Next Week’s Assignments

9.4: Nonlinear Systems of Equations and Inequalities Video Due Tues, April 6th
9.4: Nonlinear Systems of Equations and Inequalities Homework Due Tues, April 6th
9.7: Parametric Equations Video Due Fri, April 9th
9.7: Parametric Equations Homework Due Fri, April 9th

The Following Week’s Assignments

• Online Class to Review for Test Wednesday, April 14th
• Review of 9.1 – 9.4 & 9.7 Due Thursday, April 15th
• Test on 9.1 – 9.4 & 9.7 Due Friday, April 16th
  
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By the end of the lesson, you should be able to:

This week we cover Polar Coordinates. With Polar Coordinates, we identify points on an x-y coordinate system using an angle and a “radius” – the length from the origin to the point.

• Graph Points using polar coordinates
• Convert between polar and rectangular coordinates
• Convert between polar and rectangular equations
• Graph equations given in polar form

Assignment

1. Watch 7.4 Video

7-4-polar-coordinates1

►

2. Complete 7.4 Homework on CourseCompass (Due Wed, Jan 27th)

Items for this week:

• Quiz on Lessons 5.4 – 5.6: Available Friday – Tuesday
• Test on Chapter 5: Available Wednesday – Thursday
• Online class on Tuesday, October 13th at 1:30 pm

Quiz on 5.4 – 5.6

This quiz will be like the last one, in that it will have 10 questions and may be taken multiple times. The questions will be on the following topics:

Lesson 5.4

• Find points given in radian measure on the unit circle
• Find coterminal angles, complements, and supplements of angles given in radian measure
• Convert between radian and degree measure
• Find arc lengths and central angles
• Convert between linear and angular speed

Lesson 5.5

• Find function values using coordinates of points on the unit circle
• Find function values using a calculator in Radian mode
• Find the coordinates of the reflection of a point on the unit circle

Lesson 5.6

• Find amplitude, period and phase shift of sine and cosine functions, and use them to graph the function
• Given a graph, find an equation of the sine or cosine function that matches it.

Your last review quiz is available and due on Saturday! Same format as the last one. Topics are as follows:

Section 6.4
Evaluate Inverse Trig Functions
Section 6.5
Solve trigonometric equations for angle values
Section 7.1
Use the law of sines
Section 7.2
Use the law of cosines
Section 7.3
Graph a complex number and find its absolute value
Write a complex number in trigonometric form
Find the standard notation for a complex number written in trig form
Multiply complex numbers
Section 7.4
Graph a point given its polar coordinates
Convert rectangular coordinates into polar coordinates
Convert between polar and rectangular equations
Section 7.5
Find the sum of two vectors, given their magnitudes and the angle between them
Section 7.6
Find the component form and magnitude of a vector
Find the sum of vectors given in component form
Find the dot product of two vectors
Express a vector as a linear combination of unit vectors
Find the direction angle of a vector given in component form

Quiz Topics (you should be able to…)

Your next review quiz is posted. It is due Wednesday, May 8th. I am attaching the Trig Identities sheet to this post for you to use in your review. I would reccomend that you take the quiz once, print out the answers and figure out any you missed, and then take it again.

Click here for the Trig Identities PDF

 Section 5.1 

• Find trig ratio function values (sin, cos, tan, csc, sec, cot) for a given triangle
• Given one ratio value, find the other 5
• Know exact values for 30-60-90 and 45-45-90 triangles
• Use properties of cofunctions (sin of an angle is the same as the cosine of its complement, etc.)

Section 5.2

• Solve right triangles

Section 5.3

• Find reference angles
• Know the exact values of trig ratio functions at 0, 180, 270, and 360 degrees
• Given a trig ratio function value and quadrant restriction, find the angle
• Determine whether angles are coterminal

Section 5.4

• Find coterminal angles in radian form
• Find complementary and supplementary angles in radian form
• Find the location on the unit circle of a point determined by radian measure
• Convert between radians and degrees
• Know the relationship between arc length, radius and radian measure (arc length = r times theta, or the radian measure of an angle is the length of the arc, divided by the radius).
• Convert between linear speed and revolutions.

Section 5.5

• Find the exact value of a radian expression

Section 5.6

• Given a sine or cosine function, find the amplitude, period, phase shift and graph

Section 6.1

• Use basic and pythagorean identities to factor, multiply, and simplify trig expressions
• Use sum and difference identities

Section 6.2

• Use Double-angle identities

Plan Ahead:

Students,

Your first review Quiz is due Friday, May 1st. It covers the following topics:

1.1
Find the distance between two points
Find the midpoint of a line segment
1.2
Determine whether or not a graph, equation, or set of ordered pairs represents a function. It is a function if there is a unique output for each input, or if a given x-value can only yield one y-value.
Find the domain and range of a function
1.3
Find the slope between two points
1.4
Find the slope and y-intercept of a line
Determine if lines are parallel, perpendicular or neither. Parallel lines have the same slope. Perpendicular lines have negative reciprocal slope.
Find the equation of a line that passes through a given point and is parallel or perpendicular to a line
1.5
Graph piecewise functions
1.6
Perform function operations (addition, subtraction, multiplication, division of functions)
Find compositions of functions
4.1
Determine whether or not a function is one-to-one
Find the inverse of a function
- algebraically: switch the x and y and solve for y if neccesary
- graphically: reflect the original graph across the line y = x
- given a set of ordered pairs: reverse the pairs so that the y-values are inputs and x-values are outputs

Review Quiz on 5.1 – 6.3: Wednesday, May 6th

Review Quiz on 6.3 – 7.6: Saturday, May 9th

Comprehensive Final: Due 8 pm May 12th

Assignment – Due Saturday

Students,

I created an interactive lesson for you to work through for this section.

