Nebraska Open Access Mathematics - User contributions [en]

1.1 Functions

2020-07-07T16:32:35Z

Jbrummer: /* Individual Lesson Plans */

[[Covariational Reasoning | Prior Lesson]] | [[1.2 Rate of Change | Next Lesson]]

==Objectives:==
*Determine when a relationship is a function and determine its inputs and outputs
*Represent relations/functions as tables, ordered pairs, graphs, equations, etc.
*Review graphs and the vertical line test

;Definitions: function

----
==Lesson Guide==

====Determine when a relationship is a function and determine its inputs and outputs====
[15 minutes] Begin this lesson by giving examples of relationships between two sets. Examples should be easy to understand. I.e., use real world examples such as the days of the year and their average daily temperatures or the students in your class and the month they were born. Discuss how we can identify objects in the two sets and call this identification a “relation.”

A lot of this class will be about exploring the relationship between different pieces of information. For example:

{{Ex-begin}}

*The year you were born is related to your age.
*The month is related to the temperature outside.
*The price of gold is related to the supply and the demand.

If we have a very special kind of relationship called a ``function," we can use one piece of information, called the input, to completely predict the other piece of information, called the output.

It is often useful to take a few minutes and draw bubble diagrams of the examples you are going over. E.g.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/isrsxtlsgg?embed
|width=500
|height=500
|border=0
}}
{{Ex-end}}

You'll now want to define what a function is. Start by using the terms '''input''' and '''output''' to describe bubbles in your bubble diagram. Say the definition twice for emphasis.

A '''function''' is a relation (or a rule) that assigns each input to only one output.

We are not introducing function notation just yet, that comes shortly. Stress the importance of using and understanding precise language. Tell your students that if they are ever asked if a relationship is a function, they have to show that the relation either fulfills or violates the definition of a function.

{{Ex-begin}}
Is the number of letters a function of a word?

First, identify what is the input and what is the output. For example:

{| class="wikitable"
! Input: Word
! Output: Number of Letters
|-
| I
| 1
|-
| Love
| 4
|-
| Math
| 4
|-
| Class
| 5
|}
In this case, the number of letters is a function of the word, since for each input (word) has exactly one output (number of letters).

Students may notice that there are some words that have multiple accepted spellings (e.g. color and colour). Let the discussion progress naturally; have them consider whether it is still a function. For example, you may institute the rule that words with different spellings are different words, or else agree to some master list of the proper way to spell each word. Use this conversation to emphasize being precise and careful with language.

{{Ex-end}}

Take one of your previous examples of a function and identify the inputs and outputs. At this point you should also give a relationship that is a non-example of a function and
show why it violates the definition of a function.

[10 minutes] Have students do Problems 1-3.

[10 minutes] Ask students to volunteer their answers for Problems 1 and 3 and introduce function notation using these examples.

'''Note:''' Students will have difficulty in the preceding question determining which one is the input and which is the output. Explain that when it says "function of (blank)" that this means that (blank) is the input. This concept can also be explained in tandem with function notation below.

====Introduce Function Notation====

When we do have a function, we use a mathematical shortcut to talk about it. We write output = f(input), and we can use any letter to represent the function, the input, or the output. Write one of the real-world examples you discussed in class using function notation. For example, to indicate that a quantity y is a function of quantity x, we write y = f(x) and say "y equals f of x". Note: f(x) represents the output of the function f, when x is the input.

----
To indicate that the output is a function of the input, we express functions as output = f(input).
So y = f(x) means x is the input of the function and y is the output of the function. We verbally express this as “y equals f of x.”
----
It helps to draw the function machine for them to really associate that f(x) is the output.

Remind students that we can use any letter to represent the function, the input, or the output. Also, emphasize that f(x) also represents the output of the function f when x is the input.

==== Represent relations/functions as tables, ordered pairs, graphs, equations, etc.====
[5 minutes] Choose an example of a function to model in a table, as ordered pairs, and as an equation. Students will encounter functions in all of these forms, and thus should be familiar with all of them.
Make sure to plan this example before class so that you can look for an example that works well in all three cases.

{{Ex-begin}}

Words: For each gallon of paint, a painter can paint 250 ft$^2$ of wall.

Table: Note that this table is not complete (e.g. what about n=1.5?), but we can infer what f(1.5) is.

{| class="wikitable"
! n
! 0
! 1
! 2
! 3
! 4
! 5
! 6
|-
| f(n)
| 0
| 250
| 500
| 750
| 1000
| 1250
| 1500
|}

Ordered Pairs: {(0,0), (1, 250), (2,500), (3, 750), (4, 1000), (5, 1250), (6, 1500)}

Equation/Formula : <math> f(n) = 250n</math>

Graph:
[[File:Paintinggraph.png|thumb]]

Remind students how to graph functions (the inputs go along the x-axis and the outputs go along the y-axis, etc.). Tell them that this is a fourth major way to represent functions.
{{Ex-end}}

[5 minutes] Have students do Problems 4-5.

====Review graphs and the vertical line test====

[5 minutes] Now, draw several graphs and ask how one would know whether or not it represents a function. Students will most likely be reminded of the Vertical Line Test, but it is important in lecture to connect the Vertical Line Test with the formal definition of a function.
Be sure to include a graph which is not a function.

{{Ex-begin}}
If a graph fails the vertical line test it cannot be the graph of a function.

[[File:VLT2.png|thumb]]

{{Ex-end}}

[10 minutes] Have students do Problems 6-9.

[5 minutes] Have students do the synthesis problem.

[10 minutes] Finish class by having groups of students present their answer to the Synthesis Problem using the document camera. Take a few minutes at the end to use the synthesis problem and try to tie everything together.

==Comments==
----
When giving examples of graphs to test the vertical line test on, consider giving a function that is not defined at every real number. Since we don't talk about domain as a part of the definition for a function, students may assume that piecewise functions with breaks can't be functions because they aren't defined at every real number. The vertical line test can give this impression.

If you are giving them time for the survey on this day be careful about how you divide up your time.

==Individual Lesson Plans==

[[Brummer 1.1 Function Plan]] - For coordinated course

Brummer 1.1 Function Plan

2020-07-07T16:32:07Z

Jbrummer: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Determine when a relationship is a function and determine its inputs and ou..."

