Download Chapter 9:

College Pre-Calculus
Curriculum
Summer 2011
Written by:
Rosa Quagliata and Mary Sherman
Supervised by:
Maureen Welsh
1
Bellmore-Merrick
Central High School
District
BOARD OF EDUCATION
Dr. Matthew Kuschner, President
Marion Blane, Vice President
JoAnn DeLauter
Janet Goller
George Haile
Joseph Perrone
Susan Schwartz
Diane Seaman
ADMINISTRATION
Dr. Henry Kiernan
Superintendent of Schools
Dr. Mara Bollettieri
Assistant Superintendent for Personnel & Administration
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Deputy Superintendent, Business
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Assistant Superintendent, Instruction
2
College Pre-Calculus 4A/4R Curriculum
In an effort to ease the transition from Algebra 2 to Calculus the angle measure will
be used only in radians. In addition all work with exact trig values, exponents and
graphs should be done without a calculator.
Chapters 2, 10, and 12 have been separated into different units of instruction
CHECKPoint in the textbook leads to additional examples in the chapter exercises.
(Example: page 2 Example 1 of the textbook leads to a CHECKPoint Exercise 7.)
Each unit includes a day for a quiz although it is not explicitly written.
Table of Contents
Unit
Page
Chapter 9: Sequences and Series…………………………………………………….4
Appendix A31: Factoring……………………………………………………………..6
Chapter 1: Functions and Their Graphs…………………………………………….8
Chapter 2: Polynomials ……………………………………………………………...12
Chapter 2: Rational Function ……………………………………………………….14
Chapter 3: Exponential and Logarithmic Functions ……………………………...15
Chapter 4 and 5: Trigonometry and Analytic Trigonometry ……………………..17
Chapter 8: Matrices and Determinants …………………………………………….19
Chapter 10: Topics in Analytic Geometry (Conic Sections) ……………………….21
Chapter 10: Topics in Analytic Geometry (Polar Coordinates)…………………....23
Chapter 12: Limits and an Introduction to Calculus ……………………………....24
Chapter 12: Derivatives…………………………………………………………....….26
Applications of Derivatives …………………………………………………………...28
Integrals ………………………………………………………………………………..30
SUPPLEMENTAL MATERIAL………………………………………………….......31
3
Chapter 9: Sequences and Series (Approximately 8 days)
Chapter Overview
Students learned this material in Algebra 2/Trigonometry. A review of Sequences and
Series is included in order to prepare students for the SAT.
Essential Question
Why is it important to predict events that may occur in the future?
Big Idea
Patterns
Vocabulary
Terms of a sequence
Arithmetic Sequence
Common Difference
Geometric Sequence
Common Ratio
Infinite geometric series
Geometric Series
Daily Lessons
Day 1: Review key concepts of sequences
a. Writing the terms of a sequence (Example 1, 2 page 640)
b. Writing an expression for the nth term of a sequence (Example 3 page 641)
Math 4A only
c. Summation notation (Example 7 page 644)
d. Finding the sum of a series (Example 8 page 645)
Day 2: Arithmetic Sequences
a. Examples of arithmetic sequences (Example 1 page 651)
b. Finding the nth term of an arithmetic sequence (Example 2 and 4 page 652)
c. Finding arithmetic means (Classroom Discussion on page 656)
d. Writing the terms of an arithmetic sequence (Example 3 page 653)
Day 3: Finding the Sum of an Arithmetic Sequence
a. Finding the sum of a finite arithmetic sequence (Example 5 and 6a page 654)
b. Finding a partial sum of an infinite arithmetic sequence
(Example 7 page 655)
4
Day 4: Geometric Sequences
a. Finding the nth term of a geometric sequence
(Example 1, 2, 3, 4, 5 pages 661-663)
b. Finding a geometric mean
(Complete the geometric sequence 2,_,_,_,32 and Example 5 page 663)
Day 5: Finding the Sum of a Finite Geometric Sequence and an Infinite Geometric
Series
a. Finding the sum of a finite geometric sequence
(Example 6 pages 664)
b. Finding the sum of an infinite geometric series (Example 7 page 665)
Day 6: Review
Day 7: Test
5
Appendix A31: Factoring (Approximately 8 days for 4R)
(Approximately 10 days for 4A)
Chapter Overview
Review of basic factoring such as greatest common factor, difference of two squares,
trinomial factoring as well as factoring a trinomial with the leading coefficient greater
than one and factor by grouping. Student will also be able to factor the sum and
difference of cubes.
