Curriculum and Pacing Guide - Mathematics
Calculus (1202300)
Revised June 2011
Calculus Honors (1202300)
Curriculum and Pacing Guide - Mathematics
Calculus Honors (1202300)
Revised June 2011
1.
Review of the concepts from algebra and pre-calculus concerning relations /functions will probably be necessary to some extent. These
include linear, trigonometric, exponential, logarithmic, the generic functions, and piecewise functions. Topics relative to these are; slopes, domain
and range, symmetry, intercepts, intersections, and the graphs of these relations/functions. The graphing calculator (TI-89) is used to reinforce
these concepts, referring to the solving, table, and graphing capabilities of the calculator.
Chapter 1: Prerequisites for Calculus
Days Section
Topic
2
1.1
Lines
2
1.2
Functions and graphs
2
1.3
Exponential Functions
1
2
1.5
Quiz
Functions and Logarithms
3
1.6
Trigonometric Functions
1
1
14
Review
Test
Total
Days
Concepts
Increments, Slope, Point-slope Form,
Slope-intercept, Standard Form
(Skip Regression Analysis)
Function, Even and Odd Functions,
Composition, Piecewise Functions,
Recognizing Graphing Failure
Exponential Function, Exponential
Growth and Decay, Half Life
One to One, Logarithms Functions,
Inverse Functions, Identity Functions,
Change of Base Formula
Periodic Function, Period,
Trigonometric Functions, Inverse Trig,
Even and Odd
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(NV)
(TGV) Graphing Features of
calculator, Graph Viewing Skills
(TG) Graphing and Table
Features
(VGN)
(TVGN)
Data Warehouse
Questions
Calculus Honors (1202300)
2.
The concepts of a limit, as the independent variable approaches a particular value, are addressed numerically, analytically, and graphically.
This involves developing the ideas of the existence of a limit. The issues are, if the limit exist, what is it, and if it does not, why not? This
necessitates the understanding of “broken graph” oscillating, and asymptotic behaviors. Examination of limits, including one sided limits, is done
using the various algebra techniques for the types of functions, the properties of limits and special techniques for rational trig related functions.
Continuity and the Intermediate Value theorem and their applications are also part of this unit. Areas of application in this unit are finding equations
of tangent lines (and normal lines) at a point and beginning motion problems.
Chapter 2: Limits and Continuity
Days Section
Topic
Concepts
3
2.1
Rates of
Change and
Limits
Average v. Instantaneous Speed,
Properties of Limits, One sided v.
Two sided Limits, Sandwich Theorem
3
2.2
Limits
Involving
Infinity
Horizontal Asymptote, Properties of
Limits as x approaches infinity,
Vertical Asymptote, End Behavior
Model
1
3
2.3
3
2.4
1
1
15
Total
Days
Quiz
Continuity
Rates of
Change and
Tangent Lines
Review
Test
Continuity at a Point, Properties of
Continuous Functions, Composition of
Continuous Functions,
Discontinuities, Intermediate Value
Theorem, Jump Discontinuity
Average Rate of Change, Slope of a
Curve at a Point, Tangent Line,
Normal to a Curve,
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(T) See www.calculus-help.com
(T) table features of calculator
(T) trace features of calculator
Supplement Algebraic Limit Techniques
from Larson
(T) See www.calculus-help.com
(G) Graphical
Data Warehouse
Questions
129956
129750,130055,132135,1
32136,134194,134197,13
4202
(T) See www.calculus-help.com
(G) Graphical
(VN)
129952,134206
(VN)
130125,134233
Calculus Honors (1202300)
3.
The derivative is developed in this unit which involves the geometric interpretation of the tangent line at a point, leading to the limit definition
of the derivative of a function. The limit definition is used both to find derivatives at a point and to develop the basic derivative rules. Basic
derivative rules are used for first, second, and higher order derivatives and also for implicit derivatives. The derivative is also investigated in the
relationship between position, velocity, acceleration and jerk. Inverse functions are addressed with particular attention given to the natural
exponential function as the inverse of the natural logarithm function. The derivatives of the natural exponential function and the natural log
function are given. Logarithmic and exponential functions of any base are also given along with their corresponding derivatives. Various derivative
techniques are developed for natural logarithm functions techniques for rational form functions, including trigonometric.