1.1 Functions

2020-07-07T16:31:52Z

Jbrummer:

Recitation 30

2020-06-01T14:52:03Z

Jbrummer:

=Recitation 30 (Final Exam Review)=
==Objectives:==
[(i)]
* Students will be prepared for the final exam

\boxed{\parbox[c]{6.5in}{ Important Notes:

[(i)]
* Before class look at an old exam (not the current exam) pick out 5-6 problems that you believe are key problems that are exam level problems, not too hard, but also not too easy, and problems that many of your students will likely struggle with. Bring these problems to class (ideally put them on a handout for the students), and plan to go over solving these problems.
* Among these problems I strongly recommended problems like

* Definition of derivative
* Related Rates
* Optimization

[(i)]
*[ (75 minutes)] Alternate on each problem, give the students seven minutes to try the problem by themselves as if it was a test, then go over the problem as a class. The idea here is to help students to realize where they are going to struggle and what they need to study.

Recitation 29

2020-06-01T14:51:52Z

Jbrummer:

=Recitation 29 (Final Exam Review)=
==Objectives:==
[(i)]
* Students will be prepared for the Final exam

\boxed{\parbox[c]{6.5in}{ Important Notes:

[(i)]
* Make sure to look at the exam review section in the back of the course packet to see if there are any problems that are irrelevant given the pace of the course. Also check to make sure there are not problems blatantly missing based on the pace of the course.

[(i)]
*[ (75 minutes)] I find it best to tell students that for a review day if 80\% or more of the class comes with the review material completely done then I will go over the answers. This tends to help the students study with a purpose and then learn a lot from your act of going over everything in detail. I strongly recommend that you write the solution to each problem just as you would expect the solution to be written on the exam. This also helps students to see what a good solution will look like.

Recitation 28

2020-06-01T14:51:38Z

Jbrummer:

=Recitation 28 (Integration by Substitution)=
==Objectives:==
* Students will be able to integrate using substitution.
* Students will be able apply integration by substitution to applied problems.

==Lesson==
*[(5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[(25 minutes)] Have students work problem 1. Note that there are really two ways to handle definite integrals using substitution, you can either change the limits of integration or wait until you have back substituted.

*[(10 minutes)] Go over 3 of the integrals from problem 1 at the board.

*[(20 minutes)] Have students work problems 2 and 3.

*[(15 minutes)] Have students work problems 4 and 5. Wrap up class by going over problem 4 with the entire class.

Recitation 27

2020-06-01T14:51:25Z

Jbrummer:

=Recitation 27 (Exam 3 Review)=
==Objectives:==
[(i)]
* Students will be prepared for the third exam

\boxed{\parbox[c]{6.5in}{ Important Notes:

[(i)]
* Make sure to look at the exam review section in the back of the course packet to see if there are any problems that are irrelevant given the pace of the course. Also check to make sure there are not problems blatantly missing based on the pace of the course.

[(i)]
*[ (75 minutes)] I find it best to tell students that for a review day if 80\% or more of the class comes with the review material completely done then I will go over the answers. This tends to help the students study with a purpose and then learn a lot from your act of going over everything in detail. I strongly recommend that you write the solution to each problem just as you would expect the solution to be written on the exam. This also helps students to see what a good solution will look like.

Recitation 26

2020-06-01T14:51:13Z

Jbrummer:

=Recitation 26 (The Fundamental Theorem of Calculus Part 2)=
==Objectives:==

* Students will be able to use the FTC part 2.
* Students will be able to solve basic differential equations.
==Important Notes: ==

FTC Part 2 hard problems are really all about the chain rule.

==Lesson Plan==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (25 minutes)] Have students work problem 1 through 3.

*[ (25 minutes)] Have students work problems 4, 5, and 6. Many students will get stuck, you should plan a mini lecture explanation to get the class working well together.

\textbf{Example:}

Suppose we want to calculate \[\frac{d}{dx} nt_0^{x^2} \sin(t)dt.\] Let $f(x)= nt_0^x \sin(t)dt.$, then we are actually calculating \[\frac{d}{dx} f(x^2)\] which is $f'(x^2)2x$, therefore, \[\frac{d}{dx} nt_0^{x^2} \sin(t)dt=\sin(x^2)2x.\]

*[ (15 minutes)] Tell students that Problems 6-12 are good practice problems but there is no way there is sufficient time in a recitation to solve all the problems. Have students pick which two they want to solve today and tell them to complete the others in preparation for the coming exam.

*[ (5 minutes)] Solve one of problems 6 through 9 for the class.

Recitation 25

2020-06-01T14:51:00Z

Jbrummer:

=Recitation 25 (Antiderivatives)=
==Objectives:==

* Students will be able to use antiderivative analytically and numerically to evaluate integrals.
==Important Notes: ==

==Lesson Plan==
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (15 minutes)] Have students do problems 1-3. These problems are review of various items that we have already done in class. Notice that some of them are derivatives and some are antiderivatives.

*[ (15 minutes)] Have students work through problem 4. Most of these are relatively easy. However, 4.7 is difficult. Encourage students to expand the equation before trying to find the antiderivative. Note, the problem says an antiderivative to $+c$ is not actually needed. However, you should make a point of telling this to students.

*[ (15 minutes)] Have students work on problem 6.

*[ (15 minutes)] Have students work through problem 7 and 8.

*[ (10 minutes)] Come up with a short mini-lecture wrap up for the day that you can deliver in 10 minutes or less. For this time you should plan one example that covers the material they went over today in recitation.

Recitation 24

2020-06-01T14:50:42Z

Jbrummer:

=Recitation 24 (Graphical Antiderivatives)=
==Objectives:==
* Students will be able to graph an antiderivative given the graph of a function.
* Students will recognize that the graph of the antiderivative gives the value of the area under a function.
==Important Notes: ==
==Lesson Plan==
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (15 minutes)] Have students do problem 1. Again, some of the copies seem to have not been done properly. You should project an image on the board so that students can fill in values. Some students are going to try and find an equation for the line. Tell the students that the purpose is not to find an equation for the line, but to use geometry.

*[ (15 minutes)] Have students work through problem 2.

*[ (10 minutes)] Have students work on problem 3. Some students will get it right away. Other students are going to need you to explicitly show them the steps. Students will have a particularly hard time with part b when $F(0)=-2$.

*[ (15 minutes)] Have students work through problem 4. You should encourage them to graph the functions and then look at the shapes.

*[ (10 minutes)] Have students do problem 6.

*[ (5 minutes)] Go through problem 5 with the class, take your time and really show them what is going on. Many of your students will keep mixing $f(x)$ and $f(x)$. Make sure to really highlight what you are working with and what it means. I think a thorough explanation of this problem really will take 15 minutes.

