Seeing Solutions: Using Slope Fields for Differential Equations

Texas Instruments Education
27 Feb 202460:45
EducationalLearning
32 Likes 10 Comments

TLDRIn this engaging calculus session, Steve and Tom explore differential equations and slope fields using Texas Instruments technology. They delve into various equations, demonstrating how to connect equations with corresponding slope fields and solve initial value problems. The session highlights the utility of technology in visualizing and understanding complex mathematical concepts, emphasizing the importance of these topics for both teachers and students.

Takeaways
  • πŸ“˜ The session focuses on differential equations and smoke fields, which are important topics for both teachers and students preparing for the AP Calculus exam.
  • πŸ” The TI-Inspire and TI-84 calculators are used to demonstrate the plotting of slope fields and solution curves for differential equations.
  • πŸ“Š Slope fields are visual representations that help in understanding the behavior of solutions to differential equations and can be plotted using technology.
  • 🎯 When creating a slope field, the expression for the slope of the differential equation is evaluated at various points in the plane to determine the direction of the solution curves.
  • πŸ“Œ It's important to note that while slope fields can be generated using technology, students are expected to understand and be able to sketch slope fields without calculators in the AP Calculus exam.
  • πŸ€” The video script includes a discussion on how to handle the constant of integration (C) when solving differential equations and how it may change form during the process.
  • πŸ“ˆ The script provides examples of solving differential equations by separating variables and integrating both sides, highlighting the steps and considerations involved in the process.
  • πŸ“ The session also addresses the concept of domain, emphasizing the need to consider the domain when working with functions and their graphs, especially in relation to the constant of integration.
  • πŸ’‘ The use of technology, such as Mathematica, is mentioned for enhancing the visualization of slope fields and solution curves, providing additional insights into the behavior of solutions.
  • 🌐 The session encourages participants to think critically about the relationships between differential equations, their slope fields, and the resulting solution curves.
  • πŸ“ The transcript ends with a discussion on the use of vector direction fields for applications such as golf, showing the practical applications of mathematical concepts.
Q & A
  • What is the main topic of discussion in the transcript?

    -The main topic of discussion in the transcript is differential equations and smoke fields, specifically focusing on how to use the TI Inspire and TI-84 calculators to plot and analyze them.

  • What is a slope field plot?

    -A slope field plot is a graphical representation of the slopes of a differential equation at various points in the coordinate plane. It is used to visualize the behavior of solutions to the differential equation.

  • How does the speaker describe the process of plotting a slope field on the TI Inspire?

    -The speaker describes the process of plotting a slope field on the TI Inspire by first opening a graph page, selecting 'graph entry edit' from the menu, choosing 'differential equations', and then entering the differential equation into the expression field. The slope field is generated by evaluating the slope expression at each point in the visible window.

  • What is the significance of the expression x^2 / 4 in the context of the slope field plot?

    -The expression x^2 / 4 represents the slope of the differential equation at any point in the coordinate plane. It is used as an example to demonstrate how the slope field plot can differ from the expected graph of the function, in this case, a parabola.

  • What is the purpose of the 'field resolution' setting in the TI Inspire?

    -The 'field resolution' setting controls the number of line segments or the fineness of the lattice of points plotted for the slope field. Increasing the field resolution results in a more detailed and smoother slope field plot.

  • How does the speaker address the use of slope field plots in the context of the AP Calculus exam?

    -The speaker mentions that slope field questions are common on the AP Calculus exam, usually in the free response portion. However, they note that while technology like the TI Inspire can be used for practice and to check the plausibility of solutions, the actual exam expects students to sketch the slope field without the aid of technology.

  • What is the differential equation given in the example where the speaker discusses the slope being zero along a line?

    -The differential equation given in the example is dy/dx = y - x/2.

  • What is the initial condition used in the problem where the speaker finds a C value of 8/3?

    -The initial condition used in that problem is y(1) = 4.

  • How does the speaker justify the absence of the slope field in quadrants 2 and 4 for a particular differential equation?

    -The speaker suggests that the absence of the slope field in quadrants 2 and 4 is due to the square root of the product of x and y, which would be undefined for negative values in those quadrants.

  • What is the final expression for y derived by the speaker for the differential equation dy/dx = x + 1 / (x*y)?

    -The final expression for y derived by the speaker is y = (3^(8/3) * x^(4/3)) / (1 + (3^(8/3) * x^(4/3))^2).

  • What does the speaker mention about the use of technology in solving differential equations?

    -The speaker mentions that while technology, such as the TI-84 and TI Inspire calculators, can be very helpful in visualizing and solving differential equations, students are expected to understand the concepts and be able to solve problems without technology during exams.

Outlines
00:00
πŸ“˜ Introduction to Differential Equations and Smoke Fields

The paragraph introduces the topic of differential equations and smoke fields, highlighting their significance in the context of AP Calculus exams. The discussion involves Steve Kokaska and Tom Dick, who are preparing to answer questions related to this topic. The introduction also touches on the use of Texas Instruments equipment for demonstrating the solution of differential equations, with a focus on the TI-Inspire calculator's capabilities.

05:01
πŸ“Š Exploring the TI-Inspire Calculator for Differential Equations

This section delves into the specifics of using the TI-Inspire calculator to plot and analyze differential equations. The conversation explains how to navigate the calculator's menu to access the differential equation plotter and how to input equations to generate slope fields. It also discusses the importance of understanding the relationship between the differential equation and the resulting slope field, emphasizing the educational value of visual representation in grasping the concept.

10:05
πŸ” Analyzing the Characteristics of Slope Fields

The paragraph focuses on the characteristics of slope fields and how they relate to the original differential equations. It discusses the expectations students might have when visualizing slope fields and the common surprises they encounter. The explanation includes a detailed look at how the sign and magnitude of the derivative affect the slope field's appearance, using the example of x^2/4 to illustrate the concept.

15:06
πŸŽ“ Addressing Exam Expectations and Technology Use

This part of the discussion addresses the expectations for slope field questions on exams, emphasizing that while technology can aid in understanding, students will need to sketch slope fields without calculator support during the exam. It also touches on the availability of slope field programs for the TI-84 calculator and the importance of practice in using this tool for both educational and exam purposes.

20:07
πŸ€” Reflecting on the Process of Matching Differential Equations to Slope Fields

The paragraph involves a step-by-step reflection on how to match given differential equations to their corresponding slope fields. It outlines strategies for analyzing the equations, such as considering constant values, examining the first quadrant, and identifying special cases within the equation. The discussion also includes an interactive element, with the presenter solving problems and engaging the audience with questions.

25:08
πŸ“š Solving Differential Equations and Discussing Constants

This section focuses on the process of solving differential equations, with an emphasis on the importance of correctly handling constants. The explanation involves separating variables, integrating both sides of the equation, and using initial conditions to find the specific solution. The paragraph also includes a practical demonstration of solving a differential equation and the considerations involved in isolating the variable y.

30:09
🌐 Examining the Relationship Between Equations and Slope Fields

The paragraph explores the relationship between the solutions of differential equations and their corresponding slope fields. It involves a detailed analysis of how the solutions fit within the slope fields, using specific examples to illustrate the points. The discussion also raises open questions for further exploration, encouraging engagement with the material and promoting a deeper understanding of the concepts.

35:10
πŸ“ˆ Investigating the Impact of Variables on Slope Fields

This part of the discussion investigates how changes in variables affect the slope fields. It involves a detailed examination of the effects of increasing or decreasing variables on the slope of the field, and how this relates to the original differential equations. The explanation includes visual aids and interactive elements to enhance understanding and retention of the material.

40:11
🀝 Collaborative Problem-Solving and Technology Integration

The paragraph showcases a collaborative approach to solving differential equations, with a focus on integrating technology into the process. It involves a back-and-forth exchange between the presenters as they work through problems, use technology to visualize solutions, and discuss the implications of their findings. The conversation also includes audience engagement, with questions and suggestions being incorporated into the discussion.