Essential Question
How do we reorganize an expression to make it easier to adapt to situations?
Big Idea
Reorganize
Common
Pattern Recognition
Efficiency
Vocabulary
Greatest Common Factor
Trinomial
Difference of Two Squares
Cubic
Factor by Grouping
Daily Lessons
Day 1: Review of Factoring
a. Removing a common factor first (Example 6 page A31)
b. Factoring the difference of two squares (Example 7 page A31)
c. Factoring perfect square trinomials (Example 8 page A32)
Day 2: Factoring Cubes
a. Factoring the difference of cubes (Example 9 page A32)
b. Factoring the sum of cubes (Example 10 page A32)
 Check answers by distributing (Math 4A/4R)
Day 3: Factor by Grouping
a. Factor a trinomial – leading coefficient is 1 (Example 11 page A33)
b. Factor by grouping (Example 13 page A34)
c. Factor a trinomial by grouping (Example 14 page A34)
6
Day 4 and 5: Factor Completely
a. #173 – 197 page A37
Day 6 and 7: Factor Completely
a. Math 4A #198 – 212 page A37
b. Math 4A factor x 2  4 x  21  25 y 2
Day 6 and 7: Math 4R Review and Test
Day 8 and 9: Math 4A Review and Test
7
Chapter 1: Functions and Their Graphs (Approximately 18 days for 4R)
(Approximately 20 days for 4A)
Chapter Overview
This unit studies functions and their graphs. Students will be able to find intercepts,
determine symmetry, write the equation of a line, evaluate functions, determine the
domain and the range, find zeros, determine if a function is even or odd, graph Parent
Functions and their transformations, find the inverse of a function, and verify that
functions are inverses of each other.
Essential Question
Why is it important to understand how things behave?
Big Idea
Patterns
Behavior
Vocabulary
Intercepts
Symmetry
Slope
Parallel
Perpendicular
Function
Domain
Range
Even Functions
Odd Function
Vertical Shift
Horizontal Shift
Nonrigid Transformation
Inverse Function
Daily Lessons
Day 1: Intercepts and Symmetry
a. Finding the x and y intercepts (Example 4 page 16)
b. Testing for symmetry with x and y axes and the origin (Example 5 page 18)
Day 2: Writing the Equation of a Line
a. Writing the equation of a line using POINT SLOPE formula only
(Example 3 page 28) (Exercises page 71 #11-18)
b. Finding the slope of a line parallel and perpendicular (Example 4 page 29)
Day 3: Evaluating Functions
a. Evaluating functions (Example 3 page 42)
b. Evaluating piecewise functions (Example 4 page 42)
8
Day 4: Domain
a. Finding the domain of a function (Example 7 page 44)
Day 5: Evaluating a Difference Quotient
a. Evaluating a difference quotient (Example 11 page 46)
Day 6: Domain and Range, Finding Zeros
a. Finding the domain and range of a function from a graph
(Example 1 page 54)
b. Finding the zeros of a function (Example 3 page 56)
Day 7: Even and Odd Functions
a. Determining if a function is even or odd (Example 8 page 60)
*
4A only - Day 8: Average Rate of Change
a. Find the average rate of change of a function (Example 6 page 59)
b. Find the average speed (Example 7 page 59)
4A - Day 9
4R - Day 8: Parent Graphs
a. Students will be able to graph without using a calculator:
yx
y  x2
y  x3
1
y
x
y x
y
x
b. Identifying the parent graph without a calculator
(Exercises on page 71 # 19-42)
c. Determine the domain and range of these functions and introduce the concept
of horizontal and vertical asymptotes.
