Calculus III

Question 1

if a=<x,y,z> and b=<i,j,k>, what is the dot product of a and b?

Answer 1

(x)(i)+(y)(j)+(z)(k)

Question 2

is a vector plus a vector a scalar or vector?

Answer 2

Vector

Question 3

Is the dot product of a vector and a scalar a scalar or a vector?

Answer 3

Vector

Question 4

Is the dot product of two vectors a scalar or a vector?

Answer 4

Scalar

Question 5

if a=<x,y,z> and b=<a,b,c> what is the cross product of a and b?

Answer 5

(yc-zb)i - (xc-za)j + (xb-ya)k

Question 6

What is <x,y,z>=<xo+at, yo+bt, zo+ct>

Answer 6

The vector equation of a line thru a point <xo, yo, zo> and is parallel to vector <ai, bj, ck>

Question 7

What are the parametric equations of the line that passes thru point <5, 1, 3> and is parallel to the vector i-4j+2k?

Answer 7

x=5+t
y=1-4t
z=3+2t

Question 8

find 2 other points besides <5, 1, 3> on the line given the parametric equations
x=5+t
y=1-4t
z=3+2t

Answer 8

if t=1
(5, -3, 5)

if t=-1
(4, 5, 1)

Question 9

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?

Answer 9

Direction numbers

Question 9

Let P1=<x1, y1, z1> and P2=<x2, y2, z2>, what is the symmetric equation of L?

Answer 9

(x-x1)/(x2-x1) = (y-y1)/(y2-y1) = (z-z1)/(z2-z1)

Question 9

How do you find the equation of a line thru a given point P that is orthogonal to a given vector V?

Answer 9

Vx(x-Px) + Vy(y-Py) + Vz(z-Pz) = 0

Question 10

What are the parametric equations of line L thru point P0=(x0,y0,z0) and parallel to vector V=<a,b,c>.

Answer 10

r=r0+tV
r= point along line L
r0= based on P0

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

Question 11

What is the scalar equation of a plane thru point P0=(x0,y0,z0) with normal vector n=<a,b,c>

Answer 11

n(r-r0)=0
a(x-x0)+b(y-y0)+c(z-z0)=0

Question 12

Two planes are parallel if what?

Answer 12

Their normal vectors are parallel.

Question 13

How does one determine if 2 given planes (labeled a and b) with points in them are parallel or not.

Answer 13

(xa)/(xb)=(ya)/(yb)=(za)/(zb)
the planes are parallel if the statement is true.

Question 14

What is the equation cos(theta)=(n1 DotProd n2)/(||n1|| DotProd ||n2||) used for?

Answer 14

Finding acute angle between non-parallel planes.

Question 15

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

Answer 15

a) r’(t)=<3t^2, e^(-t)-te^(-t), 2cos(2t)>

b) <0, 1/(sqrt(5)), 2/(sqrt(5))>

Question 16

What is the equation T(t)=(r’(t))/(|r’(t)|)

Answer 16

The unit tangent vector

Question 17

If r(t)=<2cos(t), sin(t), 2t>, find the integral of r(t).

Answer 17

<2sin(t)+C1, -cos(t)+C2, t^(2)+C3>

Question 18

what is the equation L=Integral(A-B)[|r’(t)|dt]

Answer 18

Arc Length

Question 19

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)

Answer 19

|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

Question 20

A parametrization r(t) is called ________ on an interval I if r’(t) is continuous and r’(t) does not equal 0.

Answer 20

smooth

Question 21

what is the curvature equation?

Answer 21

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)

Question 22

Find the curvature of r(t)=<t, t^2, t^3> at (0, 0, 0)

Answer 22

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

Question 23

What are normal, tangent, and binormal vectors?

Answer 23

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

Question 24

How does one determine if 3 points exist in a straight line?

Answer 24

If the slope of line segment AB is the same as the slope of line segment BC.

Question 25

Find the domain of f(x,y)=sqrt(9-(x^2)-(y^2))

Answer 25

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}

Question 26

How would one sketch the graph of an equation of 2 or more variables?

Answer 26

Set up a systems of equations and solve for x,y,z, etc with the rest of the variables being zero.

Question 27

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?

Answer 27

Level curves

Question 28

Find the level curves of the function f(x,y)=sqrt(9-(x^2)-(y^2))
for k=0,1,2,3

Answer 28

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

Question 29

Find the limit
lim[(x,y)->(3,2)] (x^2 + 1)/(xy + 3)

Answer 29

10/9

Question 30

if a limit problem would result in an unedified solution if c was plugged in, what are your options for solving the limit?

Answer 30

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.

Question 31

Find fx and fy for f(x,y)=x^2 -2yx+y^2

Answer 31

fx=2x-2y
fy=-2x+2y

Question 32

the equation of the tangent plane to the surface z=f(x,y) at the point P(x0,y0,z0) is?

Answer 32

z-z0=fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0)

Question 33

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

Answer 33

Linearization of f at P(a,b)

Question 34

How does one do linear approximation?

Answer 34

Find the linearization of the function and use rounded values to the nearest whole number for the point coordinates.

Question 35

Evaluate the double integral shown

Answer 35

455/6

Question 36

Calculate the iterated integral.

Answer 36

(8/5)(2sqrt[2]-1)

Question 37

Find the iterated integral

Answer 37

Question 38

Evaluate the integral by reversing the order of integration

Answer 38