AP® Calculus AB College Board Syllabus 2007

Contents • Course Overview

• Chapter Outline

• Timeline

• Chapters

• Chapter Descriptions

• Section Description

• Objectives by Sections

• Teaching Strategies

• Technology and Computer Software

• Student Evaluation

• Teaching Resource

Course Overview

This course will cover all sections outlined in the AP® Calculus AB Course Description and is intended to expand the student’s comprehension of the concepts of calculus, its ideas, its methods and applications. The primary textbook is Calculus of a Single Variable, 8th edition by Larson, Hostler and Edwards. This class of calculus is taught as a whole subject and does not diverge into other mathematical ideas. The instructive methodology of this course will be done numerically, graphically, analytically and verbally. The main objectives in teaching this course are to make it possible for students to receive a deep footing in calculus skills equal in content to the best collegiate courses and to provide students with the tools needed to succeed in taking the AP® Calculus Test.

The following outline is our AP Calculus AB course, complete with a brief description of each chapter, a concise account of each section, and a timeline. This course will educate students in all topics associated with graphs, functions, limits, differentiation and integration as defined in the AB Calculus Topic Outline in the AP® Calculus Course Description.

Course Outline and Timeline

Chapter P

P Preparation for Calculus, (Review) Days:6 Although this information is not tested on the AP exam, it is used for a smooth

transition into Calculus by helping students to hone their mathematic skills needed in this program. There will be a brief review on how to find, graph and compare mathematical models for different sets of data.

P.1 Graphs and Models 1

Students will sketch the graphs of equations, find their intercepts, test graphs for symmetry with respect to an axis and the origin, and find the points of intersection of two graphs.

P.1.1 Graphs of an Equation P.1.2 Intercepts P.1.3 Symmetry P.1.4 Points of Intersection P.2 Linear Models and Rates of Change 1

Find the slope of a line passing through two points. Write the equation of a line with a given point and slope. Interpret slope as a ratio or as a rate in a real life application. Sketch the graph of a linear equation in slope-intercept form. Write equations of lines that are parallel or perpendicular to a given line.

P.2.1 Slope of a line P.2.2 Equations of lines; general, slope-intercept and point slope forms P.2.3 Ratios and Rates of Change P.2.4 Parallel and Perpendicular lines P.3 Functions and their graphs 1

This section examines coverage of notation, terminology, classification, algebraic methods and transformation of functions. Possible new concepts include odd/even function.

P.3.1 Functions and function notation P.3.2 Domain & Range P.3.3 Graphs of a function P.3.4 Transformation and Combinations of functions P.3.5 Composite functions P.3.6 Even & Odd functions P.4 Fitting Models to Data and Regression Equations 1

This section deals primarily with the use of entering data, plotting lines, and finding regression equations using calculators. This application will be useful in several calculus lessons.

P.4.1 Fitting a linear model P.4.2 Fitting a quadratic model P.4.3 Fitting a Trigonometric model Review and Assessment 2

Chapter 1 1 Limits and Their Properties Days:13

This chapter bridges the gap between pre-calculus and calculus as we look at the limits of a function and how it compares to its graph. Different limits will be studied such as their continuity, one-sided and infinite limits.

1.1 A Preview of Calculus 2 This lesson compares non-calculus to calculus examples and provides the basis for calculus by explaining the tangent line and area problem. 1.1.1 What is Calculus? Mathematics of change.

1.1.2 The Tangent Line Problem. sec( ) ( )f c x f cm

x+ Δ −

=Δ

1.1.3 The Area Problem. To find the area under the curve.

1.2 Find Limits Graphically and Numerically 2 This section shows how limits can be understood graphically, numerically and how they can fail to exist. Also included in this understanding are infinite and finite limits and their relationship to the graph of the function along with horizontal and vertical asymptotes. The formal definition of ε – δ is also given because of its important fundamental idea.

1.2.1 Introduction to Limits; lim ( )

x cf x L

→=

1.2.2 Limits that fail to exist; a) f(x) approaches a different number from the right side of c than it approaches

from the left side. b) f(x ) increases or decreases without bound as x approaches c. c) f(x) oscillates between two fixed values as x approaches c

1.2.3 Definition of a Limit, ε – δ definition of a limit. Let f be a function defined on (a,b) containing c and let L be a real number. The statement lim ( )

x cf x L

→= means that for each ε > 0 there exist a δ > 0 such that if 0 <

x c− < δ then ( )f x L− < ε

1.3 Evaluating Limits Analytically 2

Limits are taken to the next step of being solved analytically along with much emphasis placed on the indeterminate forms of solution. Also importance will be stressed with the Squeeze Theorem and the Trigonometric Limits Theorem.

1.3.1 Properties of Limits; including composite and trigonometric functions; Theorems

1.1-1.7 1.3.2 Finding limits using; cancellation and rationalization techniques, (using conjugates) 1.3.3 The Squeeze Theorem, Theorem 1.8 1.3.4 Special Trigonometric Limits, Theorem 1.9

1.4 Continuity and One-Sided limits 2

Various functions are not continuous when a limit fails. This section shows the conditions functions must meet to be continuous and how to handle limits coming from the right and limits coming from the left.

1.4.1 Continuity at a point and on an Open Interval

Definition of Continuity: A function f is continuous at c if the following three conditions are met. 1) f(c) is defined. 2) lim ( )

x cf x

→exists.

3) lim ( )x c

f x→

= f(c)

1.4.2 One-Sided Limits and Continuity on a Closed Interval;

- lim ( )x c

f x L+→

=

- lim ( )x c

f x L−→

=

- 0

lim 0n

xx

+→=

- x , greatest integer function 1.4.3 The Existence of a Limit, Theorem 1.10, definition on a closed interval 1.4.4 Continuity of a Composite functions, Theorem 1.11-1.12 1.4.5 Intermediate Value Theorem, Theorem 1.13

1.5 Infinite Limits 2

This section deals with infinite limits and how they correspond to vertical, horizontal and slant asymptotes.

