9.3: Separable Equations

Objectives:

• Solve separable differential equations.

Definitions & Formulas:
Separable Equations

Flashcards:
N/A

Examples: 
A short explanation.




Ex1. Solving a separable equation.


Ex2. More separable equations.


student example 1


Ex3. Mixing problems.


student example 2


9.3 Problem Set:
#1-5, 11-14, 16, 19, 20, 42

6.5: Average Value of a Function

Objectives:

• Interpret the Mean Value Theorem for Integrals geometrically.

• Find the average value of a continuous function.

Definitions and Formulas:

1. Average Value of a Function
2. The Mean Value Theorem for Integrals

Flashcards:

1. The Mean Value Theorem for Integrals

Examples:

An explanation of the Mean Value Theorem for Integrals

Ex1. Finding the average value of a function.

Ex2. Average velocity.

Student example


Problem Set:

# 1, 4, 7, 10, 13, 16, 22

6.3: Volumes by Cylindrical Shells

Objectives:

• Find the volume of a figure using the cylindrical shells method.

Read:
pgs. 433-436

Definitions/Formulas:
Definition of volume by cylindrical shells (formula)

Flashcards:
Definition of volume by cylindrical shells (formula)

Examples (will be posted 3/16/12):

Ex1. cylindrical shells


Ex2. more cylindrical shells


Ex3. even more cylindrical shells!


student example


Problem set 6.3
#2, 4, 5, 11, 20, 26, 33, 38, 43

6.2b: Volumes

Objectives:

• Use cross sections to find volumes of revolution (integrating with respect to y).

Read:
pg. 425 (example 3)
pg. 428 (example 6)

Definitions/ Formulas:
Definition of Volume

Flashcards:
Definition of Volume

Examples

Ex1. Volume of a solid of revolution (about the y-axis)

student example

Ex2. Volume of hollow objects of revolution (about the y-axis)

student example



Problem Set: #5, 9, 15, 21, 22, 24

6.2a: Volumes

Objectives:

• Use cross sections to find volumes of revolution (integrating with respect to x).
• Find volumes of other solids using integration.

Read:
pgs. 422 - 425 (through example 2)
pgs. 426-427
pgs. 428 - 429 (examples 7, 8)

Definitions/ Formulas:
Definition of Volume

Flashcards:
Definition of Volume

Examples

Ex1. Volume of a solid of revolution (pg. 424 ex2)

student example

Ex2. Volume of hollow objects (pg. 426 ex4)


Ex3. Volume of shifted hollow objects (pg. 426 ex5)


student example

Ex4. Volumes of other solids

student example


Problem Set: #1-4, 11, 14, 27, 30, 49, 58

6.1b: The Area Between Two Curves

Read:
Pgs. 418-419

Definitions/Formulas:
Definition 3 of the Area Between Two Curves

Flashcards:
Definition 3 of the Area Between Two Curves

Examples:

Ex1. Area between intersecting functions.

Student Example


Ex2. Integrating with respect to the y-axis.

Student Example


6.1b Problem Set:
# 2, 19, 20, 23, 29

6.1a: Area Between two Curves

Objectives:

• Determine the area betwen the graphs of two functions.

Read:

• pgs. 415 - 417

Definitions & Formulas:

• Sigma definition of the area between two curves.

• Integral definition for the area between two curves.

Flashcards:

• Integral definition for the area between two curves.

Examples:


Explanation of the Area Between Two Curves


Ex1. Basic area between two curves.


Student example



Ex2. Using your calculator to establish limits.



Student example.


6.1a problem set:
#5, 6, 12, 33, 36, 41

7.1b: Integration by Parts (Definite Integrals)

Objectives:

• Calculate definite integrals using integration by parts.

Read:

• pgs. 456 - 457

Definitions & Formulas:

• Integration by Parts formula for definite integrals.

Flashcards:

• Integration by Parts formula for definite integrals.

Examples:

Ex1. Using IBP for Definite Integrals (with substitution)

Student example


7.1b problem set:
#19, 20, 23, 24

7.1a: Integration by Parts (indefinite integrals)

Objectives:

• Define Integration by Parts

• Apply integration by parts in finding general indefinite integrals.

