10.5 Differential Equationsap Calculus

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Version #1
​The course below follows CollegeBoard's Course and Exam Description. Lessons will begin to appear starting summer 2020.
BC Topics are listed, but there will be no lessons available for SY 2020-2021

Unit 0 - Calc Prerequisites (Summer Work)
0.1 Summer Packet
Unit 1 - Limits and Continuity
1.1 Can Change Occur at an Instant?
1.2 Defining Limits and Using Limit Notation
1.3 Estimating Limit Values from Graphs
1.4 Estimating Limit Values from Tables
1.5 Determining Limits Using Algebraic Properties
(1.5 includes piecewise functions involving limits)
1.6 Determining Limits Using Algebraic Manipulation
1.7 Selecting Procedures for Determining Limits
(1.7 includes rationalization, complex fractions, and absolute value)
1.8 Determining Limits Using the Squeeze Theorem
1.9 Connecting Multiple Representations of Limits
Mid-Unit Review - Unit 1
1.10 Exploring Types of Discontinuities
1.11 Defining Continuity at a Point
1.12 Confirming Continuity Over an Interval
1.13 Removing Discontinuities
1.14 Infinite Limits and Vertical Asymptotes
1.15 Limits at Infinity and Horizontal Asymptotes
1.16 Intermediate Value Theorem (IVT)
Review - Unit 1
Unit 2 - Differentiation: Definition and Fundamental Properties
2.1 Defining Average and Instantaneous Rate of
Change at a Point
2.2 Defining the Derivative of a Function and Using
Derivative Notation
(2.2 includes equation of the tangent line)
2.3 Estimating Derivatives of a Function at a Point
2.4 Connecting Differentiability and Continuity
2.5 Applying the Power Rule
2.6 Derivative Rules: Constant, Sum, Difference, and
Constant Multiple
(2.6 includes horizontal tangent lines, equation of the
normal line, and differentiability of piecewise)
2.7 Derivatives of cos(x), sin(x), e^x, and ln(x)
2.8 The Product Rule
2.9 The Quotient Rule
2.10 Derivatives of tan(x), cot(x), sec(x), and csc(x)
Review - Unit 2
Unit 3 - Differentiation: Composite, Implicit, and Inverse Functions
3.1 The Chain Rule
3.2 Implicit Differentiation
3.3 Differentiating Inverse Functions
3.4 Differentiating Inverse Trigonometric Functions
3.5 Selecting Procedures for Calculating Derivatives
3.6 Calculating Higher-Order Derivatives
Review - Unit 3
Unit 4 - Contextual Applications of Differentiation
4.1 Interpreting the Meaning of the Derivative in Context
4.2 Straight-Line Motion: Connecting Position, Velocity,
and Acceleration
4.3 Rates of Change in Applied Contexts Other Than
Motion
4.4 Introduction to Related Rates
4.5 Solving Related Rates Problems
4.6 Approximating Values of a Function Using Local
Linearity and Linearization
4.7 Using L'Hopital's Rule for Determining Limits of
Indeterminate Forms
Review - Unit 4
Unit 5 - Analytical Applications of Differentiation
5.1 Using the Mean Value Theorem
5.2 Extreme Value Theorem, Global Versus Local
Extrema, and Critical Points
5.3 Determining Intervals on Which a Function is
Increasing or Decreasing
5.4 Using the First Derivative Test to Determine Relative
Local Extrema
5.5 Using the Candidates Test to Determine Absolute
(Global) Extrema
5.6 Determining Concavity of Functions over Their
Domains
5.7 Using the Second Derivative Test to Determine
Extrema
Mid-Unit Review - Unit 5
5.8 Sketching Graphs of Functions and Their Derivatives
5.9 Connecting a Function, Its First Derivative, and Its
Second Derivative
(5.9 includes a revisit of particle motion and
determining if a particle is speeding up/down.)
5.10 Introduction to Optimization Problems
