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Calculus
College Credit Calculus
Angel Website: http://pcps.myelearning.org/frameIndex.htm
Angela Schmit's Website: http://manila.esu4.org/angelaschmit/
Peru State College
Calculus I
Math 225--Spring 2006
Course Syllabus
Instructor: Angela Schmit
Pawnee City High School
Phone: (402) 852-2988
email: aschmit@esu6.org
Planning Hour: 4th period
Description of Course
This course includes the study of functions, limits, differentiation and its applications, integration and its applications, and logarithmic and exponential functions.
PrerequisitesTrigonometry (one full academic year)
The Objectives of the Course
- To understand the fundamentals of limits, differentiation, and integration and their applications.
- To appreciate the value of calculus.
TextsTitle: Calculus I with PreCalculus
Author: Larson, Hostetler, and Edwards
Publisher: Houghton Mifflin
Edition: 2nd
Date: 2006
Explanation of Instruction
- Lecture and discussion every day.
- Quizzes every two or three lessons.
- Tests after every unit.
- Spelling/Writing assignments as needed.
- Angel Website--On-Line Learning
Visual Aids, Utilized Films, Film Strips, Etc.
- Marker board
- Calculators (Texas Instruments--TI 83, TI 85, and TI 92) provided by the school district
- Computer Software/Technology
- Supplementary Workbooks
Units and Contents
Chapter P: Prerequisites
P.1 Solving Equations
*Identify different types of equations
*Solve linear equations in one variable
*Solve quadratic equations by factoring, extracting square roots, completing the
square, and using the Quadratic Formula
*Solve polynomial equations of degree 3 or greater
*Solve equations involving radicals
*Solve equations involving absolute values
P.2 Solving Inequalities
*Represent solutions of linear inequalities
*Use proportion of inequalities to solve linear inequalities
*Solve inequalities involving absolute values
*Solve polynomial and rational inequalities
P.3 Graphical Representation of Data
*Plot points in the Cartesian plane
*Find the distance between two points using the Distance formula
*Find the midpoint of a line segment using the Midpoint formula
*Model and solve real-life problems using a coordinate plane
P.4 Graphs of Equations
*Sketch the graph of an equation
*Find x-and y-intercepts of a graph
*Use symmetry to sketch the graph of an equation
*Find equations and sketch graphs of circles
*Use graphs of equations in solving real-life problems
P.5 Linear Equations in Two Variables
*Use slope to graph a linear equation in two variables
*Interpret slope as a ratio or rate of change
*Find the slope of a line
*Write linear equations in two variables
*Write equations of lines that are parallel or perpendicular to a given line
*Use linear equations in two variables to model and solve real-life problems
Chapter 1: Functions and Their Graphs
1.1 Functions
*Decide whether relations between two variables are functions
*Use function notation and evaluate functions
*Find the domains of functions
*Use functions to model and solve real-life problems
1.2 Analyzing Graphs of Functions
*Use the Vertical Line Test for functions
*Find the zeros of functions
*Determine intervals on which functions are increasing, decreasing, or constant
*Identify and graph linear functions
*Identify and graph step functions and other piecewise-defined functions
*Identify even and odd functions
1.3 Shifting, Reflecting, and Stretching Graphs
Recognize graphs of common functions
Use vertical and horizontal shifts to sketch graphs of functions
Use reflections to sketch graphs of functions
Use nonrigid transformations to sketch graphs of functions
1.4 Combinations of Functions
*Add, subtract, multiply, and divide functions
*Find the composition of one function with another function
*Use combinations of functions to model and solve real-life problems
1.5 Inverse Functions
*Verify that two functions are inverse functions of each other
*Use the graph of a function to decide whether the function has an inverse function
*Use the Horizontal Line Test to determine if functions are one-to-one
*Find inverse functions analytically
1.6 Mathematical Modeling
*Use mathematical models to approximate sets of data points
*Write mathematical models for direct variation
*Write mathematical models for direct variation as an nth power
*Write mathematical models for inverse variation
*Write mathematical modes for joint variation
*Use the regression feature of a graphing utility to find the least squares regression line for data
Chapter 2: Polynomial and Rational Functions