1.5.1 Infinite Limits, is a limit in which f(x) increases or decreases without bound as x

approaches c. 1.5.2 Vertical Asymptotes, Theorem 1.14 1.5.3 Properties of Infinite Limits, Theorem 1.15

Review and Assessment 3

Chapter 2 2 Differentiation Days: 24

This chapter begins by expressing the problems arriving from the need to write the function of a tangent line on a curve and continues by solving these questions. Also this chapter explains how the use of derivatives determines an instantaneous rate of change and the velocity function. Finally, the remainder of this chapter shows helpful rules that need to be memorized and how to find the derivative of a function which will be tested on the AP exam.

2.1 The Derivative and the Tangent Line Problem 5

One of the prime ideas of the study of calculus begins by observing the slope of the secant line and how, as it moves closer to the tangent line, it is defined as the derivative of the function at the point of tangency. This new function, described as the derivative at a point, is the slope of the tangent line at that point.

2.1.1 The Tangent line problem.

Definition of Tangent Line with Slope m; If f is defined on an open interval

containing c, and if the limit 0 0

( ) ( )lim limx x

y f c x f c mx xΔ → Δ →

Δ + Δ −= =

Δ Δ exist, then the line

passing through (c,f(c)) with slope m is the tangent line to the graph of f at the point (c,f(c).

2.1.2 The Derivative of a Function, The derivative of f at x is given by

0

( ) ( )( ) limx

f x xf x f xxΔ →

+ Δ′ =Δ

− , provided the limit exists.

2.1.3 Differentiability Implies Continuity, Theorem 2.1

2.2 Differentiation Rules and Rates of Change 4 This section reveals much usefulness of differentiation by applying to the principles of instantaneous rate of change of an object in motion and finding velocity which is the derivative of the position function. 2.2.1 The Constant Rule, Theorem 2.2 2.2.2 The Power Rule, Theorem 2.3 2.2.3 The Constant Multiple Rule, Theorem 2.4 2.2.4 The Sum and Difference Rules, Theorem 2.5 2.2.5 Derivative of Sine and Cosine Functions, Theorem 2.6 2.2.6 Rates of Change

Average velocity: change in distancechange in time

st

Δ=Δ

Velocity function: ( ) ( )( ) lim ( )t o

s t t s tv t s ttΔ →

+ Δ − ′= =Δ

Position function: 20 0

1( )2

s t gt v t s= + +

2.3 The Product and Quotient Rule, and to rewrite functions. 3

The concepts of derivatives are made easier by mastering the rules of this section along with higher order derivatives and their notations. Building from the previous section, the derivative of velocity then is found to be the function for acceleration.

2.3.1 The Product Rule, Theorem 2.7 2.3.2 The Quotient Rule, Theorem 2.8 2.3.3 Derivatives of Trigonometric Functions, Theorem 2.9

2.3.4 Higher Order Derivatives, [ ] [, ( ), , ( ) , xdy dy f x f x D ydx dx

′ ′ ]

Acceleration Function: ( ) ( ) ( )a t v t s t′ ′′= =

2.4 The Chain Rule 3 The Chain Rule along with the General Power Rule are very powerful rules used as building blocks of finding derivatives of functions, which are weighed heavily on the AP exam. All the rules of derivatives learned up to this point and in this section should be memorized.

2.4.1 The Chain Rule, Theorem 2.10 2.4.2 The General Power Rule, Theorem 2.11 2.4.3 Simplifying Derivatives 2.4.4 Trigonometric Functions and the Chain Rule

2.5 Implicit Differentiation 3

When it is necessary to find the derivative of a relationship that can not be written as a function or it is not possible to solve analytically, isolate y as a function of x, then the concept of implicit differentiation is used along with the chain rule.

2.5.1 Implicit and Explicit Functions 2.5.2 Implicit Differentiations, follow guidelines;

1) Differentiate both sides of the equation with respect to x.

2) Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation.

3) Factor dy/dx out of the left side of the equation. 4) Solve for dy/dx by dividing both sides of the equation by the left-hand

factor that does not contain dy/dx.

2.6 Related Rates 3 When a relationship involves one quantity that is dependent on more than one other variable, then a methodical approach using related rates is needed. Much detail is stressed on setting up each question.

2.6.1 Finding Related Rates; Guidelines for Solving Related Rates

1) Identify all given quantities and quantities to be determined. 2) Write an equation involving the variables whose rates of change either

are given or are to be determined. 3) Using the chain rule, implicitly differentiate both sides of the equation

with respect to time t. 4) After completing step 3, substitute into the resulting equation all known

values for the variables and their rate of change. Then solve for the required rate of change.

Review and Assessment 3

Chapter 3

3 Applications of Differentiation Days: 29 This chapter puts derivatives to work by making a connection between using the

first and second derivatives and extremes. The first part of this chapter deals with the first and second derivatives and how the function acts on the plane. The characteristics of limits at infinity are graphically observed and then the applications of derivative through optimization problems are applied. Newton’s Method is touched upon and then the last section deals with differentials and approximating the y-value of a function near a known point.

3.1 Extrema on an Interval 3

The valuable concepts of extrema and the Extreme Value Theorem on the open and closed interval are covered in detail along with finding the maximum and minimums. Critical values of a function are also discussed as a valuable procedure for finding relative extrema. 3.1.1 Extrema of a Function

Definition of Extrema: Let f be defined on the interval I containing c. 1) f(x) is the minimum of f on I if f(c) ≤ f(x) for all x in I. 2) f(c) is the maximum of f on I if f(c) ≥ f(x) for all x in I. The minimum and maximum of a function on an interval are the extreme values, or extrema, of the function on the interval. The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum on I.

