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apcalcab_diffeq.notebook

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February 22, 2017

AP Calculus Differential Equations

20160516

www.njctl.org

Table of Contents

Slope Fields

Differential Equations (Separable)

Introduction to Differential Equations

Modeling with Differential Equations

Introduction to Slope Fields


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February 22, 2017

A Quick Reminder

As we begin this new unit, let's review a few facts about exponential functions and logarithms. Recall the rules for exponents:

Also, when solving equations involving exponentials or logarithms, apply the opposite operation to solve. For example:

Introduction to Slope Fields

Return to Table of Contents


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February 22, 2017

Have You Ever Seen a Dinosaur?

Ask your partner/team if anybody has ever seen a real, living dinosaur.

It's fairly safe to say that nobody in class has ever seen a dinosaur in real life. However, we do have a very good idea about what dinosaurs look like. But how?

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Differential Equations

A differential equation is simply an equation that contains a derivative. You have seen many differential equations so far, typically finding them by taking the derivative of a function. In this unit we will use differential equations a little bit differently, by starting with the derivative, analyzing it, and uncovering information about the original function.

Differential Equation


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February 22, 2017

Integral Curves

What is true about the derivatives of each of these equations?

They're all the same!

An integral curve represents the possible solutions to a differential equation. Like we saw on the previous page, the solution to is , only varying by the constant. The graph below represents just 3 of these possible solutions.

Integral Curves


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February 22, 2017

Integral CurvesImagine if we drew small line segments to represent tangent lines at many points on our integral curve. The result would be the following graph:

Now, try to picture what it would look like if we erased the curves and just left our small dash marks.

Integral CurvesNotice, without the actual curves graphed we can still get a feel for the general shape of the curves! Think back to the dinosaur fossils!


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February 22, 2017

Slope FieldsIn the next section of this unit we will learn how to build/draw these graphs, known as slope fields. By using a given differential equation we will attempt to recreate the shape of the curves using small line segments of various slopes.

Slope Fields



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February 22, 2017

Slope Basics

Recall your first Algebra class when you were originally introduced to the word SLOPE.

• What words did you use to describe slope? • How can you represent slope? • What does slope look like when graphing?• How can you model slope using your hands/arms?

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Encourage discussion regarding slope, some responses may include:• steepness of a line, rise/run, m, , more/less steep, holding arm at an angle, zero slope, undefined slope, etc.

Students often have a difficult time visualizing what a "segment of slope 2" means, for example. They are used to graphing entire lines, and therefore these few slides are to encourage a better introduction to slope fields.

Slope MatchingDraw a line connecting the slope to the correct graph:

Answer


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February 22, 2017

Slope MatchingNow, similarly, draw a line connecting the slope to the

correct line segment:

Answer

Slope MatchingOnce again! Draw a line connecting the slope to the

correct line segment:Answer


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February 22, 2017

Slope SegmentsIn each box, sketch a short line segment with the given slope:

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The emphasis of this slide is to get students comfortable with sketching

segments with the given slopes, rather than graphing lines like they are used to.

Have students share ideas of strategies they use.

One helpful tip is to have students ensure that their segments are steeper/less steep

relative to the others.

What are Slope Fields?

A slope field is a graphical representation of all possible solutions to a differential equation. In other words, it is a graph of the slopes of a function at different points.

After graphing the slope field, we can visually understand the family of functions who share the same derivative.

Small dash marks are used to represent the slope at different points on the graph. These slopes vary based on the ordered pair used, but sometimes patterns arise depending on the differential equation.


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February 22, 2017

Slope FieldsLet's start with learning how to graph these slope segments. Sketch the slope field for the following differential equation:Simply plug in different coordinate values into your differential equation and make a dash mark with an accurate slope.

Try filling in the rest on your own!

Answer

Example: Given: , sketch the slope field.

(Remember that represents the slope.)

3 2 1 0 1 2 3

3

2

1

1

2

3

x

y

Substituting each ordered pair into the equation will give us the slope at that point. And each dash has the slope of the function, y, at that point. For example,

Slope Fields

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February 22, 2017

3 2 1 0 1 2 3

3

2

1

1

2

3

x

y

Slope FieldsWhy are there so many dashed lines?

Remember, there are an infinite number of functions who share the same derivative, depending on which constant is added. The slope field represents the entire family of functions who share the derivative.

3 2 1 0 1 2 3

3

2

1

1

2

3

x

y

Slope Fields

If then .