Calculus Honors (1202300)
Chapter 3: Derivatives
Days
Section
Topic
Concepts
Derivative, Derivative at a Point, Notation,
Relationship between the Graphs of F and F
Prime, One-Sided Derivatives
Non-Differentiability, Local Linearity
Differentiation Rules, Higher Order Derivatives
3
3.1
Derivative of a Function
3
4
3.2
3.3
Differentiability
Rules for Differentiation
1
4
3.4
Quiz
Velocity and Other Rates
of Change
3
3.5
Derivatives of
Trigonometric Functions
3
3.6
Chain Rule
1
3
3.7
Mid-Chapter Test
Implicit Differentiation
3
3.8
3
3.9
1
1
33
Review
Test
Total Days
Instantaneous Rates of Change, Instantaneous
Velocity, Speed, Acceleration, Linear Motion,
Free-Fall Motion
Derivative of Trig Functions, Jerk,
Suggested Approaches
(T) Technology, (V)
Verbal, (G) Graphical,
(N) Numerical
(VGN) (T)
http://www.ima.umn.ed
u/~arnold/graphics.html
(VGN)
(VN)
(T) See www.calculushelp.com
Data Warehouse
Questions
134218
130126
130054,130068,130
128,130130,134210
,134215,134217
(VG)
129924,130111,130
127,130132
(VG) Graphical
(T) See www.calculushelp.com
130038,130077,
129425,130129,130
131,130322,132149
,134235
Derivative of Composite Functions, Chain Rule
(Skip Slopes of Parameterized Curves)
(VN)
Implicit Function, Implicit Differentiation,
Power Rule for Rational Powers
(VN)
129533,130046
Derivatives of Inverse
Trigonometric Functions
Inverse, Derivatives of Inverse Trigonometric
Functions
134016,134050,134
051
Derivatives of Exponential
and Logarithmic Functions
Derivative of Exponential Functions base e and
base a, Derivative of Log Functions base e and
base a
(VGN) Supplement
Theorem 5.9 From
Larson for Derivative of
Inverse Functions
(VN)
129752,134011,134
012
Calculus Honors (1202300)
4. This unit continues the application of derivatives. Also three principle theorems are developed and used – the Extreme Value Theorem, Rolle’s
Theorem, and the Mean Value Theorem. First and second derivatives are used to determine for a given function the critical values, intervals of
increase and decrease, relative maxima and minima, points of inflection, and intervals concave up and concave down. This application is done with
and without graphing calculators. Included with this application is the examination of the relationships of the graphs of a function, the graph of its
1st derivative, and the graph of its 2nd derivative and through the use of tables. The very useful and important derivative application of solving
optimization problems, as well as linear approximations, differentials, and related rates are in this unit.
Chapter 4: Applications of Derivatives
Days
Section
Topic
5
4.1
Application of Derivatives
3
4.2
Mean Value Theorem
4
4.3
Connecting f prime and f
double prime with the graph
of f
1
5
3
4.4
4.5
Quiz
Modeling and Optimization
Linearization
6
4.6
Related Rates
1
1
29
Review
Test
Total
Concepts
Absolute Extrema, Local Extreme
Values, Critical Point, Extreme Value
Theorem,
Mean Value Theorem, Increasing
Decreasing Intervals, Antiderivative,
Rolle’s Theorem, Monotonic Functions
First Derivative Test, Definition of
Concavity, Concavity Intervals, Point of
Inflection, Second Derivative Test,
Max-Min Problems, Optimization
Linear Approximation, Differentials,
Absolute Relative and Percent Change,
(Skip Newton’s Method)
Related Rates
Suggested Approaches
(T) Technology, (V) Verbal,
(G) Graphical, (N)
Numerical
(VGN)
(VN)
(G)
(T)
www.unitedstreaming.com :
“applications of derivatives”
(V)
(VN)
Data Warehouse
Questions
129532,130450,1304
56,130457,130458
129761,130053,1303
29,130451,130453,13
4186
129515,130115,1304
60
129760,132143
132151
129753,129947,1301
21
Calculus Honors (1202300)
5.
The definite integral is developed by first examining estimates of the areas of plane regions as sums of rectangles constructed by using a
partitioning of an interval and the right, left, midpoint or any point of the partition. The definition of a definite integral can then be given as a limit
to an infinite Riemann Sum, the exact area of the plane region. The Fundamental Theorem of Calculus is developed along with the Mean Value
Theorem for Integrals leading to the Average Value of a function on an interval. The Second Fundamental Theorem is also given. The main
applications here are areas of simple plane regions, and average-value-of-a-function problems. This unit also includes estimation of plane regions
by using trapezoidal approximation.