3.1.2 The Extreme Value Theorem, Theorem 3.1 3.1.3 Definition of relative Extrema;

1) If there is an open interval containing c on which f(c) is a maximum, then f(c) is called a relative maximum of f.

2) If there is an open interval containing c on which f(c) is a minimum, then f(c) is called a relative minimum of f. The plural of relative maximum is relative maxima, and the plural of relative minimum is relative minima.

3.1.4 Definition of Critical Number. Let f be defined at c. If ( )f c′ = 0 or if ( )f c′ is undefined at c, then c is a critical number of f.

3.1.5 Relative Extrema Occur Only at Critical Numbers. Theorem 3.2 3.1.6 Finding Extrema on a Closed Interval. Guidelines; To find the extrema of a

continuous function f on a closed interval [a,b], use the following steps. 1) Find the critical numbers of f in (a,b). 2) Evaluate f at each critical numbers in (a,b). 3) Evaluate f at each endpoint of [a,b]. 4) The least of these values is the minimum. The greatest is the maximum.

3.2 Rolle’s Theorem and the Mean Value Theorem 3

The graphical interpretation from the Rolle’s and the Mean Value Theorem will help students understand that on the function there is some point where the tangent line is parallel to the line connecting the endpoints of the interval. Show that the slope of the tangent line applying the Rolle’s Theorem is equal zero. 3.2.1 Rolle’s Theorem, Theorem 3.3 3.2.2 Mean Value Theorem, Theorem 3.4

3.3 Increasing & Decreasing Functions and the First Derivative Test 2

This section shows the function increasing and decreasing on the open and closed interval using its derivative. Students will learn to find relative extrema and the change of sign at the critical point. It is also stressed that if the derivative changes from positive to negative it indicates a local maximum, and if it changes from negative to positive indicates a minimum. 3.3.1 Test for Increasing and Decreasing Functions, Theorem 3.5 3.3.2 The First Derivative Test, Theorem 3.6

3.4 Concavity and the Second Derivative Test 2

Students will learn that concavity relates to the rate of change of the function. When concavity changes, given by the second derivative, students will see this gives us the point of inflection. Finally the Second derivative Test is an alternate to finding the critical points by finding the maximum or minimum. 3.4.1 Concavity: Test for Concavity, Theorem 3.7 3.4.2 Point of Inflection, Theorem 3.8 3.4.3 Second Derivative Test, theorem 3.9

3.5 Limits at Infinity 2 This section is very important for students as it shows what the function is doing at both extremes at infinity. Furthermore the section shows what type of functions have horizontal asymptotes and show easy guidelines to find them.

3.5.1 Definition of Limits at Infinity; Let L be a real number.

1) The statement lim ( )x

f x L→∞

= means that for each ε > 0 there exist an

M > 0 such that ( )f x L− < ε whenever x > M. 2) The statement lim ( )

xf x L

→∞= means that for each ε > 0 there exist an

N < 0 such that ( )f x L− < ε whenever x < N. 3.5.2 Definition of Horizontal Asymptote; he line y = L is a horizontal asymptote of the

graph of f if lim ( )x

f x L−→ ∞

= or lim ( )x

f x L→∞

= .

3.5.3 Limits of infinity, Theorem 3.10

3.6 Summary of Curve Sketching 2 This section requires students to sketch and define the characteristics of a function using a full analysis using the following techniques and first and second derivative tests. 3.6.1 Review the following techniques;

1) x-intercept and y-intercept 2) Symmetry 3) Domain and range 4) Continuity 5) Vertical asymptotes 6) Differentiability 7) Relative Extrema 8) Concavity 9) Points of Inflection 10) Horizontal Asymptotes

3.6.2 Guidelines for Analyzing the graph of a function; 1) Determine the domain and range of the function. 2) Determine the intercepts and asymptotes of the graph. 3) Locate the x-values for which ( )f x′ and ( )f x′′ are either zero or undefined.

Use the results to determine relative extrema and the points of inflection.

Review and Assessment 3

3.7 Optimization Problems 5 Students will learn that a common application of calculus entails the determination of minimum and maximum values. The focus is to identify and isolate the function to be optimized.

3.7.1 Applied Minimum and Maximum Problems, Strategies;

1) Assign symbols to all given quantities and quantities to be determined. When feasible make a sketch.

2) Write a primary equation for the quantity that is to be maximized or minimized. 3) Reduce the primary equation to one having a single independent variable. This

may involve the use of secondary equation relating the independent variables of the primary equations.

4) Determine the domain of the primary equation. That is, determine the values for which the stated problem makes sense.

5) Determine the desired maximum or minimum value by using the second derivative test.

3.8 Newton’s Method 1

This section is used as an alternative form for approximating the real zeros of a function. 3.8.1 Newton’s Method for Approximating the Zeros of a Function; Let f(c) = 0, where f

is differentiable on an open interval containing c. Then, to approximate c, use the following steps. 1) Make an initial estimate x1 that is “close to” c. A graph is helpful.

2) Determine a new approximation 1( )( )

nn n

n

f xx xf x+ = −′

.

3) If 1n nx x +− is within the desired accuracy, let 1nx + serve as the final approximation. Otherwise, return to step 2 and calculate a new approximation.

3.9 Differentials 3

Students will learn the concept of a tangent line approximation also known as linearization of a function. This sets students up to learn about slope fields and the family of functions. Also students will learn about finding propagated error from the difference between the measurement error and measured value. 3.9.1 Linear Approximation: For dx = Δx is sufficiently small, the differential dy is a

good approximation to Δy, in that dy differs from Δy by small percentage of dx. That is Δy ≈ dy if Δx ≈ 0.