Read:

• pgs. 453-456

Definitions & Formulas:

• Integration by Parts Formulas 1 & 2

Flashcards:

• Integration by Parts Formulas 1 & 2

Examples:

A quick explanation of integration by parts.


Ex1. Applying integration by parts

Student example


Ex2. More integration by parts.

Student example


7.1a problem set:
# 1, 6, 9, 15, 29, 34, 38, 40

5.5b: The Substitution Rule for Definite Integrals

Objectives:
Apply the substitution rule when evaluating definite integrals.

Read:
pgs. 403 (starting at "Definite Integrals") - 406

Definitions & Formulas:
The Substitution Rule for Definite Integrals
Integrals of Symmetric Functions

Flashcards:
N/A

Examples:

Ex1. Evaluting definite integrals using the substitution rule.

Student example


Ex2. Evaluating definite integrals using symmetry.

Student example


5.5b Problem Set:
# 53, 58, 66, 69, 73, 78, 82

5.5a: The Substitution Rule (with indefinite integrals)

Objectives:

• Apply a u substitution to find the indefinite integral of a composition of functions.

Read:

• pg. 400 - 403 (through example 6)

Definitions & Formulas:

• The Substitution Rule

Flashcards:
N/A

Examples:

Ex1. Using "u - substitution" (with differentials)

Student Example


Ex2. More complex u - substitution

Student Example


5.5a Problem Set:
# 2, 5, 10, 16, 23, 28, 42, 47

5.4: Indefinite Integrals

Objectives:

• Find antiderivatives of given functions.

• Find specific antiderivatives given initial conditions.

Read:

• pgs. 391-396

Definitions& Formulas:

• Indefinite integral

• Net Change Theorem

Flashcards:

• Everything in the table of indefinite integrals (pg. 392)

• Net Change Theorem

Examples:

Ex1. Evaluating an indefinite integral

Student example


Explanation of Net Change Theorem


Ex2. Applying the Net Change Theorem
(in class)

Problem Set 5.4:

2, 7, 10, 12, 18, 19, 21, 24, 29, 37, 42, 47, 52, 59, 65

5.3b: The Fundamental Theorem of Calculus Part II (FTC II)

Objectives:

• Define and apply the FTC II in evaluating definite integrals.

Read:

• pg. 384 - 387

Definitions:

• The Fundamental Theorem of Calculus Part II (FTC II)

• Differentiation and Integration as inverse processes

Formulas:
N/A

Flashcards:

• The Fundamental Theorem of Calculus Part II (FTC II)

Examples:

A short explanation of FTC II


Ex1. Evaluating a definite integral

Student example


Ex2. Finding the area under a curve

Student example


5.3b Practice Set:
#19, 22, 30, 32, 44, 51, 52

5.3a: The Fundamental Theorem of Calculus (Part I)

Objectives:

• Define FTC I
• Apply FTC I in finding derivatives of "accumulation functions".

Read:

• pg. 379 - 384 (through example 4)

Definitions:

• The Fundamental Theorem of Calculus Part I (FTC I)

Formulas:N/A

Flashcards:

• FTC I

Examples:

A proof(ish) of FTC I


Ex1. Applying FTC I

Student example


Ex2. Finding the Derivative of an accumulation function

Student Example


Ex3. FTC I with the Chain Rule

Student Example


Practice Set 5.3a:
# 3, 6a, 8, 9, 14, 17, 58

Calculus with calculators

Gratuitous baby picture!

 

Hey folks, I hope class is going well. Keep taking good notes and working on the practice problems in class. I'll be back next week!

Objectives:

• Become familiar with the functions of the TI83 / TI84 calculators and their applications to Calculus.

Lesson:

1. Read through and complete the following lessons taken from the Texas Instruments website.
2. Work together in your groups to complete each lesson and corresponding self test questions in your notes composition book.
3. You will work on these lessons in class for the next two (1/31 & 2/2) class periods, and take a calculator quiz this Friday.
4. In addition to these lessons there are many more available for free at: 
   http://education.ti.com/html/t3_free_courses/calculus84_online/index.html
   if you would like to become more effective and efficient when using your calculator.