2.1 Quadratic Functions
*Analyze graphs of quadratic functions
*Write quadratic functions in standard form and use the result to sketch graphs of
quadratic functions
*Use quadratic functions to model and solve real-life problems
2.2 Polynomial Functions of Higher Degree
*Use transformations to sketch graphs of polynomial functions
*Determine the end behavior of graphs of polynomial functions using the Leading
Coefficient Test
2.3 Polynomial and Synthetic Division
*Divide polynomials using long division
*Use synthetic division to divide polynomials by binomials of the form (x-h)
*Use the Remainder Theorem and the Factor Theorem
*Use polynomial division to answer questions about real-life problems
2.4 Complex Numbers
*Use the imaginary unit I to write complex numbers
*Add, subtract, and multiply complex numbers
*Use complex conjugates to write the quotient of two complex numbers in
standard form
*Find complex solutions of quadratic equations
2.5 The Fundamental Theorem of Algebra
*Understand and use the Fundamental Theorem of Algebra
*Find all the zeros of a polynomial function
*Write a polynomial function with real coefficients, given its zeros
2.6 Rational Functions
*Find the domains of rational functions
*Find the horizontal and vertical asymptotes of graphs of rational functions
*Analyze and sketch graphs of rational functions
*Sketch graphs of rational functions that have slant asymptotes
*Use rational functions to model and solve real-life problems
Chapter 3: Limits and Their Properties
3.1 A Preview of Calculus
*Understand what calculus is and how it compares with precalculus
*Understand that the tangent line problem is basic to calculus
*Understand that the area problem is also basic to calculus
3.2 Finding Limits Graphically and Numerically
*Estimate a limit using a numerical or graphical approach
*Learn different ways that a limit can fail to exist
*Study and use a formal definition of limit
3.3 Evaluating Limits Analytically
*Evaluate a limit using properties of limits
*Develop and use a strategy for finding limits
*Evaluate a limit using dividing out and rationalizing techniques
*Evaluate a limit using the Squeeze Theorem
3.4 Continuity and One-Sided Limits
*Determine continuity at a point and continuity on an open interval
*Determine one-sides and continuity on a closed interval
*Use properties of continuity
*Understand and use the Intermediate Value Theorem
3.5 Infinite Limits
*Determine infinite limits from the left and from the right
*Find and sketch the vertical asymptotes of the graph of a function
Chapter 4: Differentiation
4.1 The Derivative and the Tangent Line Problem
*Find the slope of the tangent line to a curve at a point
*Use the limit definition to find the derivative of a function
*Understand the relationship between differentiability and continuity
4.2 Basic Differentiation Rules and Rates of Change
*Find the derivative of a function using the Constant Rule
* Find the derivative of a function using the Power Rule
* Find the derivative of a function using the Constant Multiple Rule
* Find the derivative of a function using the Sum and Difference Rules
*Use derivatives to find rates of change
4.3 The Product and Quotient Rules and Higher-Order Derivatives
*Find the derivative of a function using the Product Rule
* Find the derivative of a function using the Quotient Rule
*Find a higher-order derivative of a function
4.4 The Chain Rule
* Find the derivative of a function using the Chain Rule
* Find the derivative of a function using the General Power Rule
*Simplify the derivative of a function using algebra
4.5 Implicit Differentiation
*Distinguish between functions written in implicit form and explicit form
*Use implicit differentiation to find the derivative of a function
4.6 Related Rates
*Find a related rate
*Use related rates to solve real-life problems
Chapter 5: Applications of Differentiation
5.1 Extrema on an Interval
*Understand the definition of extrema of a function on an interval
*Understand the definition of relative extrema of a function on an open interval
*Find extrema on a closed interval
5.2 Rolle's Theorem and the Mean Value Theorem
*Understand and use Rolle’s Theorem
*Understand and use the Mean Value Theorem
5.3 Increasing and Decreasing Functions and the First Derivative Test
*Determine intervals on which a function is increasing or decreasing
*Apply the First Derivative Test to find relative extrema of a function
5.4 Concavity and the Second Dervivative Test
*Determine intervals on which a function is concave upward or concave