Recitation 23

2020-06-01T14:50:32Z

Jbrummer:

=Recitation 23 (The Fundamental Theorem of Calculus)=
==Objectives:==
* Students will use the FTC, antiderivatives, and geometry to evaluate simple integrals.
==Important Notes:==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (20 minutes)] Have students do problem 1.

*[ (15 minutes)] Have students work through problem 2. It might help to have students draw a graph.

*[ (10 minutes)] Some students are going to first try to take a derivative and then find an antiderivative. This is not the point of the problem, help students realize that all they have to do on this problem is plug in the values.

*[ (15 minutes)] Have students work through problem 4. Make sure they first find an antiderivative, you will probably need to be explicit about this.

* Strong students should continue on and solve 5-8

*[ (10 minutes)] Have different groups present their answers to problem 1. Have students evaluate the reasonableness of student answers. Note: many groups struggle to differentiate velocity and distance traveled and so this discussion can be a rich discussion for the students. If you have any time remaining go over the answer to problem 4.

Recitation 22

2020-06-01T14:50:15Z

Jbrummer:

=Recitation 22 (The Definite Integral)=
==Objectives:==
* Students will be able to approximate a Riemann sum and identify if their sum is an over or under estimate.

==Important Notes: ==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (20 minutes)] Have students do problem 1. This is a long problem and the goal in part f is to get them to develop a midpoint or average rule.

*[ (10 minutes)] Have students work through problem 2. Students often struggle to see the shapes in a problem like this.

*[ (15 minutes)] Have students work through problem 3. Don't let students take antiderivatives!

*[ (15 minutes)] Have students work through problem 4. It looks like something involving the gridlines has not printed correctly. Make sure to reprint these off and bring them to recitation with you.

*[ (10 minutes)] Have students work through problem 5.

*[ (5 minutes)] Depending on how class is going you might want to either go over a problem or have students continue working on problems 6, 7, 8 and 9.

*Note: the solution for problem 9 is
abs(\sum\limits_{i=0}^{799} \frac{5}{800} ((2+5/800 i)^2-3)- \sum\limits_{i=1}^{800} \frac{5}{800} ((2+4/800 i)^2-3))=5/800(2^2-3-(7^2-3))=0.251

Recitation 21

2020-06-01T14:49:18Z

Jbrummer:

=Recitation 21 (Parametric Equations)=
==Objectives:==
* Students will be able to recognize parametric curves.
* Students will be able to calculate rates of change involving parametric curves.

==Important Notes:==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (30 minutes)] Have students work through problems 1-3.

*[ (10 minutes)] Have students work through problem 4..

*[ (10 minutes)] Have students work through problem 5. Students really struggle to think about the same problem in multiple ways so you should expect them to really struggle with part b.

*[ (5 minutes)] Have students work through problem 6..

*[ (10 minutes)] Have students work through problem 7. Make sure students know how to do this on their graphing calculator and also by plotting points.

*[ (10 minutes)] Have students work through problem 8. Many students are likely to feel completely lost. With a student who is lost you should take a step back and have them find the tangent line to something like $y=x^2$ and then ask them what information they would need to do something similar here.

* Strong students can continue working onto problems 9 and 10.

*[ (15 minutes)] Go over problem 8 through an interactive lecture.

Recitation 20

2020-06-01T14:49:07Z

Jbrummer:

=Recitation 20 (L'Hopital's Rule)=
==Objectives:==
* Students should get some practice recognizing an applying L'Hopitals rule.

==Important Notes: ==

==Lesson Plan==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (20 minutes)] Have students work through problem 1

*[ (15 minutes)] Have students work through problem 2. This is something that they have probably not seen before but is something from their homework.

*[ (15 minutes)] Have students work through problem 3. If you have never done a problem like this then you should make sure that you also know how to make sense of it by calculating slopes.

*[ (15 minutes)] Have students work through problems 4, 5, and 6.

*[ (10 minutes)] Go over two problems that you think are meaningful.

Recitation 19

2020-06-01T14:48:53Z

Jbrummer:

=Recitation 19 (Exam 2 Review)=
==Objectives:==

* Students will be prepared for the second exam

==Important Notes: ==

==Lesson Plan==

* Make sure to look at the exam review section in the back of the course packet to see if there are any problems that are irrelevant given the pace of the course. Also check to make sure there are not problems blatantly missing based on the pace of the course.

*[ (75 minutes)] I find it best to tell students that for a review day if 80\% or more of the class comes with the review material completely done then I will go over the answers. This tends to help the students study with a purpose and then learn a lot from your act of going over everything in detail. I strongly recommend that you write the solution to each problem just as you would expect the solution to be written on the exam. This also helps students to see what a good solution will look like.

Recitation 18

2020-06-01T14:48:35Z

Jbrummer:

=Recitation 18 (Related Rates)=
==Objectives:==

* Students will set-up and solve related rates problems.

==Important Notes: ==
==Lesson Plan==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (60 minutes)] Have students work through the packet, these are challenging problems and require them to set-up models. There is not really anything to say here other then help the students where they need help but do not give away too much. Tell students that setting up the problem is the skill they need to work on, the calculus is easy. Just a warning, problem 6 should really be the speed of the tip of the shadow with reference to the pole, it is easy to make the mistake of measuring the speed relative to the person.

As a note, problem one may seem like it has nothing to do with related rates. However, implicit differentiation is the backbone of related rates problems. The purpose of problem 1 is to re-familiarize students with implicit differentiation.

*[ (10 minutes)] Go over two problems that you think are meaningful.

Recitation 8

2020-06-01T14:47:54Z

Jbrummer:

=Recitation 8 (The Power Rule)=
==Objectives:==

* Students will be able to apply the power rule.
* Student will know when it is appropriate to apply the power rule.
* Student will be able to use the power rule in applications.

== Important Notes: ==

==Lesson Plan==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (10 minutes)] Have students do Problem 1. If students have problems then feel free to reiterate the power rule.

*[ (15 minutes)] Have students do Problem 2 and 3.

*[ (15 minutes)] Have students do Problem 4. You should anticipate many students having no idea how to even start this problem. Give them hint but try to get them to realize how they go about solving the problem.

*[ (5 minutes)] Have students do Problem 5. The students have not been introduced to antiderivatives yet. We are not teaching rules of integration but instead we want student to try and think about the power rule and think backwards.

*[ (10 minutes)] Have students do Problem 6. You should anticipate many students having no idea how to even start this problem. Give them hint but try to get them to realize how they go about solving the problem.