3.9.2 Differentials: Let y = f(x) represent a function that is differentiable in an open interval containing x. The differential of x (denoted by Δx) is any nonzero real number. The differential of y (denoted by dy) is dy = f’(x)dx.

3.9.3 Error Propagation = f(x + Δ) – f(x) = Δy 3.9.4 Differentials can be used to approximate function values. For the function given by

y = f(x), use the formula ( ) ( ) ( ) ( )f x x f x dy f x f x dx′+ Δ ≈ + = + . 3.9.5 Business and Economics Applications, Terms and Formulas:

(supplementation from 6th edition of Larson, Hostetler and Edwards) Basic Terms Basic Formula x is the number of units produced or sold p is the price per unit R is the total revenue from selling x units R = xp C is the total cost of producing x units

C is the average cost per unit CCx

=

P is the total profit from selling x units P = R – C The break even point is the number of units for which R = C Marginals . dRdx

= (Marginal revenue) ≈ (extra revenue from selling one additional unit)

dCdx

= (Marginal cost) ≈ (extra cost of producing one additional unit)

dPdx

= (Marginal profit) ≈ (extra profit from selling one additional unit)

Review and Assessment 3

Chapter 4

4 Integration Days: 19 Integration is a major part of calculus by allowing the area under the function to

be calculated. Emphases will be placed on correct terminology, the family of functions, and definite integration. Students will learn the purposes of the first and second fundamental theorem of calculus. Also many methods will be discussed to evaluate integration including integration by substitution and change of the variable.

4.1 Antiderivatives and Indefinite Integration 3

This section will initiate antiderivative or indefinite integration as a general solution of differential equations. Also students will learn about finding a particular solution of a differential equation.

4.1.1 Representation of Antiderivatives, Theorem 4.1 4.1.2 Integration Formulas, p250 (know all integration rules) 4.1.3 Initial condition & Particular condition 4.1.4 Solving Differential Equations 4.1.5 Vertical motion 4.2 Area 3

Students will learn to use sigma notation to express and evaluate a sum of an area. The use of limits to find the area of a plane region will be discussed. Also covered in this section is the use of inscribed and circumscribed rectangles to approximate the area

4.2.1 Sigma Notation 4.2.2 Summation Formulas, Theorem 4.2 4.2.3 Upper and lower Sums, Theorem 4.3

a) Δx = (b-a)/n b) left end point: mi = a + (i – 1) Δx

right end point: M i = a + i Δx

c) s(n) = 1

( )n

ii

f M x=

Δ∑

4.2.4 Limit definition, 1

lim ( ) lim ( )n

in n i

s n f M x→∞ →∞

=

= Δ∑

4.3 Riemann Sums and Definite Integrations 3 This section takes the idea of the area of sums one step further by implementing the Riemann sum. Students can now see that one can make the size of the sub-intervals any width. Also students will see the beginning of definite integration and how to evaluate using the properties.

4.3.1 Riemann Sum 11

( ) ,n

i i i ii

if c x x c x−=

Δ ≤ ≤∑

4.3.2 Limit process 4.3.3 Definite integration 4.3.4 Continuity Implies Integrability, Theorem 4.4 4.3.5 The Definite Integral as the Area of a Region, Theorem 4.5 4.3.6 Additive Interval Property, Theorem 4.6

4.3.7 Properties of Definite Integrals, Theorem 4.7 4.3.8 Preservation of Inequality, Theorem 4.8

4.4 Fundamental Theorem of Calculus 3 Students can see that every section in this chapter leads up to this very important section introducing The Fundamental Theorem of Calculus and how it simplifies the definite integration process. Also students will again see inscribed and circumscribed rectangles mentioned to explain the concept of the Mean Value Theorem and how to find the average value over the closed interval. Finally we introduce The Second Fundamental Theorem of Calculus and how it is applied as an accumulation function. 4.4.1 Fundamental Theorem of Calculus, Theorem 4.9 4.4.2 Mean Value Theorem, Theorem 4.10 4.4.3 Average Value of a Function on the Interval; If f is integrable on the closed interval

[a,b], then the average value of f on the interval is 1 ( )b

af x dx

b a− ∫ .

4.4.4 Second Fundamental Theorem of Calculus, Theorem 4.11 4.5 Integration by Substitution 3

In this section the u-substitution method is introduced as a means to evaluate integration questions along with other techniques. Other techniques include change of variables, the use of The General Power Rule and recognition of patterns to simplify the evaluating process. Also students will learn to evaluate a definite integral involving an even of odd function.

4.5.1 Antidifferentiation of a Composite Function, Theorem 4.12 4.5.2 Integration by Substitution & General Power rule, Theorem 4.13

4.5.3 Change of Variables, Theorem 4.14 4.5.4 Integration of Even/Odd Functions, Theorem 4.15 4.6 Numerical integration 1

Since the Trapezoidal Rule is the only rule tested from this section, much emphasis will be given to it. Students will learn that the Trapezoidal Rule is used to approximate the value of definite integration using the idea of trapezoids instead of rectangles as subintervals. The other rules in this section will be lightly mentioned.

4.6.1 Trapezoidal rule; [ ]0 1 2 1) ( ) 2 ( ) 2 ( ) ... 2 ( ) ( )2

b

n na

b a(f x dx f x f x f x f x f xn −

−≈ + + + + +∫

4.6.2 Simpson rule; [ ]0 1 2 1( ) ( ) 4 ( ) 2 ( ) ... 4 ( ) ( )3

b

na

b anf x dx f x f x f x f x f x

n −

−≈ + + + + +∫

4.6.3 Error Analysis; Trapezoidal: 3

22

( ) max ( ) ,12b aE f x a

n− ⎡ ⎤ x b≤ ≤ ≤⎣ ⎦

Simpson: 5

44

( ) max ( ) ,180b aE f x a

n− ⎡ ⎤ x b≤ ≤ ≤⎣ ⎦

Review and Assessment 3

Chapter 5

5 Logarithmic, Exponential, and other Transcendental Functions Days: 22 Both logarithmic functions and exponential functions will be reviewed such that

students can easily recognize the graphs and properties of these functions. Also much attention will be given to the understanding of transcendental functions and their concepts. Care will be taken since many new topics will be presented to the students in this chapter.