Chapter 5: The Definite Integral
Days
Section
Topic
3
5.1
Estimating with Finite
Sums
4
5.2
Riemann Sums
6
5.3
Definite Integrals and
Antiderivatives
1
5
5.4
Quiz
Fundamental Theorem
of Calculus
3
5.5
Trapezoidal Rule
1
1
24
Review
Test
Total days
Concepts
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(VN)
Data Warehouse
Questions
(VGN)
133934
(VN)
13400,134008
Distance Traveled,
132172,132176
Rectangular Approximation
Method
Definite Integral,
(T, N) Graphing calculator: table 129757,132177,133909,13
Integrability,
feature
3912,133915
Integral Properties, Average
(VN)
129762,133879,133896,13
Value, Mean Value
3925,133938
Theorem for Integrals
Fundamental Theorem of
Calculus Part 1 and 2,
Connection to Area
Trapezoid Rule
(Skip Other Algorithms,
Simpson’s Rule and Error
Bounds)
Calculus Honors (1202300)
6.
This unit introduces slope fields and the solving of differential equations, leading to the concept of an antiderivative. Indefinite integrals are
solved through various techniques including u du substitution and pattern recognition. Exponential growth and decay model is developed from
integration of separation of variables. The Logistic Growth model is also included.
Chapter 6: Differential Equations and Mathematical Modeling
Days
Section
Topic
Concepts
1
6.1
Slope Fields and Euler’s
Method
4
6.2
Antidifferentiation by
Substitution
4
6.4
Exponential Growth and
Decay
1
3
2
1
16
6.5
Quiz
Logistic Growth
Review
Test
Total
Days
Differential Equation, Initial
Value (Skip Slope Fields and
Euler’s Method)
Properties of Indefinite
Integrals, Indefinite Integrals,
Leibniz Notation, u du
substitution
Separable Differential
Equation, Exponential
Change, Continuous and
Compound Interest, Modeling
Growth and Decay
Logistic Differential
Equations, Logistic Curve,
Logistic Growth Model (Skip
Partial Fractions)
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(VN)
Data Warehouse Questions
(VN)
130071,132164,133919,13
3949,133975,133991,1340
13,134030
(VN)
130082,130326,134034,13
4049,134224,134229
(VN)
129754,134020
Calculus Honors (1202300)
7.
This unit involves the interpretation of the integral as an accumulator and applications of finding areas and volumes. The definite Integral is
used to find the areas of regions between curves using all types of functions. These are areas on an interval, areas between curves including
curves with more than two intersections, also incorporating change of axis. The next application is volume beginning with three diminution shapes
also knows as cross sections. Also included are volumes of rotation using the disk, and washer method incorporating the change of axis.
Chapter 7: Applications of Definite Integrals
Days
Section
Topic
5
7.1
Integral as Net Change
4
7.2
Areas in the Planes
1
6
7.3
Quiz
Volume
2
1
19
Concepts
Displacement, Consumption
over time, Net Change from
Data
Area between Curves, Area
Enclosed by Intersecting
Curves, Integrating in respect
to y, Integration by Geometric
Formula
Cross Sections, Washers
(Skip Shells)
Suggested Approaches
(T) Technology, (V)
Verbal, (G) Graphical, (N)
Numerical
(VN)
Data Warehouse Questions
(VGN)
(T) Calculus in Motion
(Fee)
129749,130105,134168
(VGN)
(T) Calculus in Motion
(Fee)
Supplement from Larson
for Disk and Washer
Methods
129748,130093,134145,134152,134153
Review
Test
Total
Days
Total Instructional days: 150
This pacing guide does not count early release days as instructional days. They are to be used as review days, quiz days
or for supplemental instruction.
Major Text.
Taken from:
Finney, Damana, Waits and Kennedy. Calculus: Graphical, Numerical, Algebraic (AP Edition), 3rd Ed. Upper Saddle River, New Jersey, 2007
Finney, Damana, Waits and Kennedy. Calculus: A Complete Course, 2nd Ed. Upper Saddle River, New Jersey, 2000