*[ (10 minutes)] Have students do Problem 7. This is an application that we would expect students to be able to do on the exam. Please feel free to tell the students that they should be able to do this if it comes up.

*[ (5 minutes)] Take 5 minutes at the end of c lass to summarize everything that students did with the power rule. Students might be tempted to say they learned the power rule and they are done. Enforce in students minds the fact that the4y are not ready unless they can apply the power rule in any an all of the situations in which it might come up.

Recitation 9

2020-06-01T14:47:44Z

Jbrummer:

=Recitation 9 (Exam 1 Review)=
==Objectives:==

* Students will be prepared for the first exam

==Important Notes:==

==Lesson Plans==

* Make sure to look at the exam review section in the back of the course packet to see if there are any problems that are irrelevant given the pace of the course. Also check to make sure there are not problems blatantly missing based on the pace of the course.

*[ (75 minutes)] I find it best to tell students that for a review day if 80\% or more of the class comes with the review material completely done then I will go over the answers. This tends to help the students study with a purpose and then learn a lot from your act of going over everything in detail. I strongly recommend that you write the solution to each problem just as you would expect the solution to be written on the exam. This also helps students to see what a good solution will look like.

Recitation 10

2020-06-01T14:47:23Z

Jbrummer:

=Recitation 10 (Exponentials and Products)=
==Objectives:==
[(i)]
* Students will be able to apply the product rule and find the derivative of exponential functions.
* Students will know when to apply what rule.

\boxed{\parbox[c]{6.5in}{ Important Notes:

\textbf{Note:} In the typical flow of a course this recitation takes place after an exam. Students may want to talk about the difficulty of the exam. This is a reasonable thing to let them talk about. However, don't let this take up more then five minutes of class time.

[(i)]
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (10 minutes)] Have students do Problem 1. Remind students that this is preparation for the gateway exam and we exact students to be able to push through problems like this with relative ease. Some students may say they have not learned what to do etc. Tell these students that this is something that takes practice and they need to make sure to practice this many many times.

*[ (10 minutes)] Have students do Problem 2 and 3. You may need to remind students about point slope form.

*[ (10 minutes)] Have students do problem 4. Note: we are approximating a year using the derivative, this is admittedly a strange approximation but this is an approximation that the book makes heavy use of.

*[ (5 minutes)] Have students do Problem 5.

\textbf{Have students move onto the section on the Product and Quotient Rule}

*[ (5 minutes)] Have students do Problem 1. Note that problem 1c may not be best solved by the quotient rule and we are not using the chain rule yet so 1b should be solved using the product rule. Beware that many students will try to solve 1g using the quotient rule (afterall $5/\sqrt[3]{x}$ is a quotient right?).

*[ (10 minutes)] Have students do Problem 2 and 3.

*[ (10 minutes)] Have students do Problem 4, note this is really about differentiability implies continuity not the product or quotient rules.

*[ (5 minutes)] If you have some strong students have them work on problem 6 and 7, otherwise skip 6 and 7. Use your discretion to pick problems from 8, 9, and 10 to finish recitation.

*[ (5 minutes)] Take 5 minutes at the end of c lass to summarize everything that students did with the product and quotient rule.

Recitation 11

2020-06-01T14:46:52Z

Jbrummer:

=Recitation 11 (The Chain Rule)=
==Objectives:==
[(i)]
* Students will be able to apply the chain rule and practice the other rules they have learned.
* Students will know when to apply what rule.

\boxed{\parbox[c]{6.5in}{ Important Notes:

[(i)]
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (15 minutes)] Have students do Problem 1. Students have done a significant number of problems like this in class and should be able to breeze through these. If a student is really stuck you can refer them to their recitation problems from exponentials where they did very similar problems. At this point in the semester a student who is really lost on these problems may be someone in a lot of trouble that you may want to schedule an office hours meeting with. If this is the case you need to be very sensitive to their feelings while at the same time accounting for their need for intervention.

*[ (15 minutes)] Have students do Problem 2 and 3. Problem 2 should be easy except for the fact that it is an applied problem. Problem 3 could be a little more difficult. When most groups are done have a group present their answer to p[arts a and b of problem 3.

*[ (15 minutes)] Have students do problem 4. This is purely practice for the gateway exam. Have each group write the solution to one of the parts on the board. Tell the students to all make sure they have the same answer and are in agreement with what is on the board.

*[ (15 minutes)] Have students do Problem 5. It is important to note that they may not have seen this notation for a second derivative and you should let them know what the notation means. Also note we are looking for a general form here, this may confuse some students.

*[ (10 minutes)] Take the last 10 minutes to write down all the standard rules of differentiation for the students on the board. Tell the students that these are absolutely necessary in order to succeed in the class. Try and make this interactive if possible. The rules so far are, Power rule, Product rule, Quotient rule, Chain rule, Trig rules (Sine, Cosine, Tangent). The have also done Natural Logarithm, ArcTan and ArcSin in class.

Recitation 12

2020-06-01T14:46:38Z

Jbrummer:

=Recitation 12 (Derivatives and Implicit Differentiation)=
==Objectives:==
[(i)]
* Students will be be able to take implicit derivatives.
* Introduce logarithmic differentiation.
* Reintroduce hyperbolic trigonometry.

\boxed{\parbox[c]{6.5in}{ Important Notes:

[(i)]
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (10 minutes)] Have students do Problem 1. Problem 1 is hard in the sense that it is the first word problem they have encountered in a while.

*[ (10 minutes)] Have students do Problem 2. Likely the student have already forgotten the arc functions. However, they should be in the students notes two pages earlier. Have students refer to their notes. If a student has nothing written down in their notes then they likely are not attending class. You should point out that class counts toward their grade and is the easiest points they can earn all semester.

*[ (10 minutes)] Have students do problem 3.

*[ (20 minutes)] Have students do Problem 4 and 5. This is actually a hard problem, make sure you have worked it out before class.

*[ (10 minutes)] Have students do Problem 6, this is called logarithmic differentiation. If you have never seen this before no problem, but you might want to work out the details before you get to recitation.

*[ (10 minutes)] Take the last 10 minutes to go over problems 7-9, Hyperbolic trig is something the students have likely never seen and some instructors will skip altogether, I recommend doing 7-9 at the bard with the class. Make this interactive, but lead the class over the topics.

Recitation 13

2020-06-01T14:46:14Z

Jbrummer:

=Recitation 13 (Gateway Review)=
==Objectives:==
[(i)]
* At some point in the Semester your will be expected to do the gateway exam and gateway review. This chapter is all about the gateway.

