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

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.

Quiz Topics (you should be able to…)

 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

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

Plan Ahead:

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

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.

Assignment – Due Saturday

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.

Homework 2 Assignment

1. Read through sections 1.4 and 1.5 using the study guide below. Section 1.4 should all be review from Algebra 2, so just make sure you understand the objectives below. Section 1.5 will contain some new material.
2. Watch the video for Section 1.5.
3. Complete the homework assignment on course compass.
4. Complete the assigment at the bottom of the study guide. Email or put in my box.

Study Guide

Section 1.4: Equations of Lines

• Lines can be written in Slope- Intercept form or Point-Slope Form.
  • Slope-intercept form is f(x) = mx + b, where m is the slope of the line and b is the y-intercept (the place where the line crosses the y-axis.
    • See examples 2, 3, and 4.
  • Point-Slope form is y – y1 = m (x – x1), where m is the slope of the line and (x1, y1) is any point on the line.
    • See example 6.
• Two lines are Parallel if they have the same slope and different y-intercepts (they change at the same rate).
• Two lines are Perpendicular if the product of their slopes is -1.
  • For example, if you have two lines with slopes of -2 and 0.5, they would be perpendicular.
  • Lines are also perpendicular if one is vertical and the other is horizontal.
  • See examples 7 & 8

Section 1.5: More on Functions

Increasing and Decreasing Functions (see example 1)

• A function is increasing if it is rising from left to right. Mathematically, we say it this way:
  • A function is increasing on an open interval if for all a and b in the interval, a < b implies f(a) < f(b). In other words, if the x-value is smaller, then so is the y-value.
• Similarly, a function is decreasing if it is dropping from right to left. Mathematically, we say it like this:
  • A function is decreasing on an open interval if for all a and b in the interval,a < b implies f(a) > f(b). So if the x-value is larger, then the y-value is smaller.
• A function is constant if the values stay the same from left to right. Mathematically:
  • A function is constant on an open interval if for all a and b in the interval, f(a) = f(b). In other words, every y-value is the same in the interval.

Relative Maxima/ Minima

• We call the hills and valleys in graphs local, or relative, maxima and minima.
• The value of the maximum or minimum is the y-value, or function value. We say that the max or min occurs at the x-value.
  • For example, in the graph below, the local max is 4 and occurs at x = -1. The local min occurs at x = 1 and is equal to -4.
• 

Piecewise-Defined Functions

A piecewise function uses different output formulas for different parts of the domain. So, for example, the function could be defined by one equation for x < 0, and by another equation for x ≥ 0. See examples 5 – 7 in your textbook.

Assignment:

1. Make up a real-world piecewise defined function that increases, decreases, and is constant (not neccesarily in that order) for different parts of it’s domain.
2. Draw a graph of the function and explain in terms of the domain where the function increases, decreases, and is constant.
3. For a bonus point, write the function as a piecewise defined function with the appropriate equations.
4. Email to me (I made the diagram below just using Paint) or write out by hand and leave in my box on Wed/Thurs.

Example: T(t) is a piecewise function for Temperature as a function of time. It is defined as follows: Water is heated to boiling to make tea. The water boils for a minute and then it is taken off the heat and left on the counter to cool. A graph of the function could look like:

The function is increasing on t = (0, 2), constant on t = (2, 3) and decreasing on t > 3.

Homework 1 Assignment

1. Read through Lessons 1.2 and 1.3 using the study guide below.
2. Answer the study guide questions in blue below. Email your answers to me.
3. Watch the section videos as needed.
4. Complete the homework assignment on course compass.

Section 1.2 Objectives and Study Guide

1. Determine whether a correspondence is a relation or a function.
   • A function is a relationship between two sets of numbers where each member of the first set has only one corresponding value in the second set. You can think of a function as a machine with inputs and outputs. You put a value from the first set (called the domain) into the function; the function ouputs a specific number, a number always associated with that input; this number is a member of the second set (called the range).
   • Examples of functions:
     • The function defined by the following ordered pairs (the first number is the input, the second number is the output): (1, 2), (2, 3), (3, 2). Notice that you can have more than one input yielding the output “2″.
     • The function defined by the following relationship: The outdoor temperature as a function of time. You would write this T(t), where T = Temperature and t = time. Here the domain, or inputs, are a set of time values. The range, or outputs, are a set of temperature values. So, each time input is assigned a specific temperature value. You cannot have one time yielding two different temperatures, so this is indeed a function.
     • The function f(x) = 2x + 3 or g(x) = x^2. Each input has a unique output.
   • A relation is a relationship between two sets of numbers where each member of the first set has at least one corresponding value in the second set. So the requirements for a relation are not as strict as they are for a function.
   • Question 1: Describe a real-world function relationship as I did above. Include a description of the domain and range.
2. Find function values, or outputs, using a function or graph.
   • The output is the function value, or y-value, associated with the input, or x-value.
3. Graph functions.
   • Plot ordered pairs (x, y) and sketch the graph connecting the points.
4. Determine whether a graph is or is not a function.
   • If a relationship is a function, then the input will have no more than one output. Graphically, this means that an x-value will have only one associated y-value, or that a vertical line can only pass through a function in one place. If it were to pass through the function in two places, then there would be two y-values for the given x-value, and the relation would not be a function.
   • Question 2: Does the graph below represent a function? Explain your answer.
5. Find the domain and range of a function.
   • Remember, the domain is all possible inputs and the range is all possible outputs. So for a function like f(x) = 1/x, the domain would be everything except for zero. Zero is not in the domain because division by zero is undefined. We can express this in two ways:
     • In Set-Builder Notation: D = {x| x ≠ 0}. You read this “The Domain is equal to the set of all x-values such that x is not equal to zero.” The brackets { } denote a set. The line | means “such that”.
     • In Interval Notation: D: (-∞, 0), (0, ∞). This says “the domain includes negative infinity to zero and zero to infinity, but excludes zero.” The parentheses ( ) denote that the endpoint is NOT included. If you need to include the endpoint use the square brackets [ ].
   • Question 3: Write the domain of the function f(x) = 1/(x-2)
     • In set-builder notation.
     • In interval notation.
6. Solve Applied Problems using Functions.

Section 1.3 Objectives and Study Guide

1. Determine the slope of a line, given two points on the line.
   • Slope is found by the change in y, divided by the change in x. You can also think of this as “rise” over “run” or (y2 – y1)/(x2 – x1).
   • For horizontal lines (y = constant), the slope is always zero, because the change in y is zero.
   • For vertical lines (x = constant), the slope is always undefined, because the change in x is zero and division by zero is undefined.
2. Solve applied problems involving slope and linear functions.