4A – Day 10
4R – Day 9: Graphing Piecewise Functions
a. Graphing piecewise functions (Example 3 page 70)
9
*
4A only – Day 11: Given a Piecewise Graph, write the equation
a. Given the graph of a piecewise function write the piecewise function
(see SUPPLEMENTAL section for ditto)
4A – Day 12
4R – Day 10: Shifts
a. Vertical and horizontal Shifts (Example 1, 2 page 74)
b. Discuss domain and range
4A – Day 13
4R – Day 11: Transformations of Parent Graphs
a. Reflecting in the x and y axes (Example 3 page 76)
b. Discuss domain and range
4A – Day 14
4R – Day 12: Vertical and Horizontal Stretch and Shrink
a. Vertical stretch and vertical shrink (Example 4 page 77)
b. Horizontal stretch and horizontal shrink (Example 5 page 77)
c. Discuss domain and range
4A – Day 15
4R – Day 13: Composition of Functions
a. Finding f g x and f g 2 (Example 5 page 85)
b. Finding the domain of a composite function (4A students should be able to do
this without a calculator.)
(Exercise 6 page 86)
4A – Day 16
4R – Day 14: Inverse Functions
a. 4A – Prove that f g x  g  f x  x (Example 2 page 93)
4R – Find the inverse of f, find the inverse of g, show that f  g 1 and
g  f 1
4A – Day 17
4R – Day 15: Finding an inverse
a. Find the inverse of a function (Example 6, 7 pages 96-97)
4A - Day 18
10
4R – Day 16: Review
4A - Day 19
4R – Day 17: Test
11
Chapter 2 – Polynomials
(Approximately 9 days for 4R)
(Approximately 11 days for 4A)
Chapter Overview
Students will be introduced to higher order polynomial equations. The focus is on
synthetic division, Descartes Rule of Signs and also The Rational Zero Test. Students
will also be able to solve for all roots of a higher order polynomial equation as well as
write the polynomial equation given the roots. Students will also understand how to draw
a possible rational function given the nature of the roots.
Essential Question
Describe the nature of a roller coaster.
What real world occupations use polynomials?
Name an arch in the shape of a parabola?
Big Idea
Classifying
Shapes
Continuous
Vocabulary
Continuous
Repeated Zero
Intermediate Value Theorem (4A)
Synthetic Division
Complex Number
Imaginary Number
Complex Conjugates
Daily Lessons
Day 1: Synthetic Division
a. Synthetic division (Example 4 page 153)
b. Using the remainder theorem (Example 5 page 154)
c. Factoring a polynomial – repeated division (Example 6 page 155)
d. Applying the leading coefficient test (Example 2 page 139)
e. Math 4A only – The Intermediate Value Theorem (page 143)
Day 2: Descartes’s Rule of Signs
a. Rule of Signs (Example 9 page 173)
b. The Rational Zero Test (list p/q) (page 167)
Day 3: Solving a Polynomial with Irrational Roots
a. Use Descartes’s rule of signs and the Rational Zero Test
b. Find all real roots (Example 5 page 169)
12
Day 4: Solving a Polynomial with Imaginary Roots
a. Use Descartes’s rule of signs and the Rational Zero Test
b. Find all roots (Example 7 page 171)
Day 5: Solving a Polynomial with Repeating Roots
a. Use Descartes’s rule of signs and the Rational Zero Test
b. Find all roots (Example 8 page 172)
Day 6: Writing the Equation of a Polynomial
a. Finding a polynomial with given zeros (Example 6 page 170)
Day 7: Math 4R Review
Day 8: Math 4R Test
Day 7: Writing the Equation of a Polynomial
a. Finding a polynomial with given fractional zeros (Math 4A only)
Day 8: Sketching a possible polynomial given the Nature of the Roots
a. see ditto
Day 9: Math 4A Review
Day 10: Math 4A Test
13
Chapter 2: Rational Functions (Approximately 7 days)
Chapter Overview
The students will have a better understanding of how to find the domain of a rational
function as well as a better understanding of a vertical asymptote. The students will be
able to find the horizontal asymptote and understand that this is the behavior of the
function out at infinity.
Essential Questions
Describe the concentration of a drug into your blood stream when medicine is taken
orally.