5.11 Solving Optimization Problems
5.12 Exploring Behaviors of Implicit Relations
Review - Unit 5
Unit 6 - Integration and Accumulation of Change
6.1 Exploring Accumulation of Change
6.2 Approximating Areas with Riemann Sums
6.3 Riemann Sums, Summation Notation, and Definite
Integral Notation
6.4 The Fundamental Theorem of Calculus and
Accumulation Functions
6.5 Interpreting the Behavior of Accumulation Functions
​ Involving Area
Mid-Unit Review - Unit 6
6.6 Applying Properties of Definite Integrals
6.7 The Fundamental Theorem of Calculus and Definite
Integrals
6.8 Finding Antiderivatives and Indefinite Integrals:
Basic Rules and Notation
6.9 Integrating Using Substitution
6.10 Integrating Functions Using Long Division
​ and Completing the Square
6.11 Integrating Using Integration by Parts (BC topic)
6.12 Integrating Using Linear Partial Fractions (BC topic)
6.13 Evaluating Improper Integrals (BC topic)
6.14 Selecting Techniques for Antidifferentiation
Review - Unit 6
Unit 7 - Differential Equations
7.1 Modeling Situations with Differential Equations
7.2 Verifying Solutions for Differential Equations
7.3 Sketching Slope Fields
7.4 Reasoning Using Slope Fields
7.5 Euler's Method (BC topic)
7.6 General Solutions Using Separation of Variables
7.7 Particular Solutions using Initial Conditions and
Separation of Variables
7.8 Exponential Models with Differential Equations
7.9 Logistic Models with Differential Equations (BC topic)
Review - Unit 7
Unit 8 - Applications of Integration
8.1 Average Value of a Function on an Interval
8.2 Position, Velocity, and Acceleration Using Integrals
8.3 Using Accumulation Functions and Definite Integrals
in Applied Contexts
8.4 Area Between Curves (with respect to x)
8.5 Area Between Curves (with respect to y)
8.6 Area Between Curves - More than Two Intersections
Mid-Unit Review - Unit 8
8.7 Cross Sections: Squares and Rectangles
8.8 Cross Sections: Triangles and Semicircles
8.9 Disc Method: Revolving Around the x- or y- Axis
8.10 Disc Method: Revolving Around Other Axes
8.11 Washer Method: Revolving Around the x- or y- Axis
8.12 Washer Method: Revolving Around Other Axes
8.13 The Arc Length of a Smooth, Planar Curve and
Distance Traveled (BC topic)
Review - Unit 8
Unit 9 - Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC topics)
9.1 Defining and Differentiating Parametric Equations
9.2 Second Derivatives of Parametric Equations
9.3 Arc Lengths of Curves (Parametric Equations)
9.4 Defining and Differentiating Vector-Valued Functions
9.5 Integrating Vector-Valued Functions
9.6 Solving Motion Problems Using Parametric and
Vector-Valued Functions
9.7 Defining Polar Coordinates and Differentiating in
Polar Form
9.8 Find the Area of a Polar Region or the Area Bounded
by a Single Polar Curve
9.9 Finding the Area of the Region Bounded by Two
Polar Curves
Review - Unit 9
Unit 10 - Infinite Sequences and Series (BC topics)
10.1 Defining Convergent and Divergent Infinite Series
10.2 Working with Geometric Series
10.3 The nth Term Test for Divergence
10.4 Integral Test for Convergence
10.5 Harmonic Series and p-Series
10.6 Comparison Tests for Convergence
10.7 Alternating Series Test for Convergence
10.8 Ratio Test for Convergence
10.9 Determining Absolute or Conditional Convergence
10.10 Alternating Series Error Bound
10.11 Finding Taylor Polynomial Approximations of
Functions
10.12 Lagrange Error Bound
10.13 Radius and Interval of Convergence of Power
Series
10.14 Finding Taylor Maclaurin Series for a Function
10.15 Representing Functions as a Power Series
Review - Unit 8