Now, working backwards and finding the antiderivative, we could find the general equation for the family of functions.

What do you notice about the pattern of the slope field? What general shape do the dash marks have?

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February 22, 2017

Example: Given: , sketch the slope field.

Slope Fields

3 2 1 0 1 2 3

3

2

1

1

2

3

x

y

Answer

Slope Fields

You may have noticed some patterns while graphing the previous slope fields.

If the differential equation is expressed only in terms of x, the slopes will be the same horizontally (across each "row").

If the differential equation is expressed only in terms of y, the slopes will be the same vertically (down each "coulmn").


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February 22, 2017

Slope FieldsIf you are given a point in the original function, start at the given point and simply follow the pattern or "flow" of the slope field.

3 2 1 0 1 2 3

3

2

1

1

2

3

x

y For example, sketch a graph of the original function if it passes through the point (0,1).

Answer

Slope Fields

Example: Graph the slope field for the differential equation then graph the particular solution through the point .

Answer


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February 22, 2017

Slope Fields

• What general shape did the previous slope field take on?

• What equation could you use to represent an original function with that shape?

• Find the derivative and compare your derivative to the differential equation given on the previous page.

Answer

1

A

B

C

D

Which of the following represents the slope field for the differential equation: ?

Answ

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February 22, 2017

2

A

B

C

D


Answ

er

3

A

B

C

D


Answ

er


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February 22, 2017

4

A

B

C

D


Answ

er

When sketching the slope field for the differential equation , the slope segment at the point would have what slope?

Answer


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February 22, 2017


Answer


Answer


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February 22, 2017


Answer


Answer


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February 22, 2017

5 Which of the following differential equations matches the given slope field?

A

B

C

D

E

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Slope Field Card Match Activity

Click here to go to the Lab titled "Slope Field Card Match"

Students will work with a partner to match the card pieces from each category: Differential Equation, Slope Field, and Conclusion

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http://apcentral.collegeboard.com/apc/public/repository/ap08_calculus_slopefields_cardmatch.pdf

Helpful Hints: Do the following ahead of time to maximize student activity time.

Print each category on different colored cardstock, cut out all pieces and put them in bags for each pair of students.


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February 22, 2017

Introduction to Differential Equations


A differential equation states how a rate of change in one variable is related to other variables.

Across physics, chemistry, and mathematics, differential equations are used often. Any time a rate of change is being described, a differential equation is necessary.

Differential equations may be expressed in terms of x, or y, or a combination of the two.

For example, all of the following are differential equations:

What is a Differential Equation?

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It is worth mentioning to students that in this course we will be studying only first and second order ordinary differential equations. You may mention that there are many other types of differential equations (higher order, partial) as well as many other methods for solving. This course will cover solving by separation of variables which is tested on the AP Exam.


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February 22, 2017

Why are Differential Equations Useful?

One truth that holds across mathematics and the sciences, is that things change. When things change, they do so at a certain rate.

We saw this when studying related rates (which were, in fact, differential equations themselves!)

There are countless ways differential equations are used to represent natural phenomena including: population/bacteria growth, material decay, spring vibration, heat wave transfer, seismic waves, and fluid mechanics, just to name a few.

What is a Solution to a Differential Equation?

In earlier math courses, you learned how to solve things such as and your solution is would be a specific value, , for example.

Solutions to differential equations are not simply single values, but functions. The solution represents the function which has the given derivative.


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February 22, 2017

How do we Solve Differential Equations?

Recall from the previous unit, if then .

General Solution We call the general solution to the differential equation.

If we are given any other information about the original function, we can go a step further to find the particular solution. i.e. If

Particular Solution Substituting and solving for C, our particular solution for this differential equation would be .

*When given a solution value along with the differential equation, we call these problems Initial Value Problems (or IVPs).

Classifying Differential Equations: Order

The order or a differential equation is equal to the highest derivative present.

First Order Second Order Third Order


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February 22, 2017

Classifying Differential Equations: Ordinary vs. Partial

An Ordinary Differential Equation (or ODE) is one with derivatives with respect to one variable only. For example:

A Partial Differential Equation (or PDE) is one that has at least one partial derivative, in other words more than one independent variable. For example:

*This course will only cover Ordinary Differential Equations.

Classifying Differential Equations: Linear vs. NonLinear

A differential equation is classified as linear if the power of the dependent variable and any of its derivatives is 1, and each coefficient depends only on the independent variable, x. Otherwise we classify it as nonlinear.