Does the graph flatten out at a particular time?
How can data be displayed to understand an outcome?
Big Idea
Behavior
Numerical analysis
Vocabulary
Rational function
Vertical asymptote
Horizontal asymptote
Point of discontinuity
Daily Lessons
Day 1: Finding the domain, vertical and horizontal asymptotes of a Rational
Function
a. Find the domain of a rational function (Example 1 page 181)
b. Find the vertical and horizontal asymptotes (Example 2 page 183)
Day 2: Sketching the graph of a rational function
a. Graphing where the horizontal asymptote is y = 0 (Example 3 and 5 page
185 and 186)
Day 3: Sketching the graph of a rational function
a. Graphing where the horizontal asymptote is y   a (Example 4 page 185)
Day 4: Sketching the graph of a rational function
a. Graphing where the graph has a point of discontinuity (Example 6 page 186)
Day 5: Review
14
Day 6: Test
Chapter 3: Exponential and Logarithmic Functions (Approximately 11 days)
Chapter Overview
This chapter reviews the properties of exponents and logarithms, the graphs of
exponential and logarithmic functions, the properties of logs, and solving exponential and
logarithmic equations.
Essential Question
What tools do we need to write a research paper?
Big Idea
Strategies
Vocabulary
Natural Base e
Natural log function
Properties of logarithms/natural logs
Exponential and logarithmic graphs
Daily Lessons
Day 1: Evaluating exponents and logarithms
a. Students will be able to evaluate numerical expressions with negative and
rational exponents WITHOUT the use of a calculator.
b. Students will be able to rewrite expressions with rational exponents in radical
form.
c. Students will be able to rewrite expressions without a variable in the
denominator.
d. Students will be able to rewrite expressions that have negative exponents with
only positive exponents.
(See SUPPLEMENTAL MATERIAL for ditto)
Day 2: Graphs of Exponential Functions
a. Discuss the graphs of:
y  ax
y  a x
y  ex
y  ex
(Examples 2, 3 page 217)
Students should be able to graph these without a calculator.
b. Discuss domain, range, and horizontal asymptotes of these graphs
c. Discuss transformations of these graphs (Example 5 page 219)
15
Day 3: Evaluating Exponential and Logarithmic Expressions and Graphs of
Logarithmic Functions
a. Converting from log form to exponential form (Example 1 page 227)
b. Properties of logs on page 228 (Example 3,4 page 228-229)
c. Graphs of logarithmic functions (Example 5, 6 page 229)
Students should be able to graph these without a calculator.
Day 4: Transformations of Logarithmic Graphs, Properties of Natural Logarithms
a. Vertical and horizontal shifts of logarithmic graphs (Example 7 page 230)
b. Using the properties of the Natural Logarithm (Example 9 page 232)
c. Finding the domain of natural logarithmic functions (Example 10 page 232)
Students should be able to graph these without a calculator.
Day 5: Properties of Logarithms
a. Simplifying expressions using the properties of logarithms
(Example 3, 4 page 238)
b. Condensing logarithmic expressions (Example 6 page 239)
Days 6, 7, 8: Solving Exponential and Logarithmic Equations
a. Solve simple equations (Example 1 page 244)
b. Solve exponential equations (Examples 2, 3, 4 page 245-6)
c. Solve logarithmic equations (Examples 6, 7, 8, 9 pages 247-248)
Day 9: Review
Day 10: Test
16
Chapter 4 and 5: Trigonometry and Analytic Trigonometry
(Approximately 10 days for 4R)
(Approximately 11 days for 4A)
Overview
Students will have a better understanding of reference angles, terminal sides, special right
triangles, inverse trig functions and solving an equation for all angles that make the
equation true.
Essential Question
Other than the temperature, what other behaviors are cyclic?
