Calculus Midterm 4

Question 1

Polar Coordinate System

Answer 1

The location of a point described by its distance from a fixed point and angular direction

Question 2

What is the fixed point called?

Answer 2

Pole

Question 3

Polar Axis

Answer 3

Ray emanating from pole

Question 4

Rectangular to coordinate equations

Answer 4

x = rcos(ϴ)
y = rsin(ϴ)

Question 5

Polar to rectangular coordinates

Answer 5

r^2 = x^2 + y^2
tan(ϴ) = y/x

Question 6

Equation for line in a polar function

Answer 6

ϴ = k

Question 7

Equation for circles in a polar function

Answer 7

Centered at
Origin: r = k
x-axis: r = kcos(ϴ)
y-axis: r = ksin(ϴ)

Question 8

Calculator Function for graphing points of intersections polar

Answer 8

Polar
Simul
Graph functions
Adjust window settings

Question 9

When do polar functions intersect?

Answer 9

ONLY if the ϴ and r are the same at that point

Question 10

How to find points of intersection in polar functions

Answer 10

Set two equations equal to each other and solve for ϴ

sin(ϴ) = 1/2

(1/2, pi/6), (1/2, 5pi/6)

Question 11

Derivative of Polar Functions: dr/dϴ

Answer 11

The rate of change of radius as the angle changes

Question 12

Derivative of Polar Functions: dy/dx

Answer 12

Slope of the polar curve in the xy-plane

Question 13

Derivative of a polar function

How to find dy/dx of r = f(ϴ)

Answer 13

1. find rectangular coordinates (x and y)
2. Replace r with f(ϴ)
3. find dx/dϴ and dy/dϴ
   4/ find dy/dx

Question 14

Arc Length

Answer 14

dL = sqrt((dx/dt)^2 + (dy/dt)^2) dt

Question 15

Arc Length if ϴ is a parametric

Answer 15

r^2 + (dr/dϴ)^2

Question 16

Area of a polar curve

Answer 16

dA = 1/2r^2dϴ

Question 16

Distance vs. Displacement (definition)

Answer 16

Distance: total path covered by the objection at a time frame

Displacement: The total change in initial and final position

Question 16

Distance vs. Displacement (use)

Answer 16

Distance:
- d
- positive always
- No direction
- Scalar

Displacement:
- s
- +, -, or zero (left or right of initial point)
- has direction vector

Question 17

Distance formula

Answer 17

d = ∫b->a |v(t)| dt

Question 18

Displacement formula

Answer 18

s = ∫b->a v(t) dt

Question 19

Derivative of parametric functions

Answer 19

dy/dx = dy/dt / dx/dt

Question 20

Find the parametric equations of the direct line segment from (1,2) to (3,5)

Answer 20

first coordinate of one pair * (1-t) + first coordinate of other pair (t)

Question 21

Find Parametric equations of the line y - 4 = 1/2 (x-3)

Answer 21

x(t) = at + b
y(t) = ct + d

Slope = (c/a)
Point (b, d)

Question 22

Find the parametric functions of a circle with the radius (r)

Answer 22

Origin is the center
x(t) = rcos(t)
y(t) = rsin(t)

(h,k) is the center
x(t) = rcos(t) + h
y(t) = rsin(t) + k

Question 23

Find the parametric functions of an ellipse with the radius (r)

Answer 23

x(t) = acos(t)
y(t) = bsin(t)

Question 24

Thinking about an ellipse like…

Answer 24

A circle with different y and x radii

Question 25

Ellipse equation

Answer 25

(x/a)^2 + (x/b)^2 = 1

Question 26

Component Form

Answer 26

v = ai + bj
<a, b>
i = x-axis
j = y-axis

Question 27

Length aka norm aka magnitude of a vector

Answer 27

||v|| = sqrt(a^2 + b^2)

Question 28

Unit Vector

Answer 28

v = <a,b>

u = 1/||v|| * <a, b>

Question 29

Dot Product

Answer 29

“multiply vectors”
v = <v1, v2>
w = <w1, w2>

v * w = ||V||||w||cos(ϴ)
gives a visual
OR
v * w =
v1w1 + v2w2
gives a way to calculate it

Question 30

what is ϴ in the dot product?

Answer 30

The angle between the two vectors

Question 31

Orthogonal Projections

Answer 31

Decomposes vectors into the sum of the two orthogonal components

Question 32

||b|| (length of b in orthogonal vector)

Answer 32

(dot product of v and w / ||v||^2) * v

Question 33

Parametric equations for r(t) = <f(t), g(t)>

Answer 33

x(t) = f(t)
y(t) = g(t)

Question 34

How to graph parametric equations in the calculator

Answer 34

1. Parametric mode
2. Plug in
3. Choose a good window and t-step and t-interval

*it will show the path but not the vectors

Question 35

Find the limit of r(t) if r(t) = <x(t), y(t)>

Answer 35

find the limits of each

Question 36

r(t) is continuous at t=a if

Answer 36

lim r(t) = r(a)
t->a

Question 38

Sin(2ϴ)

Answer 38

2sin(ϴ)cos(ϴ)

Question 39

dy/dx with r and ϴ

Answer 39

dy/dϴ over dx/dϴ = dr/dϴ sin(ϴ) + rcos(ϴ) over dr/dϴ cos(ϴ) - rsin(ϴ)