Version #2
​The course below covers all topics for the AP Calculus AB exam, but was built for a 90-minute class that meets every other day.
Lessons and packets are longer because they cover more material.

Unit 0 - Calc Prerequisites (Summer Work)
0.1 Things to Know for Calc
0.2 Summer Packet
0.3 Calculator Skillz
Unit 1 - Limits
1.1 Limits Graphically
1.2 Limits Analytically
1.3 Asymptotes
1.4 Continuity
Review - Unit 1
Unit 2 - The Derivative
2.1 Average Rate of Change
2.2 Definition of the Derivative
2.3 Differentiability [Calculator Required]
​ Review - Unit 2
Unit 3 - Basic Differentiation
3.1 Power Rule
3.2 Product and Quotient Rules
3.3 Velocity and other Rates of Change
3.4 Chain Rule
3.5 Trig Derivatives
Review - Unit 3
Unit 4 - More Deriviatvies
4.1 Derivatives of Exp. and Logs
4.2 Inverse Trig Derivatives
4.3 L'Hopital's Rule
Review - Unit 4
Unit 5 - Curve Sketching
5.1 Extrema on an Interval
5.2 First Derivative Test
5.3 Second Derivative Test
Review - Unit 5
Unit 6 - Implicit Differentiation
6.1 Implicit Differentiation
6.2 Related Rates
6.3 Optimization
​ Review - Unit 6
Unit 7 - Approximation Methods
7.1 Rectangular Approximation Method
7.2 Trapezoidal Approximation Method
Review - Unit 7
Unit 8 - Integration
8.1 Definite Integral
8.2 Fundamental Theorem of Calculus (part 1)
8.3 Antiderivatives (and specific solutions)
Review - Unit 8
Unit 9 - The 2nd Fundamental Theorem of Calculus
9.1 The 2nd FTC
9.2 Trig Integrals
9.3 Average Value (of a function)
9.4 Net Change
Review - Unit 9
Unit 10 - More Integrals
10.1 Slope Fields
10.2 u-Substitution (indefinite integrals)
10.3 u-Substitution (definite integrals)
10.4 Separation of Variables
Review - Unit 10
Unit 11 - Area and Volume
11.1 Area Between Two Curves
11.2 Volume - Disc Method
11.3 Volume - Washer Method
​ 11.4 Perpendicular Cross Sections
Review - Unit 11
  1. If you want to learn differential equations, have a look at Differential Equations for Engineers If your interests are matrices and elementary linear algebra, try Matrix Algebra for Engineers If you want to learn vector calculus (also known as multivariable calculus, or calcu-lus three), you can sign up for Vector Calculus for Engineers.
  2. Maths Book back answers and solution for Exercise questions - Mathematics: Differential Calculus- Differentiability and Methods of Differentiation.
  3. Differential equations have a derivative in them. For example, dy/dx = 9x. In elementary algebra, you usually find a single number as a solution to an equation, like x = 12. But with differential equations, the solutions are function.

Learning Objectives

In this section, we strive to understand the ideas generated by the following important questions:

  • How can we use differential equations to describe phenomena in the world around us?
  • How can we use differential equations to better understand these phenomena?

Differential Equations 2019 AB4/BC4 Rain barrel: A cylindrical barrel collects rainwater, with questions relating the rates of the water height and volume, and a separable differential equation to solve explicitly for the height as a function of time t.

In our work to date, we have seen several ways that differential equations arise in the natural world, from the growth of a population to the temperature of a cup of coffee. In this section, we will look more closely at how differential equations give us a natural way to describe various phenomena. As we’ll see, the key is to focus on understanding the different factors that cause a quantity to change.

Preview Activity (PageIndex{1})

Any time that the rate of change of a quantity is related to the amount of a quantity, a differential equation naturally arises. In the following two problems, we see two such scenarios; for each, we want to develop a differential equation whose solution is the quantity of interest.

  1. Suppose you have a bank account in which money grows at an annual rate of 3%.
    1. If you have $10,000 in the account, at what rate is your money growing?
    2. Suppose that you are also withdrawing money from the account at $1,000 per year. What is the rate of change in the amount of money in the account? What are the units on this rate of change?
  2. Suppose that a water tank holds 100 gallons and that a salty solution, which contains 20 grams of salt in every gallon, enters the tank at 2 gallons per minute.
    1. How much salt enters the tank each minute?
    2. Suppose that initially there are 300 grams of salt in the tank. How much salt is in each gallon at this point in time?
    3. Finally, suppose that evenly mixed solution is pumped out of the tank at the rate of 2 gallons per minute. How much salt leaves the tank each minute?
    4. What is the total rate of change in the amount of salt in the tank?

Developing a Differential Equation

Preview activity (PageIndex{1}) demonstrates the kind of thinking we will be doing in this section. In each of the two examples we considered, there is a quantity, such as the amount of money in the bank account or the amount of salt in the tank, that is changing due to several factors. The governing differential equation results from the total rate of change being the difference between the rate of increase and the rate of decrease.

Example (PageIndex{1}): Lake Michigan

In the Great Lakes region, rivers flowing into the lakes carry a great deal of pollution in the form of small pieces of plastic averaging 1 millimeter in diameter. In order to understand how the amount of plastic in Lake Michigan is changing, construct a model for how this type pollution has built up in the lake.

Solution

First, some basic facts about Lake Michigan.