Big Idea
Cyclic
Repeating
Patterns
Vocabulary
Degree
Radian
Unit circle
Reference angles
Inverse trig functions
Exact trigonometric values
Daily Lessons
Day 1: Radian and Degree Measures
a. Sketching and finding coterminal angles (Example 1 page 282)
b. Converting from radians to degrees (Example 3 page 283)
c. Converting from degrees to radians (Example 4 page 283)
Day 2: Trigonometric Functions: The Unit Circle
a. Evaluating trigonometric functions (Example 1 page 294)
b. Evaluation trigonometric functions (Example 2 page 295)
Day 3: Right Triangle Trigonometry
a. Evaluate the special angles for the trig functions (page 301)
b. Fundamental trig identities (page 302)
c. Focus on questions on page 307 such as 21 – 30, 31(in radians), 32(in
radians), 57 – 62.
Day 4: Trigonometric Functions of Any Angle
a. Finding the reference angle (Example 4 (a, c only) page 312)
b. Using reference angles (Example 5 page 314)
17
Day 5: Inverse Trigonometric Functions
a. Evaluating inverse trigonometric functions (Example 3 page 344)
b. Using inverse properties (Example 5 (b, c only) page 345)
Day 6 and 7: Solving Trigonometric Equations
a. Collecting like terms (Example 1 page 388)
b. Extracting square roots (Example 2 page 388)
c. Factoring (Example 3 page 389)
d. Factoring an Equation of Quadratic Type (Example 4 page 390)
e. Rewriting with a single trigonometric function (Example 5 page 390)
f. Focus on the Pythagorean identity
4A Day 8: Solving Trigonometric Equations
a. Functions of multiple angles (Example 7 and 8 page 392)
Day 8 or 9: Review
Day 9 or 10: Test
18
Chapter 8: Matrices and Determinants (Approximately 12 days)
Chapter Overview
This chapter covers the addition, subtraction, and multiplication of matrices, finding a
determinant and an inverse of a matrix, solving systems of equations using matrices,
finding the area of a triangle, collinearity, and the equation of a line using matrices.
Essential Question
Why is it necessary to organize information in tables?
Big Idea
Order
Organization
Vocabulary
Order of a matrix
Square matrix
Scalar multiplication
Matrix multiplication
Identity matrix of order n
Inverse of a matrix
Determinate of a matrix
Daily Lessons
Day 1: Operations with Matrices
a. Introduce matrices and find the order of matrices (Example 1 page 570)
b. Equality of matrices (Example 1 page 584)
c. Addition of matrices (Example 2 page 585)
d. Scalar multiplication and subtraction of matrices (Example 3 page 586)
e. Solving a matrix equation (Example 6 page 588)
Days 2 and 3: Finding the Product of Two Matrices
a. 4A – Students will be able to multiply matrices of order 2x2 and a 3x3 without
the use of a calculator.
4R – Students will be able to multiply matrices of order 2x2 without the use
of a calculator.
b. Introduce the identity matrix (page 591)
Day 4: Finding the Inverse of a Square Matrix
a. Show that two matrices are inverses of each other (Example 1 page 599)
b. Formula for the inverse of a 2x2 matrix (Example 4 page 603)
19
Day 5 and 6: Solving a System of Equations using Matrices
a. Students should be able to solve a system of equations with two variables
WITHOUT a calculator. Any system larger should be solved by using a
calculator. (Example 5 page 604)
Day 7: Finding the Determinant of a Matrix
a. Definition of the determinant of a 2x2 matrix WITHOUT a calculator
(Example 1 page 609)
b. Math 4A – Find the determinant of a 3x3 matrix WITHOUT the use of a
calculator.
(Example 3 page 611)
Day 8: Applications of the Determinant
a. Finding the area of a triangle using matrices (Example 3 page 619)
b. Testing for collinear points (Example 4 page 620)
Day 9: Finding the Equation of a Line using Matrices
a. Using matrices to write the equation of a line (Example 5 page 621)
Day 10: Review
Day 11: Test
20
Chapter 10: Topics in Analytic Geometry (Approximately 10 days)
Overview
Students will have a better understanding of conic sections. The students will be
introduced to a focus of a parabola and an ellipse and a directrix for a parabola and
asymptotes for a hyperbola
Essential Question
What shape does the path of the planets around the sun make?
What roles do the figures plan in architecture?
What functions do conic sections play in the real world?