45:11
🌟 Final Thoughts and Open Questions

The conclusion of the discussion involves a wrap-up of the main points covered, along with a reflection on the use of technology in teaching and understanding differential equations. It also includes a mention of open questions raised during the session, encouraging further exploration and discussion. The paragraph ends with a note on the educational value of the content and the potential for future engagement.

Mindmap
Follow-up
Summary of Key Points
Audience Interaction
Chat Monitoring
Slope Field Analysis
Solving Differential Equations
Exam Preparation
Student Engagement
TI-84
TI-Inspire
Smoke Fields
Differential Equations
Allison Steel
Tom Dick
Steve Kokaska
Conclusion and Future
Interactive Elements
Problem Solving
Educational Strategies
Technology Demonstration
Topic Overview
Hosts and Participants
Monday Night Calculus: Differential Equations and Slope Fields
Alert
Keywords
πŸ’‘Differential Equations
Differential equations are mathematical equations that relate a function to its derivatives. In the context of the video, they are central to the discussion of slope fields and smoke fields, used to model various phenomena in calculus. The video explores how to solve these equations and how they can be represented graphically, particularly focusing on first-order differential equations that appear on the AP Calculus exam.
πŸ’‘Slope Field
A slope field is a graphical representation of the solutions to a differential equation. It consists of a grid of points, each with an arrow indicating the slope of the solution curve at that point. The slope field is integral to understanding the behavior of differential equations, as it visually communicates the rate of change and the direction of trajectories. In the video, the hosts demonstrate how to create and interpret slope fields using technology, such as the TI Inspire and TI-84 calculators.
πŸ’‘Texas Instruments Calculators
Texas Instruments is a renowned manufacturer of calculators and other technology products. In the video, the hosts use specific models of Texas Instruments calculators, namely the TI Inspire and TI-84, to demonstrate how to work with differential equations and slope fields. These calculators have built-in functionalities that allow for the plotting of slope fields and solution curves, which are essential tools for teaching and understanding calculus concepts.
πŸ’‘AP Calculus Exam
The AP Calculus Exam is a standardized test in the United States that assesses students' knowledge and skills in calculus. The video discusses the importance of differential equations and slope fields in the context of this exam, indicating that these topics often appear in the free-response section. The hosts emphasize the need for students to understand these concepts not only theoretically but also in terms of their graphical representation and application.
πŸ’‘Smoke Fields
Smoke fields, also known as vector fields, are a visual representation similar to slope fields but use vectors instead of arrows to represent the direction and magnitude of forces or rates of change at each point in space. While not explicitly discussed in the video, the concept is related to the graphical representation of differential equations. The video touches on the idea of using technology to enhance the graphical representation of these fields, which could include smoke fields.
πŸ’‘Logarithms
Logarithms are the inverse operation to exponentiation and are used to solve equations where the variable is an exponent. In the context of the video, logarithms are used to solve certain types of differential equations that involve products or quotients. The hosts demonstrate how to apply logarithms to isolate variables and solve for the function's explicit form.
πŸ’‘Integration
Integration is a fundamental operation in calculus that involves finding the function whose derivative is a given function. In the video, integration is used to find the anti-derivative of the right and left sides of a differential equation, which is a step towards solving the equation. The hosts discuss various integration techniques, such as substitution and term-by-term integration.
πŸ’‘Initial Value Problems
An initial value problem is a type of differential equation problem where the value of the function at a specific initial point is given. The video focuses on solving these problems by first finding the general solution to the differential equation and then using the initial condition to find the particular solution that satisfies the given condition.
πŸ’‘Technology in Education
The use of technology in education refers to the integration of various technological tools and devices to enhance teaching and learning experiences. In the video, the hosts leverage Texas Instruments calculators and software like Mathematica to visualize and solve differential equations, demonstrating how technology can aid in the understanding of complex mathematical concepts.
πŸ’‘Mathematica
Mathematica is a computational software program used in scientific, engineering, mathematical, and computing fields. It has capabilities for symbolic mathematics, numerical analysis, and graphics. In the video, Mathematica is used to create slope fields and visualize solution curves for differential equations, providing a powerful tool for illustrating and solving mathematical problems.
Highlights

Introduction to differential equations and smoke fields as a topic of interest for the AP Calculus exam.

Use of TI Inspire and TI-84 calculators for exploring differential equations and slope field plots.

Explanation of the slope field plot as a visual representation of the derivative of a function.

Demonstration of how to enter a differential equation into the calculator and generate a slope field plot.

Discussion on the importance of understanding the relationship between the differential equation and the corresponding slope field.

Illustration of how the slope field can help verify the solution of a differential equation.

Explanation of the process for solving separable differential equations and the inclusion of the constant term.

Use of technology to sketch slope fields and solution curves for differential equations.

Discussion on the limitations and capabilities of the calculator when it comes to graphing in different quadrants.

Application of initial value problems in differential equations and how to use them to find specific solutions.

Demonstration of how to use the initial condition to find the value of the constant in a differential equation.

Explanation of the relationship between the domain of a function and its representation on a slope field.

Discussion on the use of vector direction fields for visualizing multiple solution curves.

Explanation of how to interpret the behavior of a solution curve in relation to the slope field.

Demonstration of the process for solving a differential equation involving the log of an absolute value.

Discussion on the importance of understanding the domain and range of functions when working with differential equations.

Conclusion and wrap up of the key points discussed during the session, including the practical applications of slope fields in understanding differential equations.