downward
*Find any points of inflection of the graph of a function
*Apply the Second Derivative Test to find relative extrema of a function
5.5 Limits at Infinity
*Determine (finite) limits at infinity
*Determine the horizontal asymptotes, if any, of the graph of a function
*Determine infinite limits at infinity
5.6 A Summary of Curve Sketching
*Analyze and sketch the graph of a function
5.7 Optimization Problems
*Use calculus to solve applied minimum and maximum problems
5.8 Differentials
*Understand the concept of a tangent line approximation
*Compare the value of the differential, dy, with the actual change in y
*Estimate a propagated error using a differential
*Find the differential of a function using differentiation formulas
Chapter 6: Integration
6.1 Antiderivatives and Indefinite Integration
*Write the general solution of a differential equation
*Use indefinite integral notation for antiderivatives
*Use basic integration rules to find antiderivatives
*Find a particular solution of a differential equation
6.2 Area
*Use sigma notation to write and evaluate a sum
*Understand the concept of area
*Use rectangles to approximate the area of a plane region
*Find the area of a plane region using limits
6.3 Riemann Sums and Definite Integrals
*Understand the definition of a Riemann sum
*Evaluate a definite integral using limits
*Evaluate a definite integral using properties of definite integrals
6.4 The Fundamental Theorem of Calculus
*Evaluate a definite integral using the Fundamental Theorem of Calculus
*Understand and use the Mean Value Theorem for Integrals
*Find the average value of a function over a closed interval
*Understand and use the Second Fundamental Theorem of Calculus
6.5 Integration by Substitution
*Use pattern recognition to find an indefinite integral
*Use a change of variables to find an indefinite integral
*Use the General Power Rule for Integration to find an indefinite integral
*Use a change of variables to evaluate a definite integral
*Evaluate a definite integral involving and even or odd function
6.6 Numerical Integration
*Approximate a definite integral using the Trapezoidal Rule
*Approximate a definite integral using Simpson’s Rule
*Analyze the approximate errors in the Trapezoidal Rule and in Simpson’s Rule
Chapter 7: Exponential and Logarithmic Functions
7.1 Exponential Functions and Their Graphs
*Recognize and evaluate exponential functions with base a
*Graph exponential functions
*Recognize and evaluate exponential functions with base e
*Use exponential functions to model and solve real-life applications
7.2 Logarithmic Functions and Their Graphs
*Recognize and evaluate logarithmic functions with base a
*Graph logarithmic functions
*Recognize and evaluate natural logarithmic functions
*Use logarithmic functions to model and solve real-life applications
7.3 Using Properties of Logarithms
*Rewrite logarithmic functions with a different base
*Use properties of logarithms to evaluate or rewrite logarithmic expressions
*Use properties of logarithms to expand or condense logarithmic expressions
*Use logarithmic functions to model and solve real-life applications
7.4 Exponential and Logarithmic Equations
*Solve simple exponential and logarithmic equations
*Solve more complicated exponential equations
*Solve more complicated logarithmic equations
*Use exponential and logarithmic equations to model and solve real-life
applications
7.5 Exponential and Logarithmic Models
*Recognize the five most common types of modes involving exponential and
logarithmic functions
*Use exponential growth and decay functions to model and solve real-life problems
*Use Gaussian functions to model and solve real-life problems
*Use logistic growth functions to model and solve real-life problems
*Use logarithmic functions to model and solve real-life problems
Chapter 8: Exponential and Logarithmic Functions and Calculus
8.1 Exponential Functions: Differentiation and Integration
*Differentiate natural exponential functions
*Integrate natural exponential functions
8.2 Logarithmic Functions and Differentiation
*Find derivatives of functions involving the natural logarithmic function
*Use logarithms as an aid in differentiating nonlogarithmic functions
*Find derivatives of exponential and logarithmic functions in bases other than e
8.3 Logarithmic Functions and Integration
*Use the Log Rule for Integration to integrate a rational function
8.4 Differential Equations: Growth and Decay
*Use exponential functions to model growth and decay in applied problems
Chapter 9: Trigonometric Functions