Please click here to open the lesson. At the end of the lesson you will be given the opportunity to enter your name. Please do so – you must send me the results to recieve full credit for this assignment.

1. Read and work through the lesson on Logarithmic and Exponential Equations.
2. Email me the results.
3. Watch section videos on 4.2 and 4.3 if you need more explanation of any of the concepts.
4. Complete Homework assignment.

Thanks to everyone who has posted on the discussion board! I’ve enjoyed reading and replying to your coments…it’s definately more fun than reviewing homework answers.  If you haven’t posted yet, there are plenty of interesting questions to reply to.

Ms. Seekamp

John Napier and the Invention of Logarithms

Background and Introduction

For this lesson, we are going to step back in history. Imagine you live at the turn of the 15th century. A lot of scientific discovery is going on! Galileo Galilei is inventing the telescope and working on his heliocentric theories; Johannes Kepler is observing and calculating the laws of planetary motion, and the goundwork is being laid for Newton and Leibniz to discover Calculus in a few decades! But we have a problem. All this mathemtical and scientific discovery is hindered by the fact that arithmetic operations on large numbers have to be done by hand – there are no computers, calculators or slide rules.

Enter John Napier:

“Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers…I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.” – John Napier, 1614

Napier’s invention of logarithms, along with his other discoveries, provided scientists and mathematicians with tools to make their lives much easier. A famous mathematician named Laplace said that “by shortening the labors, [Napier]doubled the life of the astronomer.”

So, what is a logarithm, anyway!? It is an exponent. A logarithm is an exponent. Don’t ever forget that! It is the exponent you raise a base to, to get another number. For example, log₁₀100 = 2 reads “the log base 10 of 100 is 2″ and means that you must raise 10 to the second to get 100. So the logarithm, or exponent, is 2. Similarly, log₁₀1000 = 3, or you raise 10 to the third to get 1000. So the logarithm tells us how many times we have to multiply the base by itself to get the number we are looking for.

Here’s where it gets interesting. What would you have to raise 10 to, to get… 537? Well, 10² = 100 and 10³ = 1000, so you would have to raise 10 to “2.something” to get 537. If we have a way to compute logarithms, we can figure out what the “2.something” is!

Go ahead and grab your calculator, enter log 537 on a calculator and you’ll get the answer. Base 10 logarithms are called common logarithms and are usually written without the base. So log 537 means the same thing as log₁₀537.

Next on the agenda is some reading and discussion about the development of logarithms. In our next lesson we’ll start graphing and doing computations with logarithms and exponents.

Assignment

1. Read and take notes on chapters 1 and 2 of “e: The Story of a Number”
2. Take the History of the Development of Logarithms online quiz (due by midnight Tuesday).
3. Answer each of the discussion questions on the History of the Development of Logarithms discussion board (go to Communication and then click on Discussion board) (due Wednesday).
4. Respond to your fellow students answers and/or post your own question. Points will be awarded based on the thoughtfulness of your discussion (I will be moderating the discussion board on a daily basis through Saturday).

 

Introduction

This week we are studying lessons 1.6 and 4.1. We won’t have a big test on chapter 1 – instead we’ll do a couple quizzes over the next two weeks. There will be a test at the end of chapter 4 that will cover topics from both chapters.

Lesson 1.6

Lesson 1.6 is on the algebra of functions. You will learn how to add, subtract, multiply and divide functions, and find their resulting domains. This is pretty straightforward – you just add, subtract, multiply, or divide the two equations and simplify or add like terms as needed.  You will also learn how to find the composition of two functions. This means that you put one function inside the other; instead of evaluating a function at an x-value, you evaluate it at another function. For example, if f(x) = x+1 and g(x) = 2x/5, then f composed with g is written f ο g (the little circle means compose the two functions). f ο g = f [g(x)] = f [2x/5] = 2x/5 + 1. So basically, you put in the inside function wherever you see an x.

Lesson 4.1

Lesson 4.1 is on inverse functions. The inverse of a function is the function that will “undo” the function.

So if f(x) = 2x +5, f(x) is a function that multiplies the input by 2 and then adds 5. The “undoer” function would subtract 5 and then divide the result by 2. So the inverse function is f ¯¹(x) = (x-5)/2. f  with a little -1 superscript means f inverse.

Now try this:

1. Find f(1). (the output of f when you input 1)
2. Plug that value into f ¯¹(x).
3. Notice that the output is your original input.

Some things to note about inverse functions:

• If you put an output (y-value) of the original function into the inverse function, you will get out your original input (x-value).
• The x-values in f(x) are the y-values in f ¯¹(x), and the y-values in f(x) are the x-values in f ¯¹(x).
• The graph of a function and its inverse are reflections of eachother across the line y = x.
• The composition of a function and it’s inverse equals x. So f [f ¯¹(x) ] = x. This is because you are doing something to x, then undoing that process, so you just get out x.

Assignment  

1. Watch the video on section 1.6.

2. Watch the video on section 4.1.

3. Complete the homework assignment.