5.1 The Natural Logarithmic Function and Differentiation 3

Students will learn the definition of the natural logarithm and how to differentiate expressions involving them. Students will learn that, ln x, is defined in terms of an integral and they will the history of e as a natural number. All properties in this section should be memorized.

5.1.1 The Natural logarithmic Function, 1

1ln ,x

x dtt

= ∫ x > 0 and the domain of natural

logarithmic functions is the set of all real numbers. 5.1.2 Properties of the Natural Logarithmic Function, Theorem 5.1 5.1.3 Logarithmic Properties, Theorem 5.2 5.1.4 Definition of e; The letter e denotes the positive real number such that

1

1ln 1.e

e dtt

= =∫

5.1.5 Derivative of the Natural Logarithmic Function, Theorem 5.3 5.1.6 Derivative Involving Absolute Value, Theorem 5.4

5.2 The Natural Logarithmic Function and Integration 2

Emphasis on growth and decay problems using antiderivative of expressions with the form, duu∫ will be studied along with the need to recognize patterns. Again, all properties in

this section should be memorized. 5.2.1 Log Rule for Integration, Theorem 5.5 5.2.2 Integrals of Trigonometric Functions, (The six basic.)

5.3 Inverse Functions 3

Although much of inverse functions will be reviewed for students, importance will be placed on if given the inverse of a function then how to find its derivative. Also students need to verify that one function is the inverse of another. 5.3.1 Definition of Inverse Functions; A function g is the inverse of the function f if

f(g(x)) = x for each x in the domain of g and g(f(x)) = x for each x in the domain of f. The function g is denoted by f-1 read “f inverse”.

5.3.2 Reflective Property of Inverse Functions, Theorem 5.6 5.3.3 The Existence of an Inverse Function, Theorem 5.7 and monotonic on the interval. 5.3.4 Continuity and Differentiability of Inverse Functions, Theorem 5.8 5.3.5 The Derivative of an Inverse Function, Theorem 5.9

5.4 Exponential Functions: Differentiation and Integration 3

Students will learn the rules to find the derivatives and integrals of natural exponential functions. Exponential functions with other bases can be changed in terms of e. 5.4.1 Definition of the Natural Exponential Functions; the inverse of the natural

logarithmic function f(x) = ln x is called the natural exponential function and denoted by f-1(x) = ex. That is y = ex iff x = ln y.

5.4.2 Operations with Exponential Functions, Theorem 5.10

5.4.3 Properties of the Natural Exponential Function 1) The domain of ( ) xf x e= is (-∞,∞), and the range is ( 0, ∞).

2) The function ( ) xf x e= is continuous, increasing and one-to-one on its entire domain. 3) The graph of ( ) xf x e= is concave upward on its entire domain. 4) and lim 0x

xe

→∞= lim x

xe

→∞= ∞

5.4.4 The Derivative of the Natural Exponential Function, Theorem 5.11 5.4.5 Integrals Rules for Exponential Functions, Theorem 5.12

5.5 Bases Other than e and Applications 2

To define exponential functions that have bases other than e are touched upon. Much that is studied in this section is referred back to properties from previous sections. Students

need to know 1lim 1

n

ne

n→∞

⎛ ⎞+ =⎜ ⎟⎝ ⎠

.

5.5.1 Definition of Exponential Function to Base e: If a is a positive real number (a ≠ 1) and x is any real number, then the exponential function to the base a is denoted by ax and is defined by ax = e(ln a)x. If a = 1, then y = 1x = 1 is a constant function.

5.5.1 Definition of Logarithmic Function to Base a: If a is a positive real number (a ≠ 1) and x is any positive real number, then the logarithmic function to the base a is

denoted by loga x and is defined as 1log lnlna x x

a= .

5.5.2 Properties of Inverse Functions: 1) y = ax iff x = loga y 2) alog

a x = x for x > 0

3) loga ax = x for all x

5.5.3 Derivatives for Bases Other Than e. Theorem 5.13 5.5.4 Integrand involving an Exponential Function to a base other than e.

1ln

x xa dx a Ca

⎛ ⎞= +⎜ ⎟⎝ ⎠∫

5.5.5 The Power Rule for real Exponents. Theorem 5.14

Review and Assessment 3 5.6 Inverse Trigonometric Functions: Differentiation 2

Students need to know the rules for the derivative of inverse trigonometric functions and be shown through implicit differentiation. 5.6.1 Know Inverse Trigonometric. Functions; functions, domain and range 5.6.2 Know Properties of Inverse Trigonometric Functions. 5.6.3 Derivatives of Inverse Trigonometric Functions. Theorem 5.16

5.7 Inverse Trigonometric Functions: Integrations 1

This lesson will be brief as students should be familiar with integrating inverse trigonometric functions and completing the square to integrate a function. 5.7.1 Integrals Involving Inverse Trigonometric Functions. Theorem 5.17

Review and Assessment 3

Chapter 6

6 Differential Equations Days: 14

Students will be introduced to notations and terminology of differential equations and how there are other ways for solutions. Such solution will be found using slope fields and Euler’s Method of approximating. Also students need to apply the new concept of separation of variables to applied problems. The students need to understand the concepts of initial condition, general solution which had been discussed previously, and the concept of particular solution.