Transcripts
00:11

well good evening and welcome to another

00:13

edition of Monday Night calculus my name

00:16

is Steve kokaska and I'm joined this

00:17

evening by my colleague Tom Dick from

00:19

Oregon State and in the background is

00:22

our Texas Instruments expert Allison

00:25

steel uh Curtis could not make it

00:27

tonight but Allison will be monitoring

00:30

than chat as usual and we may be able to

00:33

ask her a few questions I'm not sure um

00:36

tonight's topic has to do with

00:38

differential equations and smoke Fields

00:41

this is a really enjoyable topic for me

00:43

we always get lots of good questions

00:44

about this concerning this topic and we

00:47

usually see a question on differential

00:50

equations on the AP Calculus exam

00:52

usually on the free response portion so

00:55

I think this is a very important topic

00:56

for teachers and students um we've

00:59

actually had some good questions uh sent

01:02

to Curtis prior to tonight and we'll try

01:05

to get to answer all of those and Tom

01:07

will make sure uh that I don't miss any

01:09

of those so once again Allison will be

01:12

monitoring the chat if you have any

01:13

questions and if you have any answers to

01:15

our questions uh please do correspond

01:18

with her and we're going to start

01:20

tonight with a little technology Tom's

01:22

going to look at I think the ti Inspire

01:24

first is that

01:25

correct uh yes Steve I'll take a look at

01:28

both the SL fields on the uh the Inspire

01:31

and on the 84 so okay all yours

01:35

Tommy

01:36

okay all right I'm going to start out by

01:40

uh sharing a TI Inspire screen

01:44

okay and let's see if that's uh come up

01:48

there I think it's coming up

01:50

okay and all right uh let me uh let's do

01:55

a little skape there okay so what I've

01:57

done is uh open this uh uh TI inspired

02:01

document up in a in a graphs page and

02:05

this is uh probably the graphs uh

02:07

environment that most of us would use

02:10

probably 80 90% of the time would be the

02:12

function environment uh but what I'm

02:15

going to do is go over to the

02:18

menu and uh let's go down to three graph

02:22

entry edit that's going to have our uh

02:26

different kinds of graphs that we can

02:29

look at and you can see there we are on

02:31

number one is function and what I want

02:33

to do is scroll down all the

02:37

way to this thing that says oops I'm

02:41

going to hit up Arrow it should wrap

02:43

back around there we go def differential

02:46

equations that's the plotter uh but

02:49

people should keep in mind that that is

02:51

where you're going to find What's called

02:52

the the slope field plot and that's

02:55

something that's been in the course

02:57

description now for a couple of decades

02:59

but it's a uh it was pretty radical when

03:01

it first came into the the course yes it

03:04

was

03:05

was yeah so let me go ahead and hit that

03:08

and here we see a lot different kind of

03:11

Entry screen here uh for entering our

03:15

differential equation and it's kind of

03:17

spaced a little bit funny but you you

03:19

can see a y1 there but you see way over

03:22

here this Prime so this is actually the

03:25

derivative of y1 and now we can put it

03:28

in an expression

03:30

that's going to describe the

03:34

slope of

03:36

y1 at any point and so this expression

03:40

could involve both X and Y uh I'm going

03:43

to start out just by putting in an

03:46

expression with X so let me try x

03:57

squared and I'm going to divide that

03:59

that by uh four okay so I've got x^2 / 4

04:05

and uh down below the

04:09

expression you can see there is a space

04:12

for you to put in an initial condition

04:14

that would be a single point that you

04:17

would like to be make sure the solution

04:19

curve goes through I'm not going to make

04:22

use to that right now I'm just going to

04:23

make use of this and so when I um hit

04:28

graph on this

04:30

so we enter this we get what's called a

04:33

slope field now all a slope field is is

04:36

there's a a lattice or sampling of

04:39

points throughout the plane here in the

04:42

visible window and at each of the points

04:45

here it's used that expression to

04:48

evaluate a slope and then it plots a

04:52

little segment having that slope now

04:55

whenever I've introduced slope fields to

04:58

students for the first time I often use

05:00

an example like this something that they

05:02

could very

05:03

easily uh figure out what the original

05:06

y1 form would look

05:09

like uh but they're often very surprised

05:13

when they see this slope field because

05:16

when they think of x^2 over 4 the image

05:19

that comes to their mind immediately is

05:22

a parabola and so they're expecting a

05:25

parabola to fit the slope field but

05:29

that's not really the case and we can

05:31

think about it let's see x^2 over

05:33

4 uh that is its value is always going

05:37

to be positive except at zero it would

05:40

be zero but it will never have a

05:41

negative value that means all of the

05:44

little line segments we see in the slope

05:46

field should have a positive slope and

05:49

we can see when X is close to zero well

05:52

x^2 over 4 would be very small and so

05:55

we'll have a very small positive slope

05:59

as X gets larger in either the positive

06:02

direction or large magnitude negative

06:05

Direction x^2 over4 will be a larger

06:07

number and now we're seeing a steep

06:09

positive slope so this is a slope field

06:13

that's

06:14

fitting um those slope values that were

06:17

generated there now I'm going to go

06:19

ahead and uh uh go back to the entry

06:22

thing and and show you a setting that

06:25

you may want to take advantage of notice

06:29

here it's uh kind of defaulted back for

06:32

a second differential equation I'm going

06:34

to hit the up Arrow to get back to my

06:37

original

06:38

here and what I'm going to do is Arrow

06:43

over and let's see I'm hoping that's um

06:47

what I'm trying okay I got into this

06:50

position you see this dot dot dot to the

06:53

the right there that's where I want to

06:56

go and that dot dot dot is just a Shand

06:59

has there's a lot of settings in that

07:01

dot dot dot that we can take advantage

07:04

of so I'm going to hit

07:05

enter and I get this long scroll down

07:09

menu uh you can see there's a you can

07:11

choose like Oilers method for plotting

07:14

your curves which is is a different

07:16

topic that we're not going to touch on

07:17

tonight but let me just scroll down and

07:19

show you the one setting that I changed

07:23

very often when I'm working with the

07:25

Inspire and that's the field resolution

07:28

and the field resolution is just

07:31

controlling how many line segments or

07:34

how fine of lattice of points is plotted

07:37

for your slope field and I find 14 is a

07:40

little bit coarse it's a little bit

07:43

rough and so what I tend to do is I

07:46

usually just double that so instead of

07:49

14 I'm going to put in

07:52

[Music]

