|^(3)
Find the curvature of r(t)=<t, t^2, t^3> at (0, 0, 0)
r’(t)=<1, 2t, 3t^2>
r”(t)=<0, 2, 6t>
|r’(t)|=sqrt[9t^4 + 4t^2 + 1]
r’(t) X r”(t) = <6t^2, -6t, 2>
|r’(t) X r”(t) | = sqrt[36t^4 + 36t^2 + 4]
k(t) = (sqrt[36t^4 + 36t^2 + 4]) / (sqrt[9t^4 + 4t^2 + 1])^3
r(t) = <t, t^2, t^3> = <0, 0, 0> => t=0
k(0) = 2
What are normal, tangent, and binormal vectors?
T(t) = r’(t) / |r’(t)|
N(t) = T’(t) / |T’(t)|
B(t) = T(t) X N(t)
note that T, N, and B are unit vectors
How does one determine if 3 points exist in a straight line?
If the slope of line segment AB is the same as the slope of line segment BC.
Find the domain of f(x,y)=sqrt(9-(x^2)-(y^2))
f(x,y) is defined when the value inside the square root is greater than or equal to 0.
D={(x,y) E R^2 ; x<y^2}
How would one sketch the graph of an equation of 2 or more variables?
Set up a systems of equations and solve for x,y,z, etc with the rest of the variables being zero.
Curves with the equations f(x,y)=k (where k is a constant in the range of f) of a function of two variables are called what?
Level curves
Find the level curves of the function f(x,y)=sqrt(9-(x^2)-(y^2))
for k=0,1,2,3
f(x,y)=k
sqrt(9-(x^2)-(y^2))=k
(x^2)+(y^2)=9-(k^2)
k=0, (x^2)+(y^2)=9
k=1, (x^2)+(y^2)=8
k=2, (x^2)+(y^2)=5
k=3, (x^2)+(y^2)=0
Find the limit
lim[(x,y)->(3,2)] (x^2 + 1)/(xy + 3)
10/9
if a limit problem would result in an unedified solution if c was plugged in, what are your options for solving the limit?
Take derivative of top and bottom and try again, or (if the problem has 2 variables) you can use c=(x,0), (0,y), (x,x), (y,y) and determine the limit that way.
Find fx and fy for f(x,y)=x^2 -2yx+y^2
fx=2x-2y
fy=-2x+2y
the equation of the tangent plane to the surface z=f(x,y) at the point P(x0,y0,z0) is?
z-z0=fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0)
L(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b)
is what?
Linearization of f at P(a,b)
How does one do linear approximation?
Find the linearization of the function and use rounded values to the nearest whole number for the point coordinates.
Evaluate the double integral shown
455/6
Calculate the iterated integral.
(8/5)(2sqrt[2]-1)
Find the iterated integral
Evaluate the integral by reversing the order of integration