Linear NonLinear


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February 22, 2017

6 Classify the following differential equation. Select all that apply.

A Order = 0

B Order = 1

C Order = 2

D Order = 3

E Ordinary

F Partial

G Linear

H NonLinear

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A Order = 0

B Order = 1

C Order = 2

D Order = 3

E Ordinary

F Partial

G Linear

H NonLinear

Answ

er


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February 22, 2017


A Order = 0

B Order = 1

C Order = 2

D Order = 3

E Ordinary

F Partial

G Linear

H NonLinear

Answ

er


A Order = 0

B Order = 1

C Order = 2

D Order = 3

E Ordinary

F Partial

G Linear

H NonLinear

Answ

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February 22, 2017

What is the general solution to the following differential equation?

Answer

Hint:

Remember you can find the antiderivative to arrive at the general solution!

What is the general solution to the following differential equation?

Answer

Hint:

Remember you can find the antiderivative to arrive at the general solution!


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February 22, 2017

What is the particular solution to the following differential equation if ?

Answer

Differential Equations (Separable)



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February 22, 2017

Let's look back at the previous example: if then .

Separable Differential Equations

By integrating each side (or finding the antiderivative of 6x) we were able to arrive upon the general solution of the differential equation.

But, what if we are given a differential equation such as: ?

The antiderivative is not so clear this time. And thus, we need an alternative approach to finding the original function.

Separable Differential Equations

A separable differential equation is of the form:

We call these separable because any terms involving x and any terms involving y are able to be separated on either side of the equation, to yield something like the following:

or

or


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February 22, 2017

Steps to Solve Separable Differential Equations

The following steps provide a guideline to solving separable differential equations:

1. Algebraically move all x terms to one side and y terms to the other side (including dx and dy).

2. Integrate each side, adding the constant, C, to one side of your equation.

3. If given an initial condition, substitute and solve for C.

4. If necessary, solve for y.

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Solving Separable Differential EquationsExample: Solve the following differential equation:

1. Algebraically move all x terms to one side and y terms to the other side (including dx and dy).

2. Integrate each side, adding the constant, C, to one side of your equation.

3. If given an initial condition, substitute and solve for C.

4. If necessary, solve for y.


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February 22, 2017

Why Don't We Need a Constant on Each Side?

Some of you may have wondered why we didn't add a constant to both sides of the equation when we integrated.

If we add 2 different constants, and , to each side, we can then combine like terms.

Since subtracting one arbitrary constant from another yields another constant, we are able to add only one.

Example: Find the general solution to the differential equation:

Solving Separable Differential Equations

Answer


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February 22, 2017

Example: Find the particular solution to the differential equation:

Solving Separable Differential Equations

Answer

10 Find the general solution to the differential equation:

A

B

C

D

E

Answer


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February 22, 2017

11

A

B

C

D

E

If and , then the particular solution to is

Answer

12

A

B

C

D

E

F

If and what is the value of x when y=0?

Select all that apply.

Answ

er


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February 22, 2017

13

A

B

C

D

E

Which one of the following curves solves the differential equation and passes through (0,2)?

Answ

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If and find .

Answer

Classwork, Pt. 1: Separable Differential Equations

1.

At each point (x,y) on a certain curve, the slope of the curve is . If the curve contains the point (0,4) the find its equation.

2.


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February 22, 2017



As mentioned before, there are countless ways differential equations are used to represent natural phenomena including: population/bacteria growth, material decay, spring vibration, heat wave transfer, seismic waves, and fluid mechanics, just to name a few.

In this section we will delve into these scenarios and discover how useful differential equations are in the real world.



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February 22, 2017

Exponential Growth & Decay

First we will discuss how differential equations are used to model situations involving exponential growth and decay.

Exponential growth and decay can be modeled by the differential equation:

...meaning the rate the amount increases or decreases is proportional to the amount present (with respect to time). is the proportionality constant.

Exponential Growth & DecayApply your newly gained knowledge to find the general solution to the differential equation:

Answer

Assuming the initial value is at then we arrive at the

equation:


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February 22, 2017

Exponential Growth & DecayLet's take a look at how this applies to an actual population.

Example: A population of bacteria increased from 300 to 1200 in 3 hours. Assuming the rate of increase is directly proportional to the population, find the population of bacteria at the end of 9 hours.