Big Idea
Shapes
Navigation
Modeling
Vocabulary
Directrix
Focus
Tangent
Foci
Vertices
Major axis
Minor axis
Center
Transverse axis
Conjugate axis
Daily Lessons
Day 1: Circles
a. Review completing the square
b. Finding the equation of a circle using matrices
Day 2 and 3: Introduction to Conics: Parabolas
a. Vertex at the origin (Example 1 page 735)
b. Finding the focus of a parabola (Example 2 page 735)
c. Finding the standard equation of a parabola (Example 3 page 736)
d. Writing the equation of a parabola using matrices
Day 4 and 5: Ellipse
a. Finding the standard equation of an ellipse (Example 1 page 744)
b. Sketching an ellipse (Example 2 page 744)
c. Analyzing an ellipse (Example 3 page 745)
d. Also talk about eccentricity (page 146)
21
Day 6 and 7: Hyperbola
a. Finding the standard equation of a hyperbola (Example 1 page 752)
b. Using asymptotes to sketch a hyperbola (Example 2 page 753)
c. Find the asymptotes of a hyperbola (Example 3 page 754)
Day 8: Review
Day 9: Test
22
Chapter 10: Topics in Analytic Geometry (Approximately 7 days)
Overview
Students will be exposed to a new coordinate plane known as the polar plane in which the
radius and the angle are plotted instead of the Cartesian plane where the x and y
coordinate are graphed. Students will be able to find equivalent points and reflected
points in the polar plane. Students will also be able to draw polar graphs in the polar
plane.
Essential Question
What patters do microphones release at high frequencies?
What kind of coordinates will help you navigate from point A to point B?
Big Idea
Shapes
Locations
Movements
Navigation
Vocabulary
Polar coordinates
Polar graphs
Polar equations
Daily Lessons
Day 1: Polar Coordinates
a. Plotting points on the polar coordinate system (Example 1 page 777)
b. Multiple representations of points (Example 2 page 778)
Day 2: Polar Coordinates
a. Rectangular to Polar (Example 4 page 779)
Day 3: Polar Coordinates
a. Polar to rectangular (Example 3 page 779)
Day 4: Graphs of Polar Equations
a. Graphing a polar equation by point plotting (Example 1 page 783)
Day 5: Review
Day 6: Test
23
Chapter 12: Limits and an Introduction to Calculus (Approximately 7 days)
Overview
Students will be able to determine a limit from a function and a graph. Student will be
introduced to the concept of continuity.
Essential Question
If interest is compounded continuously, why is it important to be able to predict the end
result of one’s investment?
How high can you jump?
Big Idea
Graphing
Prediction
End behavior
Closeness
Vocabulary
One sided limits
Limits at infinity
Continuous
Day 1: One Sided limits
a. Evaluating one-sided limits
(Example 6 page 865, Example 7 and 8 page 866)
b. Using a graph to find a limit (Example 5 page 852)
c. Discuss when a limit will fail to exist (Example 6 and 7 page 853)
d. Math 4A only Example 8 page 854
Day 2: Direct Substitution
a. Evaluating limits algebraically (Example 9 page 856, Example 10 page 857)
b. Techniques for evaluating limits (Example 1 page 861, Example 2 page 862)
Day 3: Direct Substitution
a. Rationalizing Technique (Example 3 page 863)
Day 4: Limits at Infinity
a. Evaluating limit at infinity (Example 1 page 882 and Example 2 page 883)
Day 5: Continuous Functions
a. Determine if a function is continuous at a point
(SUPPLEMENTAL
Exercises for section 1.4: #1 – 6 determine if the graph is continuous at c
Value.
Page 93 exercises 39 – 42: Determine if a function is continuous by
sketching a piece-wise curve.
24
Day 6: Review
Day 7: Test
25
Chapter 12: Derivatives (Approximately 10 days)
Overview
Students will be introduced to the definition of the derivative which will lead to the
power rule. Also the students will be introduced to the product, quotient and chain rule to
evaluate and find derivatives of complex functions. The students will understand the idea
of a tangent line and normal line to the graph using derivatives.
Essential Question
If the demand of a product increased, what will happen to the supply of the product?