• Module 2, lesson 2.2 (scatterplots)

• Self test questions 4, 5, 6

• Module 10, lesson 10.1 (derivative of a function at a point using the nDerive function)

• Self test question 3

• Module 12, lesson 12.1, 12.2 (Rules of differentiation)

• Self test questions 2, 3, 4, 5

• Module 13, lesson 13.4 (extreme values with the TI84)

• Self test questions: 5

• Module 17, lesson 17.1, 17.2, 17.3 (Definite Integrals)

•  Self test questions: 1, 2, 3, 4, 5

• Module 19, lesson 19.1, 19.2 (Net Area / Area between two curves)

• Self test questions: 1, 2, 3, 4

7.7: Approximate Integration

Objectives:

• Use Riemann and Trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically and by tables of values.

Read:
pgs. 495 - 500 (stop at Simpson's Rule)

Definitions:

• Error

Formulas:

• The Trapezoidal Rule

• Error Bounds (of both Trapezoidal and Midpoint Rules)

Flashcards:

• The Trapezoidal Rule

Examples:
Explanation of Trapezoidal approximation


Ex1. The Midpoint Rule (refer to 5.2a ex4)

Ex2. The Trapezoidal Rule

Student Example


Ex3. Error Bounds

Student Example


Practice Set:
7.7 #1, 8a/b,13a/b, 20

Reminder:
Quiz 5.1, 5.2, 7.7 Monday

5.2b: The Definite Integral

Read:

• pg. 373 - 376
  

Definitions:

• Properties of the Definite Integral (2 red box mid-page 373)
• Properties of the Integral ( 5 properties pgs. 373-374)
• Comparison Properties of the Integral (pg. 375)

Formulas:

 N/A

Flashcards:

• Properties of the Definite Integral (2 red box mid-page 373)
• Properties of the Integral ( 5 properties pgs. 373-374)
• Comparison Properties of the Integral (pg. 375)
  

Examples: 

Ex1. Comparison properties of integrals

Student Example

Ex2. Using properties of integrals to evaluate

Student Example

Ex3. Using the upper and lower limit properties to evaluate an integral

Student Example

Practice Problems:

5.2b: # 33, 36,41, 43, 47, 52, 57

5.2a: The Definite Integral

Read:
 pg 366 - 372 (Through the Midpoint Rule)


Defintions:

• Definition of a definite integral (also called Riemann sum)
• Theorem 3
• Theroem 4

Formulas:

•  Sigma Formulas (7 of them on pg. 369)
• The Midpoint Rule

Flashcards:

• Theorem 4
• Front: Integral
• Back: Sigma notation / delta x and x_i
•  The Midpoint Rule

Examples:

Ex1.  Expressing a Riemann sum as an integral

Student example *there is a mistake in the final answer, see the comments below for the correction*

Ex2a. Evaluating Riemann sums

Student Example

Ex2b. Evaluating integrals using Theorem 4 and the Sigma Properties

Student Example

Ex3. Evaluating integrals by interpreting areas.

Student Example

Ex4. The Midpoint Rule

Student Example



Practice Problems:
5.2a #1, 6, 8, 9, 18, 23, 26, 33, 36

5.1b: The Distance Problem

Read:
5.1b pg. 362 -363 "The Distance Problem". 

Definitions: 
N/A

Formulas:
N/A

Flashcards:
N/A

Examples:

The Distance Problem (an explanation)


Running the distance



Practice Problems:
5.1 # 12, 15, 18  

5.1a: The Area Problem

Read:
 pg. 354 - 362 (Stop at "The Distance Problem)

Define:

1. Area of a region S.
2. Sample points
3. Sigma notation
4. Sum of the first n squares formula

Examples:
A note on sigma notation.

Ex1.   Approximate Area Under a Curve (ex1 in the book).

Student Example

Ex2. Sigma Notation/ Area Under a Curve

Student Example

Practice Problems (in class Friday):
5.1 # 2, 5, 18, 19

Hello World!

Hello AP Calculus students,

This is where you will come to find the lesson plan notes, and video examples for each section. Please email me with any questions at robert.tolar@kippking.org.