\boxed{\parbox[c]{6.5in}{ Important Notes:

[(i)]
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (30 minutes)] Have students work together to go over as many of the gateway problems as they can for 30 minutes. Be sure to stop in at various groups and answer questions.

*[ (40 minutes)] Have individuals take the gateway exam. No calculators are allowed.

Recitation 14

2020-06-01T14:45:04Z

Jbrummer: /* Recitation 14 */

=Recitation 14 (Linear Approximation)=
==Objectives:==

* Students will be able to write a linear approximation to a function as a tangent line.
* Introduce logarithmic differentiation.
* Reintroduce hyperbolic trigonometry.

==Important Notes: ==
*This is a long lesson and you will probably not make it through the entire packet in this class.

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (5 minutes)] Have students work together on problem 2. Students should have seen very similar problems in lecture and should be able to do the problem without a lot of struggle.

*[ (5 minutes)] Have students work together on problems 3 and 4. These problems are basically asking for exactly the same information. We want students to recognize either way of asking this question.

*[ (10 minutes)] Have students work together on problems 5 and 6. You will almost certainly need to refer students to their notes on the MVT and point out what the hypothesis and conclusions are.

*[ (10 minutes)] Have students work together on problem 7. Students usually struggle to see exactly how to apply the MVT, you should expect to have to ask them leading questions.

*[ (10 minutes)] Have students work together on problem 8. Make sure that students test all aspects of the racetrack principle to solve this problem.

*[ (15 minutes)] Depending on how things go you might need to prepare a couple of examples to do at the end of class to wrap things up and tie it all together. Prepare examples about linear approximation instead of the theorems as the theorems are something they need to practice with on their own.

Calculus I

2020-06-01T14:44:28Z

Jbrummer:

===Recitations===
====Chapter 1====
[[Recitation 1]]

[[Recitation 2]]

[[Recitation 3]]

[[Recitation 4]]

====Chapter 2====
[[Recitation 5]]

[[Recitation 6]]

[[Recitation 7]]

====Chapter 3====
[[Recitation 8]]

[[Recitation 9]]

[[Recitation 10]]

[[Recitation 11]]

[[Recitation 12]]

[[Recitation 13]]

[[Recitation 14]]

====Chapter 4====
[[Recitation 15]] The First and Second Derivative

[[Recitation 16]] Global Extrema

[[Recitation 17]] Global Extrema Part II

[[Recitation 18]] Related Rates

[[Recitation 19]] Exam 2 Review

[[Recitation 20]] L'Hopital's Rule

[[Recitation 21]] Parametric Equations

====Chapter 5====
[[Recitation 22]]

[[Recitation 23]]

[[Recitation 24]]

====Chapter 6====
[[Recitation 25]]

[[Recitation 26]]

[[Recitation 27]]

[[Recitation 28]]

[[Recitation 29]]

[[Recitation 30]]

Calculus I

2020-06-01T14:41:47Z

Jbrummer:

===Recitations===
====Chapter 0====
[[Recitation 1]]

[[Recitation 2]]

[[Recitation 3]]

[[Recitation 4]]

====Chapter 1====
[[Recitation 5]]

[[Recitation 6]]

[[Recitation 7]]

====Chapter 2====
[[Recitation 8]]

[[Recitation 9]]

[[Recitation 10]]

[[Recitation 11]]

[[Recitation 12]]

[[Recitation 13]]

[[Recitation 14]]

====Chapter 3====
[[Recitation 15]] The First and Second Derivative

[[Recitation 16]] Global Extrema

[[Recitation 17]] Global Extrema Part II

[[Recitation 18]] Related Rates

[[Recitation 19]] Exam 2 Review

[[Recitation 20]] L'Hopital's Rule

[[Recitation 21]] Parametric Equations

====Chapter 4====
[[Recitation 22]]

[[Recitation 23]]

[[Recitation 24]]

====Chapter 5====
[[Recitation 25]]

[[Recitation 26]]

[[Recitation 27]]

[[Recitation 28]]

[[Recitation 29]]

[[Recitation 30]]

Recitation 11

2020-06-01T14:31:02Z

Jbrummer:

=Recitation 11=
==Objectives:==
[(i)]
* Students will be able to apply the chain rule and practice the other rules they have learned.
* Students will know when to apply what rule.

\boxed{\parbox[c]{6.5in}{ Important Notes:

[(i)]
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (15 minutes)] Have students do Problem 1. Students have done a significant number of problems like this in class and should be able to breeze through these. If a student is really stuck you can refer them to their recitation problems from exponentials where they did very similar problems. At this point in the semester a student who is really lost on these problems may be someone in a lot of trouble that you may want to schedule an office hours meeting with. If this is the case you need to be very sensitive to their feelings while at the same time accounting for their need for intervention.

*[ (15 minutes)] Have students do Problem 2 and 3. Problem 2 should be easy except for the fact that it is an applied problem. Problem 3 could be a little more difficult. When most groups are done have a group present their answer to p[arts a and b of problem 3.

*[ (15 minutes)] Have students do problem 4. This is purely practice for the gateway exam. Have each group write the solution to one of the parts on the board. Tell the students to all make sure they have the same answer and are in agreement with what is on the board.

*[ (15 minutes)] Have students do Problem 5. It is important to note that they may not have seen this notation for a second derivative and you should let them know what the notation means. Also note we are looking for a general form here, this may confuse some students.

*[ (10 minutes)] Take the last 10 minutes to write down all the standard rules of differentiation for the students on the board. Tell the students that these are absolutely necessary in order to succeed in the class. Try and make this interactive if possible. The rules so far are, Power rule, Product rule, Quotient rule, Chain rule, Trig rules (Sine, Cosine, Tangent). The have also done Natural Logarithm, ArcTan and ArcSin in class.

Recitation 10

2020-06-01T14:30:56Z

Jbrummer:

=Recitation 10=
==Objectives:==
[(i)]
* Students will be able to apply the product rule and find the derivative of exponential functions.
* Students will know when to apply what rule.

\boxed{\parbox[c]{6.5in}{ Important Notes:

\textbf{Note:} In the typical flow of a course this recitation takes place after an exam. Students may want to talk about the difficulty of the exam. This is a reasonable thing to let them talk about. However, don't let this take up more then five minutes of class time.