6.1 Slope Fields and Euler’s Method 3

Students need to understand that the solution of a differential is an equation that can be verified by substituting it and its derivative into the differential equation. The general solution will lead to the family of functions and will be used to arrive at their initial value. 6.1.1 General, Verifying and Particular Solution 6.1.2 Sketching Slope Fields 6.1.3 Approximating a Solution using Euler’s method

6.2 Differential Equations: Growth and Decay 4

Several days will be spent on this section. Focus will be placed on separation of variables and growth and decay problems. Students will learn how to find solutions from the classical model for growth and decay, y = Cekt. We will cover many rate of change questions with a variation of initial values, proportional constant and the final solution. Also there will be discussions of problems that can just double in time without any initial values. 6.2.1 Solving Differential Equations 6.2.2 Exponential Growth and Decay Model, Theorem 6.1 6.2.3 Newton’s Law of Cooling

6.3 Separation of Variables and the Logistic Equations 4

Students will experience that differential equation can be separated into a general form as M(x) + N(y) dy/dx = 0 in order to find the solution which is called separation of variables. Applying this idea starts by writing and then solving a differential equation. The next steps are to find the general solution and then a particular solution. Homogeneous differential equations will be discussed to achieve a degree of comparison. Finally using the idea of having a family of functions, we can apply Orthogonal Trajectories to find the family of functions that are perpendicular to that family of functions. 6.3.1 Separation of Variables 6.3.2 Finding a Particular Solution Curve 6.3.3 Homogeneous Differential Equation: Definition; A Homogeneous differential

equation is an equation of the form M(x,y) dx + N(x,y) dy =0 where M and N are homogeneous functions of the same degree.

6.3.4 Change of Variable for Homogeneous Equations. Theorem 6.2

6.3.5 Finding Orthogonal Trajectories 6.3.6 Deriving the General Solution

Review and Assessment 3

Chapter 7

7 Applications of Integration Days: 12

The two sections of chapter 7 will extend the concept of area under the curve to the area between two curves and the volume found by revolving around an axis. The emphasis of the topics in this chapter will be the need for students to know the proper set up for each problem and to look for a pattern to solve each question easily.

7.1 Area of a Region between Two Curves 4 Using a plethora of functions to find the area between two curves with respect to the x-axis and the y-axis will be discussed in this section. Students will need to know how to find the intervals on which the area is to be found and how to find which function is on top of the other to perform the proper definite integration. 7.1.1 Area of a Region between Two Curves. If f and g are continuous on [a,b] and g(x)

≤ f(x) for all x in [a,b], then the area of the region bounded by the graphs of f and g

and the vertical lines x = a and x = b is Area = [ ( ) ( )]b

af x g x dx . −∫

7.1.2 Area of a region between Intersecting Curves 7.1.3 Horizontal Representative Rectangles 7.1.4 Integration as an Accumulation Process

7.2 Volume I 5

We will take the idea of area found in section 7.1 and expand it by rotating it around an axis whether this axis be x, y or any horizontal or vertical line to find the volume. Solids volume can be found using the disc method. Also a hole can be subtracted from a solid volume using the washer method. Finally students will find the volume of a solid with a known cross section. 7.2.1 The Disk Method: To find the volume of a solid of a revolution with the disk

method, use one of the following; Horizontal Axis of Revolution Vertical Axis of Revolution

Volume = V= 2[ ( )]b

aR x dxπ ∫ Volume = V = 2)][ (

d

cπ ∫ R y dy

7.2.2 The Washer Method: 2 2([ ( )] [ ( )] )b

aR x r x dy π −∫

7.2.3 Solids with known Cross Sections

Review and Assessment 3

Chapter 8

8 Integration Techniques Days: 4

Students will review many integrations pointed out in this section. Students are expected to know how to integrate functions that are derived from differential equations.

8.1 Basic integration Rules 2 Students will review measures taken to fit an integral to most basic integration rule. 8.1.1 Fitting Integrands 8.1.2 Substitution using a2 – u2 8.1.3 Trigonometric Identities

Review and Assessment 2

Objectives by Section

Chapter 1 1.1 Decide whether problems can be solved using pre-calculus or calculus. 1.1 Understand that the tangent problem and area problem are central to calculus. 1.2 Complete tables to estimate limits. 1.2 Graph to find limits. 1.2 Find the limit using ε – δ definition to prove that the limit is L. 1.2 Use graphing utility to estimate the limit. 1.3 Learn the limit theorems and find the limits. 1.3 Find if the limit exits. 1.3 Use strategies graphically, numerically, or by analytical analysis to find the limit. 1.3 Use position function to find velocity questions. 1.3 Evaluate a limit by applying rationalizing techniques and by applying the Squeeze Thm. 1.4 Find if limit exists and explain why if it does not exist. 1.4 Explain the continuity of a function on a closed or open interval. 1.4 Find the x values of a function at which it is not continuous. 1.4 Describe the continuity of a composite function. 1.4 Give details of the interval at which the one-sided limit is continuous. 1.4 Verify the Intermediate Value Theorem applies to a function and find the value c. 1.5 Determine whether the function approaches ∞ or -∞ . 1.5 Find whether the graph of the function has a vertical asymptote. 1.5 Use a graphing utility to graph the function to determine a one-sided limit.

Chapter 2 2.1 Estimate the slope of a graph at a point using a graphing utility. 2.1 Find the slope of the tangent line of a function at a point.