9.1 Radian and Degree Measure
*Describe angles
*Use radian measure
*Use degree measure
*Use angles to model and solve real-life problems
9.2 Trigonometric Functions: The Unit Circle
*Identify a unit circle and its relationship to real numbers
*Evaluate trigonometric functions using the unit circle
*Use the domain and period to evaluate sine and cosine functions
*Use a calculator to evaluate trigonometric functions
9.3 Right Triangle Trigonometry
*Evaluate trigonometric functions of acute angles
*Use the fundamental trigonometric identities
*Use trigonometric functions to model and solve real-life problems
9.4 Trigonometric Functions of Any Angle
*Evaluate trigonometric functions of any angle
*Use reference angles to evaluate trigonometric functions
9.5 Graphs of Sine and Cosine Functions
*Sketch the graphs of basic sine and cosine functions
*Use amplitude and period to help sketch the graphs of sine and cosine functions
*Sketch translations of the graphs of sine and cosine functions
*Use sine and cosine functions to model real-life data
9.6 Graphs of Other Trigonometric Functions
*Sketch the graphs of tangent functions
*Sketch the graphs of cotangent functions
*Sketch the graphs of secant and cosecant functions
*Sketch the graphs of damped trigonometric functions
9.7 Inverse Trigonometric Functions
*Evaluate the inverse sine function
*Evaluate the other inverse trigonometric functions
*Evaluate the compositions of trigonometric functions
9.8 Applications and Models
*Solve real-life problems involving right triangles
*Solve real-life problems involving directional bearings
*Solve real-life problems involving harmonic motion
Chapter 10: Analytic Trigonometry
10.1 Using Fundamental Trigonometric Identities
*Recognize and write the fundamental trigonometric identities
*Use the fundamental trigonometric identities to evaluate trigonometric functions
and to simplify and rewrite trigonometric expressions
10.2 Verifying Trigonometric Identities
*Develop a strategy for verifying trigonometric identities
*Verify trigonometric identities
10.3 Solving Trigonometric Equations
*Use standard algebraic techniques to solve trigonometric equations
*Solve trigonometric equations of quadratic type
*Solve trigonometric equations involving multiple angles
*Use inverse trigonometric functions to solve trigonometric equations
10.4 Sum and Difference Formulas
*Use sum and difference formulas to evaluate trigonometric functions
*Use sum and difference formulas to verify identities and solve trigonometric
equations
10.5 Multiple-Angle and Product-to-Sum Formulas
*Use multiples-angle formulas to rewrite and evaluate trigonometric functions
*Use power-reducing formulas to rewrite and evaluate trigonometric functions
*Use half-angle formulas to rewrite and evaluate trigonometric functions
*Use product-to-sum formulas to rewrite and evaluate trigonometric functions
Chapter 11: Trigonometric Functions and Calculus
11.1 Limits of Trigonometric Functions
*Determine the limits of trigonometric functions
11.2 Trigonometric Functions: Differentiation
*Find and use the derivatives of the sine and cosine functions
*Find and use the derivative of other trigonometric functions
*Apply the First Derivative Test to find the minima and maxima of a function
11.3 Trigonometric Functions: Integration
*Integrate trigonometric functions using trigonometric identities and u-substitution
*Use integrals to find the average value of a function
11.4 Inverse Trigonometric Functions: Differentiation
*Differentiate inverse trigonometric functions
*Review the basic differentiation rules for elementary functions
11.5 Inverse Trigonometric Functions: Integration
*Integrate functions whose antiderivatives involve inverse trigonometric
functions
*Use the method of completing the square to integrate a function
*Review the basic integration rules involving elementary functions
11.6 Hyperbolic Functions
*Develop properties of hyperbolic functions
*Differentiate and integrate hyperbolic functions
*Develop properties of inverse hyperbolic functions
*Differentiate and integrate functions involving inverse hyperbolic functions
Chapter 12: Topics in Analytic Geometry
12.1 Introduction to Conics: Parabolas
*Recognize a conic as the intersections of a plane and a double-napped curve
*Write the standard form of the equation of a parabola
*Use the reflective property of parabolas to solve real-life problems
12.2 Ellipses and Implicit Differentiation
*Write the standard form of the equation of an ellipse
*Use implicit differentiation to find the slope of a line tangent to an ellipse
