DATE
Unit 2 Packet Name: __________________________
Unit Circle Review/Polar Coordinates (Homework)
Use the circle to answer the following.
1. At which point(s) shown is the cosine negative?
B _________________________________
2. What is the positive radian measure of [pic]?
_________________________________
3. What is the sine of the angle that would be at point G?
______________________
4. Give the polar coordinates of E. _________________
F
5. What is the counterclockwise angle measure, in radians, of [pic]? What is the clockwise measure of the same angle?
______________________ ______________________
6. At which labeled point does sine=cosine? Give the degree and radian measure of this angle.
_________________________________
7. Which point labeled on the circle corresponds to the angle of [pic]? ____________________
8. Which point labeled on the circle corresponds to the angle [pic]? ____________________
9. On the graph, plot the following and label: 10. On the graph, plot [pic] using the
following points:
| r | [pic] |
| | 0 |
| | [pic] |
| | [pic] |
| | [pic] |
| | [pic] |
| | [pic] |
| |[pic] |
| |[pic] |
A) [pic]
B) [pic]
C) [pic]
D) r = 3
E) [pic]
[pic]
Polar Practice
NO CALCULUATOR!!!
1. Convert the RECTANGULAR coordinate TO a POLAR coordinate. [pic]
2. Convert the POLAR coordinate TO a RECTANGULAR coordinate. P: [pic]
3. Convert the polar equation to rectangular form. [pic]
4. Write the polar equation of a circle with radius [pic] and center [pic].
5. Convert the rectangular equation x = 9 to a polar equation.
6. Represent[pic] in at least four different ways.
7. Convert the polar equation to rectangular form. [pic]
8. Convert the polar equation to rectangular form. [pic]
9. Integrate: [pic]
10. Integrate: [pic] using the identity: [pic]
BC Calculus
Polar Lab Wrap Up
Let’s see what you have learned…Without a calculator, match the following graphs with their equation.
GRAPHS:
[pic] [pic] [pic]
____________ ______________ ______________
[pic] [pic]
____________ ______________ ______________
Functions:
A) [pic] B) [pic] C) [pic]
D) [pic] E) [pic] F) [pic]
AP Calculus BC Polar Lab Follow Up Practice
Complete each problem. When finished, each problem should have an equation, graph, domain
(where graph starts repeating) and description of symmetry (relative to the x-axis and y-axis).
| |[pic] | |[pic] |
| | | | |
|Equation: | |Equation: [pic] | |
| | | | |
| | | | |
|Domain: | |Domain: | |
| | | | |
| | | | |
|Symmetry | |Symmetry | |
| |[pic] | |[pic] |
| | | | |
|Equation: | |Equation: [pic] | |
| | | | |
| | | | |
|Domain: | |Domain: | |
| | | | |
| | | | |
|Symmetry | |Symmetry | |
| |[pic] | |[pic] |
| | | | |
|Equation: | |Equation: [pic] | |
| | | | |
| | | | |
|Domain: | |Domain: | |
| | | | |
| | | | |
|Symmetry | |Symmetry | |
| |[pic] | |[pic] |
| | | | |
|Equation: | |Equation: [pic] | |
| | | | |
| | | | |
|Domain: | |Domain: | |
| | | | |
| | | | |
|Symmetry | |Symmetry | |
| |[pic] | |[pic] |
| | | | |
|Equation: | |Equation: [pic] | |
| | | | |
| | | | |
|Domain: | |Domain: | |
| | | | |
| | | | |
|Symmetry | |Symmetry | |
| |[pic] | |[pic] |
| | | | |
|Equation: | |Equation: [pic] | |
| | | | |
| | |Domain: | |
|Domain: | | | |
| | | | |
| | |Symmetry | |
|Symmetry | | | |
| |[pic] | |[pic] |
| | | | |
|Equation: | |Equation: [pic] | |
| | | | |
| | |Domain: | |
|Domain: | | | |
| | | | |
| | |Symmetry | |
|Symmetry | | | |
| |[pic] | |[pic] |
| | | | |
|Equation: | |Equation: [pic] | |
| | | | |
| | |Domain: | |
|Domain: | | | |
| | |Symmetry | |
| | | | |
|Symmetry | | | |
Self-Study Lengths of Polar Curves
What is the circumference (“length around”) the circle at the right? _______
What is the length of the arc in quadrant I? _______
This is TOO EASY!!
Today, you will find out how to find the length around other curves like:
First revisit the formula for finding the distance between two points:
Find the length of this line segment:
Think of the cardioid as a set of really small line segments with lengths that could be determined by the
distance formula and then add all of these tiny lengths together to find the length of the curve. Imagine
making the segments smaller and smaller so that we have more and more of them. Sound like a
similar calculus topic? But of course. . . . . Riemann Sums and Integration!
Now, account for the fact that we are using polar coordinates instead of rectangular coordinates and we have…
Example: Find the length of
Now, we need to know what the boundaries of integration will be. Think back to your discover lab and to our discussions and what is the smallest θ-interval (domain) that we can use to get the entire cardioid curve?
Use your calculator to do the integration. So, …
Your Turn to Try: (YOU can use your calculator to do the integrating! )
1. Find the length of the curve: Answer:
2. Find the length of the specified portion of the polar curve: Answer:
3. Find the length of the curve, , where a is an unknown constant. Answer:
-----------------------
C
H
D
A
G
E
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[pic]
h {h¤J3B*[pic]OJQJ^Jph!h {höf”B*[pic]OJQJ^Jphhöf”CJmHnHu[pic]7jhØE[pic]B*[pic]CJOJQJU[pic]^JaJmHnHphu[pic]&hÞ4chÄEß5?CJOJQJ\?^JaJjhÄEßU[pic]mHnHu[pic] höf”5?CJOJQJ\?^JaJ&hÞ4chöf”5?CJOJQJ\?^JaJ h4¿5?CJOJ EMBED Unknown [pic]
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Over for Practice
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