2.1 Use the limiting process to find the derivative. 2.1 Find the equation of the tangent and its perpendicular bisector. 2.1 Use the alternate form of the derivative to find the derivative at x = c if it exists. 2.1 Determine whether a function is differentiable and continuous at a given x point. 2.2 Find the derivative of a function using the new theorems. 2.2 Show how a function might need to be re-written in order to find its derivative. 2.2 Determine the point at which a function has a horizontal tangent line or m = 0. 2.2 Use a graphing utility to apply linear approximation. 2.2 Apply the standard position function to find the vertical motion of an object. 2.2 Use the derivatives to find the rate of change. 2.3 Find the derivative of a function using the product and quotient rules. 2.3 Re-write a function to find the derivative. 2.3 Find the derivative of a trigonometric function. 2.3 Verify the results of a derivative at a point. 2.3 Use derivatives to determine whether an x exists in an interval. 2.3 Find the second derivative of a function and apply it to acceleration. 2.3 Clarify that a function satisfies the differential equation. 2.4 Apply the chain rule to find the derivative of a function. 2.4 Use a graphing utility to graph a function and its derivative and describe its behavior. 2.4 Find the equation of a tangent line to a function at a given point. 2.4 Solve problems involving Harmonic Motion and the Doppler Effect. 2.4 Use a graphing utility to find linear regression of modeling data. 2.5 Compare the difference between implicit and explicit form of expression. 2.5 Apply the guidelines for implicit differentiation to find the derivative of a function. 2.5 Calculate the slope of the tangent line of a progressive function. 2.6 Locate the required values of dy/dt and dx/dt. 2.6 Find that the rate of change of an object is related to the rate of change of other variables.

Chapter 3 3.1 Uncover the value of the derivative over an indicated extremum.

3.1 Decide if they exist and locate the relative extrema on an indicated interval. 3.1 Find any relative extrema and critical numbers of a given function on the closed interval. 3.1 Sketch a function on the closed interval and locate the absolute extrema. 3.1 Use a graphing utility to graph a function and find the absolute extrema. 3.1 Employ maximum and minimum to horizontal distances in application problems. 3.2 Determine if Rolle’s Theorem works and apply it on the closed interval. 3.2 Consider if the Mean Value Theorem is applicable on the closed interval and apply. 3.2 Sketch the graph of an arbitrary function and use the Mean Value Theorem. 3.2 Prove that the equation has exactly one real solution. 3.3 Identify on the open interval where the function is increasing or decreasing. 3.3 Use the First Derivative Test to identify the critical points and relative extrema. 3.3 Apply symmetry, extrema and zeros to draw the function plus show how they differ. 3.3 Identify functions and how the intervals can move in a positive or negative direction. 3.3 Use regression capabilities of a graphing utility to model data. 3.4 Find the point of inflection of a function and test for concavity. 3.4 Use the Second derivative Test to identify the relative extrema. 3.4 Sketch the graph of a function having various characteristics. 3.4 Make use of maximum and minimum to application problems. 3.4 Draw on the first derivative to determine how approximation changes. 3.5 Employ the definition of a horizontal asymptote to the limit of a function.

3.5 Discover the limit at infinity. 3.5 Find the infinite limit at infinity. 3.5 Use a graphing utility to identify any horizontal asymptotes. 3.5 Sketch the graph of the function using intercepts, extrema and any asymptotes. 3.5 Use regression capabilities of a graphing utility to model data. 3.6 Draw the function showing any asymptotes, intercepts, extrema and points of inflection. 3.6 Generate a function whose graph has specific characteristics. 3.7. Find positive numbers that satisfy definite requirements. 3.7 Apply minimum and maximum to find values of a variety of problems. 3.8 Complete tables using iterations of Newton’s Method of a function. 3.8 Analyze a function to estimated the x value and then minimize the approximation to zero. 3.9 Utilize the derivative rules to find the differentials of functions and composite functions. 3.9 Use differentials to find the difference between measurement error and the measured value. 3.9 Apply differentials to approximate function values for the tangent line.

Chapter 4 4.1 Write the general solution of differential equations and integrals. 4.1 Re-write functions and apply the basic integration rules to find an indefinite integral. 4.1 Discover the particular solution after integrating and applying an initial condition value. 4.1 Plot the slopes of the family of functions onto directional fields. 4.1 Use integration to find the standard position when given the velocity. 4.2 Complete to write a sum using sigma notation. 4.2 Evaluate sums and re-write expressions using the properties of summation. 4.2 Use summation to find the upper and lower sums to find the area of a region. 4.2 Utilize the limiting process to find the area of a region between a function and the x-axis. 4.3 Apply the Riemann sum to show partitions can be subintervals of unequal widths. 4.3 Write the limit as a definite integral on an interval. 4.3 Find the area of an integral using the properties of definite integration. 4.4 Use the Fundamental Theorem of Calculus to find the area of an integral. 4.4 Evaluate the value of c guaranteed by the Mean Value Theorem over the interval. 4.4 Find all values of x in the closed interval for which a function equals its average value. 4.4 Integrate to find F as a function of x by using the Second Fundamental Thm. of Calculus. 4.5 Use pattern recognition and u-substitution to find the indefinite integration. 4.5 Re-write an integral in terms of u completely using the process of change of variable. 4.5 Evaluate an integral using the properties of even and odd functions. 4.5 Apply integration to rates to find inverse proportional problems. 4.6 Approximate a definite integral using the Trapezoidal and Simpson Rule. 4.6 Use a graphing utility to find a definite integral. 4.6 Investigate the approximate error in the Trapezoidal and Simpson’s Rule.

Chapter 5 5.1 Apply properties of logarithms to expand a term or write a logarithm as a single quantity. 5.1 Evaluate a logarithm by using a graphing utility 5.1 Find the derivative of natural logarithmic functions. 5.1 Understand the definition of the number e and where it is used on the graph. 5.1 Verify that a function is a solution of a differential equation. 5.2 Use the Log Rule to integrate rational functions. 5.2 Use integration rule on trigonometric functions. 5.2 Plot the slopes of the family of logarithmic functions onto directional fields. 5.2 Find the derivative of a logarithmic function using integration.