07:54

28 and then I'll say

07:57

okay and let's see let me

08:03

uh escape to get the

08:06

uh um entry line off of there and now

08:09

you're seeing a slope field for the

08:11

exact same differential equation but

08:14

it's much finer and also I think now you

08:16

can see like I said x^2 over 4 what

08:20

function would have that as a derivative

08:22

well it's going to be what a cubic

08:24

actually X cubed over 12 I think if I

08:26

did my arithmetic right but you can see

08:29

that a cubic would fit the slope field

08:31

very nicely in fact we can try that out

08:35

by going back to the

08:37

menu go back to graph entry

08:40

edit and now get the function on there

08:44

and without erasing the slope

08:46

field we could actually enter a function

08:49

that we think might fit the slope field

08:52

so I'm going to go ahead and see if

08:54

I if my uh calculation in my head was

08:58

right I'm going to put in

08:59

x

09:04

cubed and I'm going to divide

09:07

that by 12 whoops I think I might have

09:11

accidentally entered didn't I okay so

09:14

let

09:15

me get back

09:20

up sorry about that okay let me get back

09:23

in the entry I wanted to

09:26

divide by 12 mhm by the way when I look

09:30

at the what I did plot I can tell that's

09:33

not going to work plain old X cubed is

09:36

clearly not it's actually crossing a lot

09:40

of these line segments at an angle

09:42

that's just not fitting so let me amend

09:45

that put in X cubed over

09:48

12 and now I'm seeing a much better fit

09:52

to my SL field so that's just kind of

09:55

the basics of smoke field and and how

09:57

you can plot over it

09:59

uh there is also a uh this this is um a

10:04

functionality that's resident on the

10:07

Inspire does you don't have to have cast

10:09

or anything it's it's on all the

10:11

inspires uh however on the

10:14

ti84 this is the one spot where if

10:17

there's one program that you load onto

10:20

your 84 for calculus I think this would

10:22

be the one you'd want to program for

10:24

slope field and I don't know if Steve

10:27

mentioned this but slope field questions

10:30

uh on the

10:31

exam uh are almost always going to be on

10:36

the non-calculator part of the exam so

10:39

when those are assessed they're going to

10:42

for example expect you maybe to sketch a

10:44

slope field on your own but there in

10:47

that setting it wouldn't make sense for

10:49

people to have access to something that

10:51

would just generate the slope wheel for

10:53

you right uh nevertheless they're a

10:55

fantastic tool they're a great thing to

10:57

have technology generated but when

10:59

they're testing whether you understand

11:00

what a slope field is they're going to

11:03

do that on the non- calculated part of

11:04

the exam so I'm going to do a uh just a

11:08

quick stop share and then share back

11:11

again Steve if that's okay yeah sure and

11:14

switch over to the

11:18

ti84 and let's see if I'm able to bring

11:21

that up okay okay here we go there's the

11:25

ti84 and on the ti84 uh

11:30

there's not a a resident slope field

11:33

program uh but there is one that you can

11:35

download for free off of the ti

11:38

education.com

11:40

site um and we'll make sure people have

11:42

some information we'll communicate that

11:44

to Curtis and Allison so we can make

11:47

sure people know how to get that if I go

11:49

to my programs here this ti4 already has

11:53

one in

11:54

there and it's uh just named SLP FLD

11:59

short hand for slope field so that's the

12:02

program and the way this slope field

12:05

program works is I'm going to go to y

12:10

equals and it's going to expect to see

12:13

the differential equation for dydx

12:16

stored in

12:18

y1 now that means if you wanted to put

12:21

in something like an expression in terms

12:24

of X and Y let's do one like that say

12:28

negative

12:30

x

12:31

/

12:36

y that's going to look really peculiar

12:39

to you because you never see why in the

12:42

y equals menu as in there but keep in

12:45

mind that this is just an

12:47

expression and y1 if I have any

12:50

numerical value for x and y stored y1

12:53

will

12:54

evaluate using both of those values okay

12:59

uh so this is where we're storing our

13:01

differential equation so dydx is equal

13:04

tox over

13:07

Y

13:08

and if I now quit out of

13:11

there uh I can go to

13:16

programs and there's my slope field I

13:19

want to execute that

13:21

program

13:24

and let's see I've got a little bit

13:27

different window than a St standard one

13:29

but it's good enough okay so here's a

13:32

whatever window I had it noticed it's

13:34

doing a nice slope feel for me just like

13:36

on the Inspire but it was a program that

13:39

I had to load on there uh this might

13:43

look like a ripples of a you know like

13:45

throwing a stone into a a pond and

13:48

seeing the ripples come out um that's

13:52

actually not by accident at all if you

13:54

think about that Negative X over y

13:55

that's that's a derivative you may have

13:58

encountered when you doing implicit

14:01

differentiation if you do implicit

14:03

differentiation on a circle equation

14:05

like an X2 + y^2 = 9 or something like

14:09

that the implicit derivative turns out

14:11

to Bex over y so we can see the solution

14:14

Curves in this case are going to be

14:17

parts of circles so that's kind of cool

14:20

and again uh if you did want to plot a

14:24

function on top of

14:26

this you could just use use Y2 or Y3 or

14:30

any any of the other uh

14:34

equations y equals things notice the the

14:37

soap Field's already on there it's not

14:39

selected but I'll go ahead in Y2 I'm

14:43

going to put in

14:45

the square

14:48

root of I mentioned x^2 + y^2 = 9 well

14:53

let me solve that for y so that y would

14:55

be the square root of the quantity 9 -

14:58

x^2

15:06

and let me graph

15:09

that and boom there it is there's a nice

15:11

curve and it's following the fit of the

15:13

slope field so um again uh slope Fields

15:19

uh on the exam it's probably going to be

15:20

on the non- calculated part of the exam

15:23

but for practice uh and and for

15:26

especially for seeing whether the

15:28

solution to a differential equation is

15:31

uh plausible what I mean makes sense is

15:34

you could do a slope field for it and

15:35

then take your solution and plot it on

15:37

top of the slope field and see if it

15:39

fits if you see any uh discrepancies

15:42

that's a Surefire tip that something's

15:45

gone ay in your calculation somewhere

15:48

that you need to take a look at all

15:50

right I think I've wagged my tongue

15:52

enough Steve I'm G to turn back over to

15:54

you and so you can dive into your

15:56

questions so while

15:59

we're go ahead while we're doing that I

16:02

did just want to give a shout out we've

16:04

got quite a few joining us from Miss

16:06

Connelly's class as well as Mr uh the

16:08

famous Mr Crowley's class this evening

16:12

all right

16:13

great shout out to Mark hey Tom before

16:17

before you leave the screen yes sir one

16:19

quick question uh so what do you do in

16:23

order to sort of erase the slope

16:25

field is there some special command for

16:27

that so now all done with this and I

16:30

want to graph another function I'm going

16:32

on to another problem how do I sort of

16:35

erase this and start over yeah no no

16:37

good point uh actually you know any

16:41

change at all you would make to the

16:44

window

16:45

would get rid of the slope field so for

16:48

example if I was to just do a

16:51

zoom decimal here

16:55

yes I see okay I've got a fresh screen

16:58

it went ahead and still graphed my that

17:01

function I had entered into Y2 okay but

17:04

slope field for me to generate a new

17:06

slope field in a new window I also need

17:09

to rerun the program gotcha so if I

17:11

change if I say oh the slope field I

17:14

really want to look at it at a different

17:16

part or a different window I'd go change

17:18

my window but then I would need to go to

17:20

quit to the kind of the home calculation

17:22

screen pull up the program just run it

17:25

again gotcha okay yeah no that's

17:28

excellent question is if you're wanting

17:30

to wipe the Slate clean you know just

17:33

change the window and then you could

17:34

even go back to the old window you had