  • The volume of the lake is (5 times 10^{12}) cubic meters.
  • Water flows into the lake at a rate of (5 times 10^{10}) cubic meters per year. It flows out of the lake at the same rate.
  • Each cubic meter flowing into the lake contains roughly (3 times 10^{−8}) cubic meters of plastic pollution.

Let’s denote the amount of pollution in the lake by (P(t)), where (P) is measured in cubic meters of plastic and (t) in years. Our goal is to describe the rate of change of this function; in other words, we want to develop a differential equation describing (P(t)).

First, we will measure how (P(t)) increases due to pollution flowing into the lake. We know that (5 times 10^10) cubic meters of water enters the lake every year and each cubic meter of water contains (3 times 10^{−8}) cubic meters of pollution. Therefore, pollution enters the lake at the rate of

(left(5 cdot 10^{10} dfrac{m^3 text{water}}{text{year}}right)cdotleft(3 cdot 10^{-8} dfrac{m^3 text{plastic}}{m^3 text{water}}right) = 1.5 cdot 10^{3})

Second, we will measure how (P(t)) decreases due to pollution flowing out of the lake. If the total amount of pollution is (P) cubic meters and the volume of Lake Michigan is (5 cdot 10^{12}) cubic meters then the concentration of plastic pollution in Lake Michigan is

(frac{P}{5 cdot 10^{12}}) cubic meters of plastic per cubic meter of water.

Since (5 cdot 10^{10}) cubic meters of water flow out each year, then the plastic pollution leaves the lake at the rate of

(left(dfrac{P}{5cdot10^{12}} dfrac{m^3 text{plastic}}{m^3 text{water}}right)cdotleft(5cdot10^{10} dfrac{m^3 text{water}}{text{year}}right)= dfrac{P}{100}) cubic meters of plastic per cubic meter of water.

The total rate of change of (P) is thus the difference between the rate at which pollution enters the lake minus the rate at which pollution leaves the lake; that is,

(begin{align} dfrac{dP}{dt} &= 1.5 cdot 10^3 − dfrac{P}{100} &= dfrac{1}{ 100} (1.5 cdot 10^5 − P). end{align} )

We have now found a differential equation that describes the rate at which the amount of pollution is changing. To better understand the behavior of (P(t)), we now apply some of the techniques we have recently developed.

Since this is an autonomous differential equation, we can sketch (dP/dt) as a function of (P) and then construct a slope field, as shown in Figure (PageIndex{1}).

Figure (PageIndex{1}): Plots of ( dfrac{dP}{dt}) vs. (P) and the slope field for the differential equation (dfrac{dP}{dt} = dfrac{1}{100} (1.5 cdot 10^5 − P)).

These plots both show that (P = 1.5 cdot 10^5) is a stable equilibrium. Therefore, we should expect that the amount of pollution in Lake Michigan will stabilize near (1.5 cdot 10^5) cubic meters of pollution. Next, assuming that there is initially no pollution in the lake, we will solve the initial 6 and we assume that each cubic meter of water that flows out carries with it the plastic pollution it contains value problem

(dfrac{dP}{dt} = dfrac{1}{100} (1.5 cdot 10^5 − P)), (P(0) = 0).

Separating variables, we find that

( dfrac{1}{1.5 cdot 10^5− P} dfrac{dP}{dt} = dfrac{1}{100} .)

Integrating with respect to (t), we have

(int dfrac{1}{1.5 cdot10^5− P} dfrac{dP}{dt} dt = int dfrac{1}{100} dt )

and thus changing variables on the left and antidifferentiating on both sides, we find that

(int dfrac{dP}{1.5 cdot 10^5 P } = int dfrac{1}{100} dt )

(- ln |1.5 cdot 10^5 - P | = dfrac{1}{100}t + C)

Finally, multiplying both sides by −1 and using the definition of the logarithm, we find that

(1.5cdot 10^5 − P = Ce^{−t/100}. tag{7.1}label{7.1} )

This is a good time to determine the constant (C). Since (P = 0) when (t = 0), we have

( 1.5 cdot 10^5 -0 = Ce^0 = C.)

In other words, ( C = 1.5 times 10^5)

Using this value of (C) in Equation ((ref{7.1})) and solving for (P), we arrive at the solution

(P(t) = 1.5 cdot 10^5 (1 - e^{-t/100}))

Superimposing the graph of P on the slope field we saw in Figure (PageIndex{1}), we see, as shown in Figure (PageIndex{2}) We see that, as expected, the amount of plastic pollution stabilizes around (1.5 cdot 10^5) cubic meters.