*Use properties of ellipses to model and solve real-life problems
*Find the eccentricity of an ellipse
12.3 Hyperbolas and Implicit Differentiation
*Write the standard form of the equation of a hyperbola
*Find the asymptotes of a hyperbola
*Use implicit differentiation to find the slope of a line tangent to a hyperbola
*Use properties of hyperbolas to solve real-life problems
*Classify a conic from its general equation
12.4 Parametric Equations and Calculus
*Evaluate a set of parametric equations for a given value of the parameter
*Sketch the curve that is represented by a set of parametric equations
*Rewrite a set of parametric equations as a single rectangular equation
*Find a set of parametric equations for a graph
*Find the slope of a tangent line to a curve given by a set of parmetric equations
12.5 Polar Coordinates and Calculus
*Understand the plar coordinate system
*Rewrite rectangular coordinates and equations in polar form and vice versa
*Find the slope of a tangent line to a polar graph
12.6 Graphs of Polar Equations
*Graph a polar equation by point plotting
*Use symmetry, zeros, and maximum r-values as graphing aids
*Recognize special polar graphs
12.7 Polar Equations of Conics
*Write equations of conics in polar form
*Use equations of conics in polar form to model real-life problems
Chapter 13: Additional Topics in Trigonometry
13.1 Law of Sines
*Use the Law of Sines to solve oblique triangles (AAS, ASA, or SSA)
*Find the areas of oblique triangles
*Use the Law of Sines to model and solve real-life problems
13.2 Law of Cosines
*Use the Law of Cosines to solve oblique triangles (SSS or SAS)
*Use Heron’s Area Formulas to find the area of a triangle
*Use the Law of Cosines to model and solve real-life problems
13.3 Vectors in the Plane
*Represent vectors as directed line segments
*Write the component forms of vectors
*Perform basic vector operations and represent them graphically
*Write vectors as linear combinations of unit vectors
*Find the directions angles of vectors
*Use vectors to model and solve real-life problems
13.4 Vectors and Dot Products
*Find the dot product of two vectors and use the Properties of the Dot Product
*Find the angle between two vectors
*Determine whether two vectors are orthogonal
*Write a vector as the sum of two vector components
*Use vectors to find the work done by a force
13.5 Trigonometric Form of a Complex Number
*Plot complex numbers in the complex plane
*Write the trigonometric forms of complex numbers
*Multiply and divide complex numbers written in trigonometric form
*Use DeMoivre’s Theorem to find powers of complex numbers
*Find nth roots of complex numbers
Chapter 14: Systems of Equations and Matrices
14.1 Systems of Linear Equations in Two Variables
*Use the method of substitution to solve systems of equations in two variables
*Use the method of elimination to solve systems of linear equations in two
variables
*Interpret graphically the numbers of solutions of systems of linear equations to
two variables
*Use systems of equations in two variables to model and solve real-life problems
14.2 Multivariable Linear Systems
*Recognize and write multivariable linear systems in row-echelon form
*Use Gaussian elimination to solve systems of linear equations
*Solve nonsquare systems of linear equations
*Use systems of linear equations in three or more variables to model and solve
application problems
14.3 Systems of Inequalities
*Sketch the graphs of inequalities in two variables
*Solve systems of inequalities
*Use systems of inequalities in two variables to model and solve real-life
problems
14.4 Matrices and Systems of Equations
*Identify the order of a matrix
*Perform elementary row operations on matrices
*Use matrices to solve systems of linear equations
14.5 Operations with Matrices
*Determine whether two matrices are equal
*Add and subtract matrices and multiply matrices by real numbers
*Multiply two matrices
*Use matrix operations to model and solve real-life problems
14.6 The Inverse of a Square Matrix
*Verify that two matrices are inverses of each other
*Use Gauss-Jordan elimination find the inverses of matrices
*Use a formulas to find the inverses of 2x2 matrices
*Use inverse matrices to solve systems of linear equations
14.7 The Determinant of a Square Matrix
*Find the determinants of 2x2 matrices
*Find minors and cofactors of square matrices
*Find the determinants of square matrices
*Use the determinant to find the equation of a line through two points
Calculus Projects
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