17:36

but now you won't have the slope field

17:38

there anymore okay all right okay very

17:42

good all right I'll stop the share it's

17:44

all your Steve all right well let's take

17:46

a look if we can here at some of the

17:49

problems and

17:51

solutions that we posted on differential

17:54

equations and slope Fields so I'm going

17:57

to try to write a little bit here here

17:58

and this first problem involves trying

18:00

to connect uh the differential equation

18:04

with the corresponding slope field and I

18:06

think Tom correct me if I'm wrong but

18:08

this is the kind of a problem that one

18:10

might see a student might see in the

18:11

multiple choice portion of the exam

18:14

might abely might get a differential

18:17

equation and say okay which of these

18:19

four slope Wheels corresponds to this te

18:22

okay so when I solve these problems I

18:25

mean I'm always looking for some sort of

18:27

prescriptive way to do this kind of a

18:29

stepbystep process and I don't know if

18:31

there is one but here are a couple of

18:33

things that I do in trying to figure

18:35

this out um I might look at what happens

18:39

when I hold say y

18:42

constant and I might look along for

18:44

example a horizontal line and I might

18:46

ask well does the does the slope change

18:51

do those slopes of the line segments

18:54

change as X

18:56

varies well if they don't well that

19:00

would suggest that X is not involved in

19:02

the differential

19:04

equation and similarly I might look

19:06

along a line where X is constant and see

19:09

well are those changing as y varies if

19:14

they're

19:15

not then that suggests to me that y

19:18

might not be

19:19

involved now another thing that I

19:21

frequently do is I take a look at the

19:24

first quadrant where both X and Y are

19:27

positive that somehow seems to be the

19:30

easiest the most intuitive for all of us

19:33

and it's almost easy to plug in a couple

19:35

of values and see what's going on so I

19:38

always look in the first

19:39

quadrant and finally the last thing that

19:42

I do is I kind of look for any special

19:44

circumstances in the differential

19:46

equation for example are there any

19:48

places where I know that the derivative

19:51

is going to be zero and can I can I

19:53

identify that pattern in the slope

19:56

field so I'm actually going to start

19:59

there by looking at

20:01

a if I look at that differential

20:03

equation one of the first things I

20:05

thought about is H when will that

20:07

derivative be zero well let's see if I

20:09

can scribble a little bit up here Tom

20:11

let me know if you can see this if I

20:13

scribble a little bit up here if y were

20:15

equal to X

20:17

over2 then that derivative would be

20:20

zero and yal X over2 is a straight line

20:24

through the

20:25

origin so if I can find any one of these

20:28

slope Fields where in fact I have zero

20:32

slope along that line yal over2 I'm

20:35

feeling pretty good about this and in

20:38

fact if I look real closely over here

20:41

I'm going to scribble a little to I

20:43

think there is a line in there there it

20:46

is YX

20:49

over2 where in fact the slope is

20:52

zero now I can also argue now that I've

20:56

got that I can also argue think about

20:58

what's going on above that line is y

21:00

bigger than x over2 is it less than x

21:02

over2 and how does that affect the slope

21:05

and take a look at those line segments

21:07

but boy that's a very good start on that

21:09

one so a corresponds to slope field

21:13

two we going to look at

21:16

B here's what I saw over here in B I saw

21:19

a product and I thought well how do I

21:22

make that product zero the only way to

21:25

do that is if one of the factors is zero

21:27

so that means if x is0 or y is equal to

21:32

-2 the derivative is going to be zero so

21:36

I'm looking down here at slope field

21:38

three right here where Y is equal to

21:42

two I have those line segments with

21:44

slope zero as as expected in that

21:48

differential equation and right here

21:50

sorry about this kind of funny Arrow but

21:52

right there along the Y AIS where X is

21:55

equal to zero if you can see those

21:57

little GRE green line segments they have

22:00

slope zero also that's pretty

22:03

cool now I think another thing you can

22:06

see here is probably in the first

22:08

quadrant pretty easily where both X and

22:10

Y are positive in fact where X is

22:13

positive and Y is greater than minus two

22:16

the product there is going to be

22:17

positive and the slope going to be

22:19

positive and that's over in this area

22:21

here and I'm feeling pretty good about

22:24

this I think B is connected with slope

22:27

field three

22:30

cool see how about this one well here's

22:34

what I saw over here in this one once

22:36

again I was thinking about well when is

22:38

that derivative going to be zero well I

22:41

I can kind of see that equation on the

22:44

right hand side and I'm thinking you

22:46

know that's kind of related to a circle

22:48

I'm going to scribble a little bit off

22:50

to the side here x^2 + y^2 = 4 well I

22:56

know that if I pick an X and A Y

22:58

that satisfy that equation it's on that

23:02

Circle and that's going to make that

23:03

derivative

23:05

zero and so it seems to me that on this

23:08

circle x^2 + y^2 = 4 man those line

23:12

segments better have slope zero now I

23:16

didn't take a very fine resolution as

23:19

Tom was talking about but I think you

23:21

can kind of see along this

23:26

circle the line segments have slope

23:30

zero what happens outside that Circle

23:34

well outside that

23:36

Circle x^2 + y^2 is going to be greater

23:40

than four and therefore the derivative

23:42

is positive and so look at all those

23:44

line segments outside that Circle they

23:48

all have positive slope I'm feeling good

23:50

about this one and inside the circle

23:53

that expression x^2 + y^2 is less than 4

23:59

and so I better have some line segments

24:02

that are negative and in fact they are

24:04

which is pretty

24:06

cool so C goes with

24:09

four ah

24:12

finally this is do this one Steve I

24:15

think I can do this one what's that I

24:18

think I can do this one you can you can

24:21

match this

24:22

one well this is kind of a cool one

24:24

maybe I should have done this one first

24:26

y Prime is the cosine e to Theus XY I

24:29

I'm not sure exactly how to start on

24:31

this one except maybe looking in the

24:33

first

24:34

quadrant I mean what happens when both X

24:37

and Y are positive and what happens when

24:39

both X and Y increase get

24:43

large well let's see when both X and Y

24:46

are large that product is large and this

24:48

is very large negative what does that

24:50

mean for E to that value well that makes

24:53

e to that expression e to that value

24:56

small close to

24:58

Z and what's the cosine of zero a value

25:02

close to zero son of a

25:04

gun that's going to give me a

25:07

one and so it seems as X and Y get large

25:12

increase without

25:14

bound that argument of the cosine is

25:17

getting very close to zero and so the

25:19

derivative should be getting close to

25:21

one and in fact that's what's happening

25:23

over here and that's exactly what

25:26

happens over here because say in the

25:28

third

25:29

quadrant X and Y have the same

25:33

sign so their product is positive and

25:36

then there's that extra negative there

25:38

in the

25:39

exponent which once again drives that

25:42

argument close to

25:44

zero and we get a slope of

25:47

one that's really cool over here in the

25:50

second and the fourth quadrant I think

25:53

if you consider say look along a

25:56

horizontal lines

25:58

and let X vary let X increase I think

26:01

you can argue that there's going to be

26:03

some sort of cycle here we know that the

26:06

cosine cycles and there's going to be

26:08

some sort of cycle here uh as X

26:10

increases it's not quite evident in the

26:13

second and the fourth quadrant in my

26:15

drawing again my resolution isn't that

26:17

great but I'm pretty sure d goes with

26:21

one Co Steve yes sir um like on your

26:26

part A there

26:28

on the exam will they include that Roman

26:30

numeral too right after

26:33

it I don't think so Tom okay we could

26:37

Lobby for

26:38

[Laughter]