There are many important lessons to learn from Example (PageIndex{1}). Foremost is how we can develop a differential equation by thinking about the “total rate = rate in - rate out” model. In addition, we note how we can bring together all of our available understanding (plotting ( dfrac{dP}{dt}) vs. (P), creating a slope field, solving the differential equation) to see how the differential equation describes the behavior of a changing quantity.

Of course, we can also explore what happens when certain aspects of the problem change. For instance, let’s suppose we are at a time when the plastic pollution entering

Figure (PageIndex{2}):The solution ( P(t)) and the slope field for the differential equation (dfrac{dP}{dt}= dfrac{1}{100} (1.5 times 10^5 − P)).

Lake Michigan has stabilized at (1.5 times 10^5) cubic meters, and that new legislation is passed to prevent this type of pollution entering the lake. So, there is no longer any inflow of plastic pollution to the lake. How does the amount of plastic pollution in Lake Michigan now change? For example, how long does it take for the amount of plastic pollution in the lake to halve?

Restarting the problem at time (t = 0), we now have the modified initial value problem

(dfrac{dP}{dt} = − dfrac{1}{100} P), (P(0) = 1.5 cdot 10^5.)

It is a straightforward and familiar exercise to find that the solution to this equation is (P(t) = 1.5 cdot 10^5 e^{−t/100}.) The time that it takes for half of the pollution to flow out of the lake is given by (T) where (P(T) = 0.75 cdot 10^5). Thus, we must solve the equation

Differential Equations Solver

10.5 Differential Equationsap Calculus

(0.75 times 10^5 = 1.5 times 10^5 e −T/100 , )

or

(dfrac{1}{2} = e^{−T/100} .)

It follows that

(T = −100 ln left(dfrac{1}{2}right) approx 69.3), years.

In the upcoming activities, we explore some other natural settings in which differential equation model changing quantities.

Activity (PageIndex{1}): Accrued Savings

10.5 Differential Equations Ap Calculus Multiple Choice

Suppose you have a bank account that grows by 5% every year. Let (A(t)) be the amount of money in the account in year ( t).

  1. What is the rate of change of (A) with respect to (t)?
  2. Suppose that you are also withdrawing $10,000 per year. Write a differential equation that expresses the total rate of change of (A).
  3. Sketch a slope field for this differential equation, find any equilibrium solutions, and identify them as either stable or unstable. Write a sentence or two that describes the significance of the stability of the equilibrium solution.
  4. Suppose that you initially deposit $100,000 into the account. How long does it take for you to deplete the account?
  5. What is the smallest amount of money you would need to have in the account to guarantee that you never deplete the money in the account?
  6. If your initial deposit is $300,000, how much could you withdraw every year without depleting the account?

Differential Equations Examples

Activity (PageIndex{2}): Morphine

A dose of morphine is absorbed from the bloodstream of a patient at a rate proportional to the amount in the bloodstream.

10.5 Differential Equations Ap Calculus Solver

  1. Write a differential equation for (M(t)), the amount of morphine in the patient’s bloodstream, using (k) as the constant proportionality.
  2. Assuming that the initial dose of morphine is (M_0), solve the initial value problem to find (M(t)). Use the fact that the half-life for the absorption of morphine is two hours to find the constant (k).
  3. Suppose that a patient is given morphine intravenously at the rate of 3 milligrams per hour. Write a differential equation that combines the intravenous administration of morphine with the body’s natural absorption.
  4. Find any equilibrium solutions and determine their stability.
  5. Assuming that there is initially no morphine in the patient’s bloodstream, solve the initial value problem to determine (M(t)). What happens to (M(t)) after a very long time?
  6. To what rate should a doctor reduce the intravenous rate so that there is eventually 7 milligrams of morphine in the patient’s bloodstream?

Summary

In this section, we encountered the following important ideas:

  • Differential equations arise in a situation when we understand how various factors cause a quantity to change.
  • We may use the tools we have developed so far—slope fields, Euler’s methods, and our method for solving separable equations—to understand a quantity described by a differential equation.

Contributors and Attributions

10.5 Differential Equationsap Calculus Transcendentals

Matt Boelkins (Grand Valley State University), David Austin (Grand Valley State University), Steve Schlicker (Grand Valley State University)