[(i)]
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (10 minutes)] Have students do Problem 1. Remind students that this is preparation for the gateway exam and we exact students to be able to push through problems like this with relative ease. Some students may say they have not learned what to do etc. Tell these students that this is something that takes practice and they need to make sure to practice this many many times.

*[ (10 minutes)] Have students do Problem 2 and 3. You may need to remind students about point slope form.

*[ (10 minutes)] Have students do problem 4. Note: we are approximating a year using the derivative, this is admittedly a strange approximation but this is an approximation that the book makes heavy use of.

*[ (5 minutes)] Have students do Problem 5.

\textbf{Have students move onto the section on the Product and Quotient Rule}

*[ (5 minutes)] Have students do Problem 1. Note that problem 1c may not be best solved by the quotient rule and we are not using the chain rule yet so 1b should be solved using the product rule. Beware that many students will try to solve 1g using the quotient rule (afterall $5/\sqrt[3]{x}$ is a quotient right?).

*[ (10 minutes)] Have students do Problem 2 and 3.

*[ (10 minutes)] Have students do Problem 4, note this is really about differentiability implies continuity not the product or quotient rules.

*[ (5 minutes)] If you have some strong students have them work on problem 6 and 7, otherwise skip 6 and 7. Use your discretion to pick problems from 8, 9, and 10 to finish recitation.

*[ (5 minutes)] Take 5 minutes at the end of c lass to summarize everything that students did with the product and quotient rule.

Recitation 14

2020-06-01T14:29:46Z

Jbrummer: Created page with " =Recitation 14= ==Objectives:== * Students will be able to write a linear approximation to a function as a tangent line. * Introduce logarithmic differentiation. * Reintrod..."

=Recitation 14=
==Objectives:==

* Students will be able to write a linear approximation to a function as a tangent line.
* Introduce logarithmic differentiation.
* Reintroduce hyperbolic trigonometry.

==Important Notes: ==
*This is a long lesson and you will probably not make it through the entire packet in this class.

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (5 minutes)] Have students work together on problem 2. Students should have seen very similar problems in lecture and should be able to do the problem without a lot of struggle.

*[ (5 minutes)] Have students work together on problems 3 and 4. These problems are basically asking for exactly the same information. We want students to recognize either way of asking this question.

*[ (10 minutes)] Have students work together on problems 5 and 6. You will almost certainly need to refer students to their notes on the MVT and point out what the hypothesis and conclusions are.

*[ (10 minutes)] Have students work together on problem 7. Students usually struggle to see exactly how to apply the MVT, you should expect to have to ask them leading questions.

*[ (10 minutes)] Have students work together on problem 8. Make sure that students test all aspects of the racetrack principle to solve this problem.

*[ (15 minutes)] Depending on how things go you might need to prepare a couple of examples to do at the end of class to wrap things up and tie it all together. Prepare examples about linear approximation instead of the theorems as the theorems are something they need to practice with on their own.

Recitation 13

2020-06-01T14:29:41Z

Jbrummer: Created page with " =Recitation 13} ==Objectives:== [(i)] * At some point in the Semester your will be expected to do the gateway exam and gateway review. This chapter is all about the gatewa..."

=Recitation 13}
==Objectives:==
[(i)]
* At some point in the Semester your will be expected to do the gateway exam and gateway review. This chapter is all about the gateway.

\boxed{\parbox[c]{6.5in}{ Important Notes:

[(i)]
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (30 minutes)] Have students work together to go over as many of the gateway problems as they can for 30 minutes. Be sure to stop in at various groups and answer questions.

*[ (40 minutes)] Have individuals take the gateway exam. No calculators are allowed.

Recitation 12

2020-06-01T14:29:33Z

Jbrummer: Created page with " =Recitation 12} ==Objectives:== [(i)] * Students will be be able to take implicit derivatives. * Introduce logarithmic differentiation. * Reintroduce hyperbolic trigonomet..."

=Recitation 12}
==Objectives:==
[(i)]
* Students will be be able to take implicit derivatives.
* Introduce logarithmic differentiation.
* Reintroduce hyperbolic trigonometry.

\boxed{\parbox[c]{6.5in}{ Important Notes:

[(i)]
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (10 minutes)] Have students do Problem 1. Problem 1 is hard in the sense that it is the first word problem they have encountered in a while.

*[ (10 minutes)] Have students do Problem 2. Likely the student have already forgotten the arc functions. However, they should be in the students notes two pages earlier. Have students refer to their notes. If a student has nothing written down in their notes then they likely are not attending class. You should point out that class counts toward their grade and is the easiest points they can earn all semester.

*[ (10 minutes)] Have students do problem 3.

*[ (20 minutes)] Have students do Problem 4 and 5. This is actually a hard problem, make sure you have worked it out before class.

*[ (10 minutes)] Have students do Problem 6, this is called logarithmic differentiation. If you have never seen this before no problem, but you might want to work out the details before you get to recitation.

*[ (10 minutes)] Take the last 10 minutes to go over problems 7-9, Hyperbolic trig is something the students have likely never seen and some instructors will skip altogether, I recommend doing 7-9 at the bard with the class. Make this interactive, but lead the class over the topics.

Recitation 11

2020-06-01T14:28:44Z

Jbrummer: Created page with " =Recitation 11} ==Objectives:== [(i)] * Students will be able to apply the chain rule and practice the other rules they have learned. * Students will know when to apply wha..."

=Recitation 11}
==Objectives:==
[(i)]
* Students will be able to apply the chain rule and practice the other rules they have learned.
* Students will know when to apply what rule.

\boxed{\parbox[c]{6.5in}{ Important Notes:

[(i)]
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (15 minutes)] Have students do Problem 1. Students have done a significant number of problems like this in class and should be able to breeze through these. If a student is really stuck you can refer them to their recitation problems from exponentials where they did very similar problems. At this point in the semester a student who is really lost on these problems may be someone in a lot of trouble that you may want to schedule an office hours meeting with. If this is the case you need to be very sensitive to their feelings while at the same time accounting for their need for intervention.

*[ (15 minutes)] Have students do Problem 2 and 3. Problem 2 should be easy except for the fact that it is an applied problem. Problem 3 could be a little more difficult. When most groups are done have a group present their answer to p[arts a and b of problem 3.

*[ (15 minutes)] Have students do problem 4. This is purely practice for the gateway exam. Have each group write the solution to one of the parts on the board. Tell the students to all make sure they have the same answer and are in agreement with what is on the board.

*[ (15 minutes)] Have students do Problem 5. It is important to note that they may not have seen this notation for a second derivative and you should let them know what the notation means. Also note we are looking for a general form here, this may confuse some students.