5.2 Use the rules of integration to show equivalent trigonometric identities. 5.3 Show that two functions are inverse functions graphically and analytically. 5.3 Use a graphing utility to determine if a function is one-to-one or monotonic on the domain. 5.3 Determine the existence of an inverse function. 5.3 Evaluate the derivative of an inverse function. 5.3 Use the Horizontal line test to determine if a function is one-to-one on the domain. 5.4 Apply the operations and properties with exponential functions to solve expressions. 5.4 Find the derivative and indefinite integral of natural exponential functions. 5.4 Use the rule of integration to integrate natural exponential functions. 5.4 Find the equation of the tangent line to the graph of the function at a given point. 5.4 Show that a function y = f(x) is a solution of the differential equation. 5.4 Evaluate the definite integral of the natural exponential function and verify. 5.5 Solve expressions with bases other than e. 5.5 Find the derivatives and integrate functions with bases other than e. 5.5 Complete tables to determine the gain or loss compounded by time. 5.5 Determine the exponential function that fits experimental data collected over time. 5.5 Develop recognition of the six inverse trigonometric functions. 5.6 Learn the rules to find the derivative of inverse trigonometric functions. 5.6 Write and calculate the expressions of inverse trigonometric functions. 5.6 Use inverse trigonometric functions to find rate of changes. 5.7 Learn to find the integration of inverse trigonometric functions. 5.7 Plot the slopes of the family of inverse trigonometric functions onto directional fields. 5.7 Utilize completing the square to integrate a function. 5.7 Find the area on the interval using integration of inverse trigonometric functions.

Chapter6 6.1 Apply the initial condition to find the particular condition of a differential equation. 6.1 Sketch a solution using a directional field. 6.1 Find a general solution of a differential equation using integration. 6.1 Approximate solutions of differential equations with specific values using Euler’s Method. 6.2 Apply the exponential growth and decay model to rate change problems. 6.2 Exercise separation of variables to solve a simple differential equation. 6.2 Complete tables to find the growth, decay and make predictions of different models. 6.2 Use regression capabilities of a graphing utility to find exponential models. 6.3 Answer differential equations problems using separation of variables. 6.3 Solve homogeneous and logistic differential equations. 6.3 Find a particular solution that satisfies the initial condition. 6.3 Apply differential equations to model and find applied problems.

Chapter 7 7.1 Find the area of a region between two curves of functions using integration. 7.1 Find the area of a region between two intersecting functions using integration. 7.1 Use integration as an accumulation process to find the area of rectangles under the curve. 7.1 Apply the area process to estimate the cumulative differences. 7.2 Use the disk method to find the volume of a solid revolving around the x and y-axis. 7.2 Use the washer method to find the volume of a solid revolving around the x and y-axis. 7.2 Find the volumes of solids with known cross section. 7.3 Use the shell method to find the volume of a solid revolving around the x and y-axis. 7.3 Make comparisons between the disk and shell method to find the best solution. 7.3 Model data using regression capabilities of a graphing utility.

Chapter 8

8.1 Use extended integration rules to best fit an integrand for integration. 8.1 Sketch a solution using a directional field. 8.1 Use a graphing utility to find the area of a definite integration.

Teaching Strategies For the Teacher The teacher should begin by explaining to students the expectations necessary to be successful on

the AP Calculus Exam. Specific chapter strategies can be discussed in each individual chapter planners. The following are some guidelines that will ensure a quality learning environment:

Each lesson will be prepared with notes that can be copied and the use of good quality examples. All assignment problems will be worked out on paper to eliminate misunderstanding. Identify all rules, properties and theorems that need to be committed to memory. Help foster organization of notes and a time line to prepare for the AP® Calculus Test. Encourage good grades, motivate healthy study habits and inspire class involvement.

For Students

A prelude to the AP Calculus course is for the students to understand their expectations. Students need to ask themselves what they need to know to be successful. The following are some suggestions that students should observe to be successful:

Knowledge of required tools on their calculator and how to access them quickly. Know integration and differentiation rules and how to find the tangent line. Know the connection between f’, f’’ and f’’’ and how to use position function to acceleration. Know how to minimize loss of scores by reading the questions carefully and guessing cautiously. Know techniques by eliminating the wrong answers. Recall major theorems to resolve many questions. Work free-response questions in an easy fashion using clear, neat and organized answers. You are not required to simplify answers so save time and eliminate possible mistakes. Effectively know how to find area and volume using integration.

Technology and Computer Software

-TI 83 – TI 89 -HM (Houghton Mifflin) mathSpaceTM Student CD -Several lessons are prepared for presentation using Power Point. -HM ClassPrep, Instructional Resourses CD for test, quizzes and worksheets.

Student Evaluation Four nine week grades consist of compiling scores from homework assignments, quizzes, group projects, and chapter tests. Most assignments come from the textbook exercises of each section. The text book has a plethora of questions and problems to choose from for daily assignments. The instructor resource guide provides a host of testing options. A specific number of tests are subject to change based on time constraints, chapter content and as deemed necessary by the instructor. All nine weeks represent 90 percent of the student’s grade and the final test represents 10 percent of the grade. Homework is checked at random and assessed on completeness. Quizzes will occur during every section and are assessed ten to thirty percent of a test grade. Tests will have either multiple-choice or completion format and students can use a calculator for most work. Previous and simulated AP Exams will be experienced during the year allowing for additional experience and assessment.

Teaching Resources

• Larson, Hostetler and Edwards, Calculus of a Single Variable, 8th Edition, Boston, New York: Houghton Mifflin, 2006.

• Cade, Lucia, and Caldwell, Fast Track to a 5: Preparing for the AP Calculus AB and Calculus BC Examinations, Boston, New York: Houghton Mifflin, 2006.

• McMullin, AP* Teacher’s Resource Guide, Calculus Eight Edition, Advanced Placement*Edition, Boston, New York: Houghton Mifflin, 2007.

• Lipp, Little Books of BIG IDEASTM, Calculus, Slope-Fields, New Jersey, The Peoples Publishing Group, Inc.: 2006.