How does the volume of business affect the rate of employment?
Big Idea
Rates of change
Velocity
Acceleration
Approximation
Vocabulary
Slope
Definition of the derivative
Power rule
Product rule
Quotient rule
Chain rule
Tangent line
Normal line
Day 1: Definition of the Derivative
a. Evaluating a limit from calculus (Example 9 page 867)
b. Introduce power rule
Day 2: Slope of a Graph
a. Visually approximating the slope of a graph (Example 1 and 2 page 872,
Example 3 and 4 page 874 and Example 5 page 875)
b. Use definition of the derivative or power rule to calculate the slope
Day 3: Product Rule (SUPPLEMENTAL page 127)
Day 4: Quotient Rule (SUPPLEMENTAL page 129)
Day 5: Chain Rule (SUPPLEMENTAL page 137)
Day 6: Chain with Product and Quotient
(SUPPLEMENTAL – EXERCISES for Section 2.4)
Day 7: Equation of the tangent line
a. Exercises on page 878 # 43 – 50
b. Exercise on page 879 # 55 – 70
26
Day 8: Equation of the normal line
a. Use the same Exercises on page 878 #43-50. Change instructions to find the
equation of the normal line.
Day 9: Review
Day 10: Test
27
Applications of Derivatives (Approximately 8 days for 4R)
(Approximately 9 days for 4A)
Overview
Students will be introduced to finding the maximum, minimum, intervals of increasing,
intervals of decreasing, point of inflection, intervals of concave up and intervals of
concave down of a function using the number line test. Students will also be introduced
to perimeter, area and volume problems using calculus.
Essential Question
Why is it important for a company to maximize the amount of production while
minimizing the cost of the item?
Big Idea
Optimization
Vocabulary
Increasing
Decreasing
Extrema
Points of inflection
Concave up
Concave down
Day 1: Increasing and Decreasing Functions
a. Discuss the connection between increasing and decreasing and f (x) .
b. Discuss maximum and minimum points.
(SUPPLEMENTAL pages 179-186)
Day 2 and 3: First Derivative Test
a. Use the first derivative test to determine max and min points.
b. Use the first derivative test to determine intervals of increasing and
decreasing. (SUPPLEMENTAL pages 179-186)
Day 4 and 5: Concavity and the Second Derivative Test
a. Define concavity
b. Discuss the connection between concavity and f (x).
c. Define a point of inflection
d. Discuss the second derivative test
e. Use the second derivative test to find points of inflection and intervals of
concave up and concave down. (SUPPLEMENTAL pages 189-194)
Day 6: Optimization – Maximize/Minimize - Area/Perimeter
MATH 4A only
(SUPPLEMENTAL pages 213-214 and EXERCISES for Section 3.7)
Day 7: Optimization – Volume of a box
28
(SUPPLEMENTAL pages 213-214 and EXERCISES for Section 3.7)
Day 7 or 8: Review
Day 8 or 9: Test
29
Integrals and Application of Integrals (Approximately 6 days for 4R)
(Approximately 7 days for 4A)
Overview
The students will be introduced to finding the anti-derivative of simple functions.
Students will understand the difference between an indefinite integral and a definite
integral. Students will also be introduced to finding the area under one and two curves.
Essential Question
What is the total distance traveled from point A to point B?
Big Idea
Distance
Total
Vocabulary
Anti-derivative
Area
Indefinite Integral
Definite integral
Day 1 and 2: Finding Anti-derivatives
a. Use the formula to find an antiderivative
(SUPPLEMENTAL pages 251-253 and EXERCISES for Section 4.1)
Day 3 and 4: Evaluating Definite Integrals and Area
a. Evaluate a definite integral (SUPPLEMENTAL pages 282 and 289)
b. Use an integral to find the area under a curve.
Day 5: Find the Area of a Region Between Two Curves
MATH 4A only
a. Determine the area between two curves
(SUPPLEMENTAL Section 6.1 pages 403-406 and EXERCISES for
Section 6.1)
Day 5 or 6: Review
Day 6 or 7: Test
30
SUPPLEMENTAL
MATERIAL
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