*[ (10 minutes)] Take the last 10 minutes to write down all the standard rules of differentiation for the students on the board. Tell the students that these are absolutely necessary in order to succeed in the class. Try and make this interactive if possible. The rules so far are, Power rule, Product rule, Quotient rule, Chain rule, Trig rules (Sine, Cosine, Tangent). The have also done Natural Logarithm, ArcTan and ArcSin in class.

Recitation 10

2020-06-01T14:28:38Z

Jbrummer: Created page with " =Recitation 10} ==Objectives:== [(i)] * Students will be able to apply the product rule and find the derivative of exponential functions. * Students will know when to apply..."

=Recitation 10}
==Objectives:==
[(i)]
* Students will be able to apply the product rule and find the derivative of exponential functions.
* Students will know when to apply what rule.

\boxed{\parbox[c]{6.5in}{ Important Notes:

\textbf{Note:} In the typical flow of a course this recitation takes place after an exam. Students may want to talk about the difficulty of the exam. This is a reasonable thing to let them talk about. However, don't let this take up more then five minutes of class time.

[(i)]
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (10 minutes)] Have students do Problem 1. Remind students that this is preparation for the gateway exam and we exact students to be able to push through problems like this with relative ease. Some students may say they have not learned what to do etc. Tell these students that this is something that takes practice and they need to make sure to practice this many many times.

*[ (10 minutes)] Have students do Problem 2 and 3. You may need to remind students about point slope form.

*[ (10 minutes)] Have students do problem 4. Note: we are approximating a year using the derivative, this is admittedly a strange approximation but this is an approximation that the book makes heavy use of.

*[ (5 minutes)] Have students do Problem 5.

\textbf{Have students move onto the section on the Product and Quotient Rule}

*[ (5 minutes)] Have students do Problem 1. Note that problem 1c may not be best solved by the quotient rule and we are not using the chain rule yet so 1b should be solved using the product rule. Beware that many students will try to solve 1g using the quotient rule (afterall $5/\sqrt[3]{x}$ is a quotient right?).

*[ (10 minutes)] Have students do Problem 2 and 3.

*[ (10 minutes)] Have students do Problem 4, note this is really about differentiability implies continuity not the product or quotient rules.

*[ (5 minutes)] If you have some strong students have them work on problem 6 and 7, otherwise skip 6 and 7. Use your discretion to pick problems from 8, 9, and 10 to finish recitation.

*[ (5 minutes)] Take 5 minutes at the end of c lass to summarize everything that students did with the product and quotient rule.

Recitation 9

2020-06-01T14:28:25Z

Jbrummer: Created page with " =Recitation 9= ==Objectives:== * Students will be prepared for the first exam ==Important Notes:== ==Lesson Plans== * Make sure to look at the exam review section in..."

=Recitation 9=
==Objectives:==

* Students will be prepared for the first exam

==Important Notes:==

==Lesson Plans==

* Make sure to look at the exam review section in the back of the course packet to see if there are any problems that are irrelevant given the pace of the course. Also check to make sure there are not problems blatantly missing based on the pace of the course.

*[ (75 minutes)] I find it best to tell students that for a review day if 80\% or more of the class comes with the review material completely done then I will go over the answers. This tends to help the students study with a purpose and then learn a lot from your act of going over everything in detail. I strongly recommend that you write the solution to each problem just as you would expect the solution to be written on the exam. This also helps students to see what a good solution will look like.

Recitation 8

2020-06-01T14:28:19Z

Jbrummer: Created page with " =Recitation 8= ==Objectives:== * Students will be able to apply the power rule. * Student will know when it is appropriate to apply the power rule. * Student will be able t..."

=Recitation 8=
==Objectives:==

* Students will be able to apply the power rule.
* Student will know when it is appropriate to apply the power rule.
* Student will be able to use the power rule in applications.

== Important Notes: ==

==Lesson Plan==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (10 minutes)] Have students do Problem 1. If students have problems then feel free to reiterate the power rule.

*[ (15 minutes)] Have students do Problem 2 and 3.

*[ (15 minutes)] Have students do Problem 4. You should anticipate many students having no idea how to even start this problem. Give them hint but try to get them to realize how they go about solving the problem.

*[ (5 minutes)] Have students do Problem 5. The students have not been introduced to antiderivatives yet. We are not teaching rules of integration but instead we want student to try and think about the power rule and think backwards.

*[ (10 minutes)] Have students do Problem 6. You should anticipate many students having no idea how to even start this problem. Give them hint but try to get them to realize how they go about solving the problem.

*[ (10 minutes)] Have students do Problem 7. This is an application that we would expect students to be able to do on the exam. Please feel free to tell the students that they should be able to do this if it comes up.

*[ (5 minutes)] Take 5 minutes at the end of c lass to summarize everything that students did with the power rule. Students might be tempted to say they learned the power rule and they are done. Enforce in students minds the fact that the4y are not ready unless they can apply the power rule in any an all of the situations in which it might come up.

Recitation 7

2020-06-01T14:28:05Z

Jbrummer: Created page with " =Recitation 7 (Interpreting the Derivative)= ==Objectives:== * Students will be able to interpret the derivative of a function in real-world terms using units. ==Importa..."

=Recitation 7 (Interpreting the Derivative)=
==Objectives:==

* Students will be able to interpret the derivative of a function in real-world terms using units.

==Important Notes: ==

==Lesson Plan==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (10 minutes)] Have students do Problem 1. Units are really important here and you should emphasize to students that units are a powerful tool for making sense of the derivative. The Table method may be something you are unfamiliar with. To use the table method you want to form a table of $\frac{f(x_h)-f(x)}{h}$ values for smaller and smaller $h$.

*[ (15 minutes)] Have students do problems 2-5, Reiterate that students will need to have the table of increasing, decreasing etc. committed to memory. However, they don't have to memorize the table, they just need to be able to recreate the ideas in the table.

*[ (5 minutes)] Go over problem 5 with your entire recitation.

*[ (10 minutes)] Have students do problem 6-8.

*[ (15 minutes)] Have students do problem 9. This is a hard problem, and not all parts are always possible on an unrestricted domain. Work with students and encourage students to share their answers and double check their neighbors answers.

*[ (10 minutes)] Have students do problem 10.

*[ (10 minutes)] Use remaining time to have students do problem 11 and then share their answers with the rest of the class.