26:42

that good

26:44

question all right I have a couple of

26:47

problems here in in two where we're just

26:49

going to solve the differential equation

26:51

these are not initial value problems

26:53

we're simply going to solve the initial

26:55

solve the differential equation um a

26:58

couple of the questions that was sent to

26:59

Curtis dealt with including this

27:02

constant so I'm going to try to make

27:04

sure that I'm very careful as I speak

27:05

about these constants and when they

27:07

arrive when they should be put into this

27:09

equation when they should

27:12

appear all right so the only kind of

27:14

differential equation that students need

27:16

to know how to solve are separable and

27:18

what does that mean well I can separate

27:19

the variables I can bluntly put all the

27:22

Y's on one side and all the x's on the

27:25

other so okay here's my differential

27:28

equation I'm going to get all my y's on

27:30

the left and all my x's on the right

27:32

hand side here we go in this step I'm

27:36

going to integrate both

27:37

sides on the left hand side I get a log

27:40

of the absolute value of y question we

27:44

often hear Tom and I often hear as well

27:46

do you need to really need to worry

27:48

about including that absolute value

27:50

symbol and the answer is yes especially

27:53

especially when we're dealing with

27:56

initial value problems yes you need to

27:58

include it we'll get rid of that or

28:01

we'll take care of eliminating those

28:03

absolute values as we eventually solve

28:05

this for why but yes you definitely need

28:07

it at this

28:09

step on the right hand side this is an

28:12

integration almost by inspection but

28:14

it's really substitution with u equal to

28:17

minus 2x I'll make maybe one more step

28:19

here so du is minus

28:23

2dx so that's an integration by

28:25

substitution I think that gives me a

28:27

min-2 eus 2X and this is the step right

28:31

here where this constant C has to come

28:35

in so where I find an anti-derivative of

28:38

the left side I find one of the right

28:40

side I've got to add that constant plus

28:42

C not a silly question but sometimes we

28:45

get this one well really when I find the

28:48

antiderivative on the left and one on

28:50

the right there's really two constants

28:52

agreed but I've combined all of those

28:54

and brought it over to the right hand

28:56

side

28:58

so now what do I do to isolate y what do

29:00

I do to get an explicit expression for y

29:04

well I'm going to exponentiate both

29:07

sides on the left hand side e and log

29:11

are inverse functions so I'm left with

29:13

the absolute value of y on the right

29:17

hand side kind of hard to see but that

29:19

expression minus

29:21

one2 eus 2x + C is all in the

29:25

exponent I'm going to use some

29:27

properties here of exponents so that

29:29

I've got an e to the c times this

29:32

expression and e to the C I will just

29:36

abbreviate or use some other notation

29:38

and call that a different constant

29:42

C1 now how do I isolate

29:45

why well I think about this

29:50

as that argument can be either positive

29:53

or

29:55

negative and if it's positive in my mind

29:58

I'm On One branch of this solution or

30:00

one piece of the solution if it's

30:02

negative I'm on another piece of the

30:06

solution so it's either going to be a

30:08

plus or minus y when I evaluate that

30:11

absolute value and we will incorporate

30:14

that sign into this constant C1 and

30:18

write that as a different constant

30:22

C2 now I didn't ask for this but I

30:26

actually created or included in the

30:28

solution files here the slope field

30:31

associated with this differential

30:33

equation and teachers students you might

30:36

think about this does this make sense

30:38

with the differential equation up above

30:41

um can you justify can you reason out

30:45

that this equation for y can you draw

30:49

that on this slope field it might not be

30:52

fine enough for some of some of this but

30:55

but does it make sense that that

30:58

equation somehow fits in this slope F so

31:02

we'll leave that as kind of an open

31:05

question here's B let's try this one

31:09

this is kind of a cool one too I like

31:11

this one so do ydx is e to xus 2 over

31:15

the cosine of Y I've got to separate the

31:17

variables I'm going to bring all the Y's

31:19

to the left once again all the xes to

31:21

the right I'm going to integrate both

31:24

sides here we go the integral of the

31:26

cosine of Y that's an integration by

31:28

inspection one of our common

31:30

anti-differentiation rules that's just

31:32

the sign of Y on the right hand side

31:36

integrate term by term so that's just e

31:39

to x minus 2X and here's where my

31:42

constant C comes

31:45

in and this isn't so bad I mean I want

31:48

to solve for why in order to do that I'd

31:50

have to take the inverse sign of both

31:54

sides so here's some things to think

31:57

about and then I'm going to hand it over

31:58

to Tom here I was thinking about this I

32:01

drew the slope field for this and I was

32:03

thinking about this and well wait a

32:05

minute what's the domain of this

32:11

function do I need to worry about that

32:13

or or do I need to think about that

32:16

here's a question for the chat

32:18

Allison what's got to be

32:20

true about this argument right there

32:24

what has to be true about that argument

32:28

in order me for me to be able to

32:31

evaluate the inverse sign or the arc

32:34

sign and the reason that I thought a

32:37

little bit about this was because I

32:38

didn't think originally that this slope

32:41

field actually made

32:43

sense so I started out by thinking about

32:46

the

32:47

domain and I'm G to turn it over to

32:50

TA see if he can justify a little bit of

32:53

this with

32:55

technology okay all all right let's give

32:57

it a

33:02

shot okay I'll U go ahead and use

33:07

the uh 84 on this one I think okay

33:11

sounds good

33:13

and so I'm going to go to my y equals

33:17

menu and uh clear out y1 and remember on

33:21

this slope field program on the 84

33:23

that's y1 is where we'll store our

33:25

differential equation

33:27

uh and uh let's see this was actually a

33:30

fraction so I'm going to do this uh

33:33

Alpha of the XT Theta key get a nice

33:36

little fraction there and let's see I

33:39

think it was e to the X

33:44

yep minus two correct that's my

33:48

numerator and then down at the do excuse

33:50

me in the denominator we've got the

33:52

cosine of

33:54

Y again it's a very weird looking

33:57

expression to have in y1 but that's just

33:59

our placeholder for the differential

34:01

equation

34:02

expression and let me go ahead and clear

34:04

Y

34:06

2 and now I'm going to

34:10

um go back to the calculator screen and

34:15

bring up the slope field program now

34:18

since we had run the slope field program

34:20

before I can either pull it out of

34:22

programs but since there before if I

34:24

just hit enter it's going to run it

34:26

again with this new differential

34:29

equation in

34:31

there and hopefully we'll get a slope

34:33

field that looks not dissimilar to the

34:37

one

34:38

that okay it might look a little bit

34:40

different just because of the way the

34:42

lattice is laid out and stuff but this

34:45

isett good pretty good yeah and now uh

34:49

we're GNA check out and see if Steve's

34:51

solution makes sense going to go back to

34:54

the Y equals and go to Y

34:57

2 uh we didn't have an initial condition

35:01

uh so to get a particular solution I'm

35:03

going to need a pick a value uh for C uh

35:07

so what we could do let's see I think

35:09

the the general function was the inverse

35:12

sign of e to the

35:20

x -

35:22

2x

35:24

mhm and then inside the parentheses

35:27

that's where that plus C showed up so we

35:29

could put in a value here uh we've got

35:32

to be a little bit careful about the

35:35

plus C because as Steve said the domain

35:38

of the inverse sign that's going to be

35:41

limited to between Nega one and one

35:44

right whatever I have in here it's got

35:46

to be something that I'll get some

35:48

values between negative one and one um

35:51

so I'm going to put it let's see e to

35:53

the zero is already

35:55

one uh but we are am subtracting X so

35:59

you know what I'm going to put in maybe

36:00

one half cross my

36:04

fingers so that's one value of the C

36:08

that we could try out and let's graph

36:13

it and I don't think I picked a very

36:16

good value no I don't see it Tom I don't

36:19

see it it might be off the screen so

36:21

I'll tell you what let's go back to y

36:23

equals I I'm uh I'm G to make the the

36:26

plus C the C value just zero I think see

36:30

yeah let's try

36:32

that okay that's a perfectly good value

36:35

of C now whether that's going to be one

36:38

that we'll see some urve for oh you know

36:42

what we lost our slope field because I

36:45

did a new graph there so let me just run

36:49

the slope field again okay regenerate

36:55

it doesn't take too

37:03

long and now go to y

37:06

equals uh turn on Y2

37:10

again and now we'll graph and this time

37:12

it should graph on top of the slope

37:15

field oh there we go how about that yeah

37:18

great and it looks like a pretty good

37:20

fit though it's you know it's in a

37:22

pretty there's not that many line

37:24

segments there so something we could do

37:26

do is we could change our window to

37:30

maybe just being the first

37:32

quadrant so I'm going to make the X-Men

37:38

zero the Y men

37:42

zero now when I do that what I'm going

37:45

to need to do is run the slope field

37:47

again to generate a new slope field for

37:49

the first

37:53

quadrant go so when that runs Tom does

37:56

that turn off off all the other y's and

37:59

Y it does it automatically turns off all

38:01

the Y's and then it's just referring to

38:03

why not y1 itself isn't even selected it

38:07

just it uses y1 inside the program just

38:10

to evaluate what the slope should be

38:13