Answer

Exponential Growth & Decay Helpful Hint

When working with exponential growth/decay problems, it is typically more beneficial to solve for rather than , because it is usually easier to resubstitute .


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February 22, 2017

Exponential Growth & Decay Half Life

Sm151 is a radioactive isotope of Samarium. It's decay can be

modeled by the differential equation , where t is

measured in years. What is the halflife of Sm151?

Let's try another example:

Answer

Hopefully students recognize the pattern and can arrive at the equation:

If not, work through the integration another time, or encourage students to review previous slides.

We need to find the time for half of the isotope to be present.

Exponential Growth & DecayExample: The population of a herd of sheep is 900, and growing exponentially. At years, the population is 1800 sheep. Write an equation to represent the population of sheep at any time.

Answer


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February 22, 2017

14

A B C D E

*1998 AP Calculus AB Exam #84

Population y grows according to the equation where k is a constant and t is measured in years. If the population doubles every 10 years, then the value of k is

Answ

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A B C D E

Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria doubles in three hours, in how many hours will the number of bacteria triple?

Answ

er


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February 22, 2017

16

A

B

C

D

E

How long would it take $6000 to grow to $18000 at 5% compounded continuously? Round your answer to the nearest tenth of a year.

Answer

17

A

B

C

D

E

The decay equation for a radioactive element is where t is in days. About how long will it take for the amount of substance to decay to 37% of its original value?

Answer


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February 22, 2017

Simple Inhibited Growth

Growth in the real world is sometimes limited by some factor making the equation unrealistic for very large values of t. If we allow this natural maximum to be called, , the growth of the quantity cannot occur beyond this value.

Our equation can be modified to reflect this:

We refer to this situation as simple inhibited growth. And solving the new differential equation, we get:

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Assuming the initial value occurs when t=0...

Simple Inhibited Growth

*Taken from the 2012 AP Exam

The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time , when the bird is first weighed, its weight is 20 grams. If is the weight of the bird, in grams, at time days after the bird is weighed then,

Use separation of variables to find if .

Answer


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February 22, 2017

A tank initially holds 200 gallons of brine solution containing 1 lb of salt. At another brine solution containing 1lb of salt per gallon is poured into the tank at a rate of 4 gal/min, while the well stirred mixture leaves the tank at the same rate. Find the amount of salt in the tank after 10 minutes.

Answer

A big book company notices that the maximum number of a new book sold is 80 per day, and the rate of growth of sales is proportional to the difference of current level sales and the maximum sales. In the first hour of sales, 6 books were sold. After 3 hours, 40 books were sold. Approximate the time at which 60 books will be sold.

Answer


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February 22, 2017

The Siberian Tiger is an endangered species. According to the latest estimates, there are only about 1,000 tigers left in the wild on the entire planet (2016). Suppose a population of tigers has a constant birth rate of 12 per thousand tigers per year, and every year there are 16 per thousand tigers that die or are killed. Assume also that approximately 8 per thousand tigers per year are taken to sanctuary every year. What will be the approximate number of tigers living in the wild by 2020? A

nswer

Newton's Law of Cooling

What happens to the temperature if you...

leave a cup of hot coffee out on the counter at room temperature?

put leftover dinner in the refrigerator to save for later?

put an ice cube in a soda and let it sit for some time?

Choose one scenario above and make a quick sketch of the temperature over time.

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February 22, 2017


is the temperature of the object at time t

is the surrounding temperature

is the temperature at t=0

is time

Keep them all straight what does each mean?

Newton's Law of Cooling deals with the rate at which an object will change temperature when introduced to a new environment of constant temperature.

Solving the differential equation, we get:

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Example: A bowl of hot soup is brought into a room which is 20℃. The

soup was originally 80℃, and after 2 minutes it has cooled to 60℃. How many minutes pass before the soup cools to 40℃?

Answer


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February 22, 2017

18

A

B

C

D

E

The temperature of a glass of lemonade is initially 5℃ when

brought into a 30℃ room. After 5 minutes the lemonade has warmed to 10℃. What is the temperature of the lemonade after 10 minutes?

Answer

19

A

B

C

D

E

Continuing from the previous question, at what time will

the temperature of the lemonade be 25℃?Answer


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February 22, 2017

A casserole is taken out of a oven and placed on a counter in a kitchen. After 8 minutes the temperature of the casserole drops to . How long will it take to cool to ?

Answer

Classwork, Pt. 2: Modeling with Differential Equations

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