Recitation 6

2020-06-01T14:27:58Z

Jbrummer: Created page with " =Recitation 6 (The Derivative Function)= ==Objectives:== * Students should be able to interpret and estimate instantaneous rate of change as the tangent line using a graph..."

=Recitation 6 (The Derivative Function)=
==Objectives:==

* Students should be able to interpret and estimate instantaneous rate of change as the tangent line using a graph.
* Students will be able to use the limit definition of a derivative.

==Important Notes: ==

*(5 minutes) Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*(10 minutes) Have students do Problem 1. You may need to do one or two points on a document camera to help the students get started.

*(10 minutes) Have students do Problem 2 and 3. Don't get caught up in differentiability at this point. The idea is to focus on what the derivative function looks like where it is defined. Particularly on problem 3 we expect students to finish with a piecewise function.

*(10 minutes) Have students do problem 4. Make sure they do all the calculations including expanding the cubic. These problems should help them realize the utility of the rules they are learning and prepare them to solve a problem just like this on the exam. In fact, you can tell the students that nearly every exam for the past several years has had a problem just like this on it.

*(15 minutes) Have students do Problem 5 and then present their methods for solving problem 5.

*(10 minutes) Have students do problem 6. Make sure they simplify before taking their derivatives. Nobody should be using the product or quotient rule at this point.

*(10 minutes) Have students do problem 7 and 8.

*(10 minutes) Have students do problem 9 and 10.

== Notes: ==
* If this is being paired with the derivative graph matching activity then skipping problems 3 and 9 saved some time and didn't seem to skip any important concepts.
* Students were very confused on Problem 6. If you work that problem then expect to work it together or at least discuss with the groups.

Recitation 4

2020-06-01T14:27:21Z

Jbrummer: Created page with " =Recitation 4 (Continuity)= ==Objectives:== ==Important Notes:== ==Lesson Guide== *(5 minutes)] Welcome the class to recitation. Ask students if they have filled in the not..."

=Recitation 4 (Continuity)=
==Objectives:==
==Important Notes:==
==Lesson Guide==
*(5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*(10 minutes)] Have students do Problem 1 don't let students get away with a simple one sentence answer. Make students really understand the claims they are making. We have set aside ten minutes for this, use the whole time.

*(10 minutes)] Have students do Problem 2 and make students draw their graphs on the board.

*(10 minutes)] Have students do Problems 3 and 4. For problem 4, make sure they really understand the definitions. Note: you should look in the book at the definitions before going to class. Don't worry about the formal definition of continuity, instead something like "you don't have to pick up your pencil" is actually reasonable here.

*(5 minutes)] Go over problem 5 with the students. Take your time and really help the students here.

*(5 minutes)] Have students do Problem 6.

*(10 minutes)] Have students do Problem 7. Make sure students really appeal to continuity, it is far more important that they understand what the intermediate value theorem says not that they can just rattle off a title.

*(20 minutes)] Have students do Problem 8. Then ask someone to come up to the board and draw a graph of each function.

*(10 minutes)] Guide students through problem 9.

*'''Note:''' If your lecturer is using quizzes you will need to cut out some material to fit in the quiz. You should plan to give 20 minutes for the quiz.

Recitation 3

2020-06-01T14:27:06Z

Jbrummer: Created page with " =Recitation 3 (Trigonometric Functions)= ==Objectives:== ==Important Notes:== * We have grouped problems 1 and 2 together. However, problem 2 is noticeably harder than p..."

=Recitation 3 (Trigonometric Functions)=
==Objectives:==
==Important Notes:==

* We have grouped problems 1 and 2 together. However, problem 2 is noticeably harder than problem 1.
* For problem 3 it is useful to have students work on a unit circle. If you can use both a unit circle plus bubble diagrams of a functions students seem to do well. Also note that 3.3 has no solution.

==Lesson Guide==
*(5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*(5 minutes)] Have students do Problems 1 and 2.

*(10 minutes)] Have students do Problem 3.

This problem is tricky but there is a way to do it that seems to make some sense to students. To solve something like problem 3.2 it seems to help to have students consider something like:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/jlujngan3g?embed
|width=800
|height=800
|border=0
}}

You then appeal to the unit circle to finish completing the problem.

*(10 minutes)] Have students do Problem 4.

*(15 minutes)] Go over problem 5 with the students. Take your time and really help the students here.

*(15 minutes)] Have students do Problem 6.

*(5 minutes)] Have multiple students show their solutions to problem 6 on the board.

*If time permits have students work on problem 7.

Recitation 2

2020-06-01T14:26:15Z

Jbrummer: Created page with " =Recitation 2= ==Objectives:== ==Important Notes:== * If students try to use something other than a exponential function on problem 6 ask them if there is a better func..."

=Recitation 2=
==Objectives:==

==Important Notes:==

* If students try to use something other than a exponential function on problem 6 ask them if there is a better function they could use. *If they insist that linear is a good function then allow them to use a linear function but ask them where their model will break down.

==Lesson Guide==

Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (15 minutes)] Have students do Problems 1 and 2.

*[ (10 minutes)] Ask three different student to write their ''solutions'' on the board.

*[ (25 minutes)] Have students do Problems 3-4.

*[ (25 minutes)] Have students do Problems 5-6.

Recitation 1

2020-06-01T14:26:02Z

Jbrummer: Created page with "=Recitation 1= ==Objectives:== * Students will be able to answer the questions "What is a function?" using correct notation to provide meaningful interpretation * Students w..."

=Recitation 1=
==Objectives:==

* Students will be able to answer the questions "What is a function?" using correct notation to provide meaningful interpretation
* Students will be able to correctly read and interpret graphs
* Students will be able to correctly apply laws of exponents/logarithms
* Make sure students understand the expectations of the CRA, how we expect them to prepare, and impact of the material on their learning

==Important Notes:==

* Problem 5: algebra with inequalities and absolute value
* Problem 6: they will have no idea where to start
* Problem 9: what piecewise functions are and how to evaluate them
* Problem 9.5: evaluating functions at expressions
* Problem 11: no one remembers log rules
* Problem 14: inverse trig functions and drawing triangles

==Rough Schedule==
*[ (10 minutes)] Introduce yourself, talk about something interesting you have done, discuss your expectations for the course, talk about group work.

*[ (5 minutes)] Have students do Problems 1-3.

*[ (5 minutes)] Ask three different student to write their \emph{solutions} on the board.

*[ (10 minutes)] Have students do Problems 11,14,5,6 .

*[ (5 minutes)] Have a student come to the board and present their answer to one of these

*[(40 minutes)] Give the CRA