these little line segments okay and when

38:15

I go back to Y notice that Y2 was turned

38:18

off yep need to turn it back on and

38:22

graph again and I get a little bit more

38:25

of now I so you can see I change the

38:27

window a lot each time I change the

38:29

window I do need to run the slope field

38:31

program again but this is looking like a

38:33

pretty good

38:34

fit pretty

38:36

cool all

38:38

right

38:40

so did you want to look at anything else

38:43

on this Steve or should I just Let's see

38:45

we had a suggestion okay try a c of

38:50

minus5 okay be happy I don't know if

38:53

that'll work try that one and see if

38:55

that does you know what I'm going to

38:56

make got my

38:58

Y3 okay so let's make

39:04

uh I'll just go ahead and enter it again

39:08

since that plus C was inside the

39:13

parentheses so minus

39:15

2X and the suggestion was minus .5 yes

39:24

okay and let's graph that

39:28

hey nice and notice it has a bit wider

39:32

domain to it yes uhuh very cool great

39:37

and so you can see that you can actually

39:39

do multiple solutions for different

39:41

values of the constant on the same

39:44

backdrop of the slope field yep very

39:47

nice okay stop the share there back to

39:51

you Steve all right

39:54

deal here we go so there was one

39:57

question that came in here about that

39:59

constant C in that one problem we did

40:02

Tom I guess it was a we a change from a

40:05

c to a C1 to a C2 and the question is do

40:08

students have to do that on the exam and

40:10

and the answer is probably not uh

40:13

they're very we are very forgiving about

40:15

that c uh as it changes quote unquote

40:20

you can just keep the same symbol C you

40:23

don't necessarily have to have a

40:25

subscript I don't think students would

40:27

uh lose a point for that in my opinion

40:30

yeah and it was kind of it is kind of a

40:33

subtle thing I mean your C1 Steve I

40:36

don't know if you want to back up to

40:37

that page

40:41

um I mean your your

40:44

C1 is really an

40:48

arbitrary positive or non- negative

40:50

constant yeah so you got an absolute

40:53

value of y is is an arbitrary positive

40:57

constant times this e to the one2 well

41:00

that means y itself is either plus or

41:04

minus that arbitrary positive constant

41:06

well you could just say that's just an

41:08

arbitrary constant period which that's

41:11

your C2 so the C1 was arbitrary positive

41:15

but C2 is just plain arbitrary it took

41:18

care of both positive and negative and

41:20

that's why you were able to drop that

41:21

absolute value sign y I think the way on

41:24

the exam it kind of gets around that is

41:27

you you often have some initial

41:29

condition you have to work with and

41:31

that's the thing that resolves the

41:32

matter and then it's cut and dryed

41:35

whether you're using the positive or

41:36

negative branch of

41:38

your correct all right okay I hope we

41:42

can I hope we do have the opportunity to

41:44

look at one of those here let's see um

41:48

let me take a look at C we'll solve one

41:51

more or maybe two more quick

41:52

differential equations and then we'll

41:53

take a look at an initial value problem

41:55

if we can

41:57

once again I'm going to get all the Y's

41:58

on one side of the equation all the x's

42:00

on the right hand side I'm going to

42:01

integrate both sides this one's pretty

42:04

easy this is a one one of our

42:07

anti-differentiation rules common

42:09

anti-differentiation rules that's an arc

42:12

tangent on the right hand side this is

42:15

again as a substitution with = x^2 + 1

42:19

and du = 2x

42:22

DX so let's see I integrate on the left

42:25

hand side add the tangent inverse on the

42:27

right hand side I had this expression in

42:30

X there's my plus C and again not too

42:33

bad to solve for why uh what's the

42:36

inverse of the inverse tangent well the

42:38

tangent I could take the tangent of both

42:40

sides and here's my expression can I

42:43

take the tangent of any

42:46

argument yeah I guess I can so I don't

42:48

have to worry too much about that domain

42:51

um does that function y fit nicely into

42:54

this slope field well again we'll kind

42:56

of lead that uh for teachers and

42:58

students to think about I provided that

43:00

slope field here you might try to draw

43:02

this function by looking at that slow

43:04

field good so see that tangent function

43:09

it will have just some isolated places

43:11

where it's not defined right

43:14

because but there won't be some whole

43:16

stretch of numbers where it's undefined

43:18

I think that's what yep all right well

43:22

let's take a look at three here we're

43:23

going to solve some initial value

43:25

problems um I like this one this

43:27

produced kind of a weird slope feeli

43:30

although this did not ask to do that I

43:33

had to see what was going on here so

43:35

here's the differential equation dydx is

43:37

x + 1 / the of XY we have this condition

43:41

y of 1 is equal to four I'm going to

43:43

start out in the usual way here's all my

43:46

y's on that side here's the x's on the

43:48

right hand side and looking ahead just a

43:50

little bit um how am I going to be able

43:52

to integrate that well the technique is

43:55

to write that as as two separate terms

43:57

or two separate fractions and to write

44:00

each of those terms as X raised to some

44:03

power so I did that over here now I'm

44:07

going to integrate both

44:09

sides don't make a mistake on this one

44:12

Tom on the left hand side 2/3 wider the

44:15

three halves on the right hand side we

44:17

integrate term by term and here's the

44:19

plus c um it is at this step now right

44:24

at this next step is where we recommend

44:26

that students use the initial condition

44:30

look can you go on and solve this for

44:33

y and then once you have isolated why

44:36

use the initial condition at that point

44:39

absolutely but in my opinion not only is

44:42

that harder but I think it introduces

44:45

more chance of an error so the best

44:48

thing to do is once you have found the

44:50

anti-derivative of both sides included

44:53

Your plus C use the initial condition

44:56

right away so here's the yal 4 here's

45:00

the xal 1 and saving a little bit of

45:04

time here I think that gives me a c

45:06

equal to

45:07

8/3 so I went back up here plugged in a

45:11

c equal to 8/3 and I multiply both sides

45:14

of that by three Hales is an expression

45:17

almost with Y

45:19

isolated how do I do that well how do I

45:22

get y all alone well I raise both sides

45:25

of this equation to the 2/3 power and

45:28

here's kind of a weird looking

45:30

expression or

45:33

why now here's what I did down at the

45:35

bottom and I may need a little help with

45:36

this

45:37

Tom so when I drew this slope field I

45:40

was kind of I I didn't know what was

45:43

going on initially

45:45

here but I only got a slope field in

45:48

quadrant one and

45:51

three I wasn't quite sure why so we'll

45:54

leave that as a question question for

45:56

participants why isn't there any slope

46:01

field for this differential equation in

46:04

quadrants 2 and

46:06

four

46:08

h well the second thing I did was I

46:11

wanted to actually draw this particular

46:14

solution curve that goes through the

46:17

0.14 so here it is here's the

46:20

0.4 and it looks like

46:24

this so my next question

46:27

is where X is equal to

46:31

zero is that actually on the solution

46:34

curve or should there be an open circle

46:36

there I

46:38

wonder so that's something else you can

46:40

think

46:41

about Allison we'll let you monitor the

46:44

chat and let us know if we get any

46:45

answers to those questions okay or those

46:48

are good ones maybe for :l tomorrow

46:50

morning in

46:52

class I liked that one Tom take a look

46:55

at could

46:56

be oh this has some good things in it

47:00

too so here we have dydx is equal to y -

47:04

2 1 + x^2 we have another initial value

47:07

problem y of 1 is three this is a

47:09

separable differential equation I can

47:11

get all the Y's on one side all the x's

47:13

on the other and this one involves the

47:16

log of an absolute value and let me make

47:17

sure that I do this correctly so on the

47:19

left hand side this is the log of the

47:21

absolute value of Yus 2 do you need the

47:23

absolute value yes you do you have to

47:26

worry about that on the right hand side

47:28

this is a common anti-differentiation

47:31

formula this is the arc tangent tangent

47:33

inverse of X and here's my plus

47:36

C okay at this step at this point we

47:40

want to use the initial condition what

47:42

was it y of 1 is equal to 3 so Y is

47:47

equal to 3 there's the xal to 1 and that

47:52

gives me a c = to minus piun 4 but it's

47:56

also told me something

47:59

else when this is the way that I think

48:02

about this when I plug in that y equal

48:06

to 3 this argument here this argument

48:10

inside the absolute value is

48:13

positive and that tells me I am on the

48:15

branch of this

48:17

solution where that argument is

48:21

positive if you think about for example

48:24

the graph of the log of the absolute

48:27

value of x just think about that I'm

48:29

going to scribble a little

48:32

bit if you think about that

48:35

graph it's not defined that function is

48:38

not defined at x equals z and there are

48:41

two pieces two branches to

48:44

that one branch corresponds to where X

48:47

is positive one corresponds to where X

48:50

is

48:51

negative in my mind that's kind of

48:54

what's going on

48:56

and I need the branch in this case where

49:00

the argument is

49:01

positive that means in my next step I

49:05

can simply drop the absolute

49:08

values how do you take the absolute

49:11

value of a positive argument it is just

49:14

that argument I don't need that extra

49:16

negative

49:18

sign so okay I've got the log of Yus 2

49:23

is equal to this expression with an arc

49:25

tangent over there I'm going to solve

49:27

this for y by first exponentiating and

49:30

then adding a two to both sides there's

49:32

my expression for y and there's some

49:35

cool things going on here I

49:38

think in this slope field and this

49:43

equation uh there's the SL f it kind of

49:47

looks like that the that

49:51

the line segments have slope zero at X

49:56

two wonder if I could justify that I

49:59

think I could if I look back up at the

50:01

differential

50:02

equation and I drew in this curve this

50:06

curve that satisfies the initial

50:08

condition of

50:10

13 now don't be misled here if you look

50:14

at this very

50:17

quickly it seems like there might be

50:20

some horizontal asmp tootes on the graph

50:22

of

50:24

y and it's seems like their two

50:27

horizontal ASM tootes might be at yal 2

50:31

and yal

50:34

4 but they're

50:37

not and so a question is can you find

50:40

the horizontal Asm

50:42

tootes on this graph of

50:46

y I wonder if the slope field might be

50:50

just a little bit misleading in finding

50:53

those okay we'll leave that as an open

50:56

question I like that one have we got

50:58

time for one more I think

50:59

so maybe a couple

51:02

more all right and C holy Toledo This is

51:06

complicated on the right hand side I'm

51:07

going to bring all the Y's to the left

51:09

all the X's to the right hand side oh

51:12

it's not so bad I guess the integral of

51:14

e to the x is just e to the X and this

51:16

isn't bad on the left hand side either

51:19

that's a simple substitution with Y = 1

51:22

+ y^2 I believe the derivative of of U

51:26

is 2 ydy the 2y is part of the integr

51:31

product except for that constant two so

51:34

we should be able to do this I integrate

51:37

both sides there's my constant C I use

51:40

the initial condition it turns out that

51:42

c is equal to

51:44

one so I'm going to go back up here and

51:47

I'm going to plug in a c equal one

51:48

there's my expression in X and Y can I

51:51

solve that can I isolate y and by the

51:53

way sometimes you cannot

51:56

not here I think I can I'm going to

51:59

square both sides I'm going to subtract

52:01

one from both sides and then I'm going

52:04

to take the square root of both sides

52:07

whoops there's my expression for y and I

52:10

did the same sort of thing down here I

52:13

drew a slope field and I drew in the

52:17

solution curve that goes through that

52:19

point what was it what was that initial

52:22

condition again I think it was Zero

52:24

theare of

52:26

three so there's another question I have

52:29

here you know as I looked at that

52:30

solution curve it looks an awful lot

52:32

like y equal e to the X yes granted it

52:35

does not go through the 01 but it still

52:38

looks an awful lot like y equal e to the

52:40

X and I wonder if you can make an

52:43

argument to explain that to justify that

52:47

why does that solution curve look a lot

52:49

like yal e to thex boy lots of open

52:51

questions here you're G to have to be

52:53

writing lots to Curtis this week I think

52:57

all right I'm going to take a look at

52:58

four Tom and maybe we'll call it a night

53:00

this is a good one here so you Ju Just

53:05

wanted to throw out that uh you know

53:07

that problem you had uh a couple back

53:10

where when you did the slope field it

53:12

only showed up in the first quadrant yes

53:16

thir quadrant yes I I believe in the

53:19

chat uh Wanda Burns pointed out that

53:22

that square root of the product of X and

53:24

Y

53:26

might be the culprit there and indeed it

53:29

is yep very good one yep there's a

53:32

square root of x times Y and in the

53:34

second and the fourth quadrant that

53:36

product would be negative and of course

53:37

the square root would be undefined that

53:39

is exactly why we don't have any part of

53:41

slope field in there very cool very

53:45

cool all right this was a nice one and I

53:47

I actually discovered something Tom on

53:49

Mathematica when I was doing this one

53:51

and produced an extra graph here just

53:53

for the heck of it I'm the the first

53:55

thing I asked is to use technology to

53:57

sketch a slope field for the

53:58

differential equation so I don't know

54:00

maybe we could end if you want to take a

54:01

look at this one eventually but this is

54:03

what I got using

54:05

Mathematica and I use this slope field

54:08

to sketch the solution curve that goes

54:09

through the0 21 so there's the point on

54:12

the graph 21 and I did my very best to

54:15

sort of sketch following those line

54:18

segments following those line segments

54:21

to draw a rough sketch of the solution

54:23

curve something sort of very interesting

54:25

seems to be happening here I hope that I

54:28

can explain this before I end this

54:30

problem but it looks like as we go over

54:32

here in the third quadrant it looks like

54:34

that curve is

54:36

following the line yal

54:40

X that seems awfully strange to me but

54:43

it seems to be doing

54:45

that I was looking something up in

54:48

mathematic to try to make my slopes

54:50

feels give them a couple of more options

54:53

make them look a little nicer

54:55

and I happen to cross a an option that

54:59

allows you to draw well what I would say

55:01

are several solution Curves in here and

55:04

I don't know if you can see this Tom but

55:06

you might be able to see the green sort

55:09

of arrows in there those correspond to

55:12

my solution curve that I drew up above

55:15

that's the curve that goes through the

55:17

point 2 one that was kind of cool gives

55:20

me even a better picture of what's going

55:23

on and those arrows actually

55:25

those vectors actually represent those

55:27

little line seg very cool yeah I believe

55:31

somebody in the chat was asking about uh

55:35

using slope fields to do vectors and I

55:37

think those are called Direction fields

55:39

and similar idea there yeah very cool

55:44

yeah I think I think when I did that

55:46

when I produced that if I let me go back

55:48

to that for one

55:50

second I think when I produced that one

55:53

of the options that I gave

55:55

uh Mathematica was something that would

55:59

produce those vectors so that they all

56:00

have the same

56:03

length but there is another option that

56:05

allows you to adjust the the length of

56:08

the magnitude of that

56:11

Vector U so that can be useful in in

56:14

other

56:17

applications all right so we're still

56:19

working on the same problem let's see if

56:20

we can solve this differential equation

56:22

I'm going to bring all the Y's to the

56:24

left all the X's to the right hand side

56:28

well this doesn't look too bad I've

56:29

separated the variables simple

56:32

substitution on both sides U is equal to

56:34

minus X U is equal to minus y I get this

56:38

expression we bring in the plus C we Ed

56:41

the initial

56:42

condition and I found this value of C

56:45

now that was a little cumbersome for me

56:47

to

56:48

write so what I did is I just said well

56:51

that's my value of c and I'm going to

56:53

assume that and solve for y here so that

56:56

I didn't have to write it out all the

56:58

time I did that I exponentiated both

57:01

sides I took the log of both

57:04

sides and I think I am left

57:09

with this

57:13

expression I multiplied through by a

57:15

minus one by the way in order to make

57:17

that a little bit easier to work with

57:19

and there's my expression for y and what

57:22

I did here is I actually used that

57:25

exess explicit value of c and I thought

57:29

a little bit about the domain here and I

57:33

think I think the domain here might be

57:36

all reals

57:40

okay so I'm G to add this

57:43

graph to my slope field in son of a gun

57:47

it looks an awful

57:48

lot like what I drew on the previous

57:51

page and I wonder if we can justify this

57:55

part over here what's going on over

57:58

there can we justify that at all I

58:03

wonder so as X

58:07

increases

58:09

negatively what's happening inside here

58:13

well as X gets bigger and bigger let's

58:16

see this is going to be a small number

58:18

so that's

58:21

inconsequential as X gets bigger and

58:24

bigger

58:26

let's see if that's inconsequential then

58:29

this is just like the log of e to the

58:31

minus X and those are inverse function

58:35

right so that's just going to give me a

58:37

minus X so it's like I've got a minus

58:40

Min -

58:42

x which is just X and remember X is

58:47

negative and so this is really just the

58:51

line yal X over here in the third

58:56

quadrant that's really cool so I saw

59:00

that on the slope field I wasn't sure

59:03

that it really made sense but I think we

59:05

can actually justify

59:07

that pretty cool

59:10

Tom well Steve I was looking at your

59:12

original differential equation so up the

59:15

page there a bit yes

59:19

um on the line y equal

59:23

x

59:25

my derivative will be exactly one how

59:29

about that because my e to the negative

59:31

x e to the Y will cancel out and give me

59:34

one but Y equals X is a line of slope

59:38

one so it it's all kind of falls into

59:42

place so kind of cool how about that I

59:46

think we're just about out of time

59:48

Steve we we're going to post these

59:51

problems uh with Allison and Curtis

59:53

tomorrow we do have one overtime problem

59:56

I've seen several of these in the

59:58

Facebook group recently Tom and this is

60:00

a very common sort of free response

60:02

question so uh I've got this one here in

60:05

the OT portion of our presentation and

60:07

we'll post this question and answer

60:10

tomorrow great so great thank you very

60:13

much for joining us final wrap up before

60:16

we close up I just wanted to share the

60:18

last of the chat was um noting that the

60:20

vector Direction fields are very

60:22

informative for uh golf and in fact

60:25

there was some conversation about seeing

60:27

uh that type of it's fun to see uh part

60:30

of math teacher Job showing up on

60:33

TV how about that thanks Allison great

60:37

job thanks Tom thank you Steve okay very

60:44

good

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