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# Prepare for Calculus

This course covers the topics shown below.
Students navigate learning paths based on their level of readiness.
Institutional users may customize the scope and sequence to meet curricular needs.

Curriculum

•Real Numbers
Fractions
Simplifying a fraction
Ordering fractions
Addition or subtraction of fractions with different denominators
Fraction multiplication
Fraction division
Fractional part of a circle
¨
Proportion and Percent
Converting a fraction to a percentage
Converting a percentage to a fraction
Converting between percentages and decimals
Percentage of a whole number
Word problem on percentage: Problem type 1
Word problem on percentage: Problem type 2
Word problem on percentage: Problem type 3
Basic word problem on rates
Solving a proportion: Basic
Word problem on proportions: Problem type 1
¨
Integers and Signed Numbers
Absolute value of a number
Operations with absolute value
Integer subtraction
Integer multiplication and division
Signed fraction multiplication
Evaluating expressions with exponents: Problem type 1
Exponents and order of operations
¨
Number Systems
Integers and rational numbers
Rational and irrational numbers
Properties of real numbers
¨
·
Equations and Inequalities
Linear Equations
Evaluation of a linear expression in two variables
Additive property of equality: Problem type 2
Multiplicative property of equality: Problem type 2
Solving a linear equation: Problem type 1
Solving a linear equation: Problem type 2
Solving a linear equation: Problem type 3
Solving a linear equation with several occurrences of the variable: Problem type 1
Solving a linear equation with several occurrences of the variable: Problem type 2
Solving a linear equation with several occurrences of the variable: Problem type 3
Solving a linear equation with several occurrences of the variable: Problem type 4
Solving a linear equation with several occurrences of the variable: Problem type 5
Solving a word problem using a linear equation: Problem type 1
Solving a word problem using a linear equation: Problem type 2
Solving a word problem using a linear equation: Problem type 3
Solving a word problem using a linear equation: Problem type 4

Linear Inequalities and Absolute Values
Solving a linear inequality: Problem type 2
Solving a linear inequality: Problem type 3
Solving a linear inequality: Problem type 4
Solving an equation involving absolute value: Basic
Solving an equation involving absolute value: Advanced
Solving an inequality involving absolute value: Basic
¨
Systems of Linear Equations
Solving a system of linear equations
¨
Solving equations written in factored form
Finding the roots of a quadratic equation with leading coefficient greater than 1
Solving a quadratic equation needing simplification
Solving a word problem using a quadratic equation with rational roots
Solving a word problem using a quadratic equation with irrational roots
¨
Graphs and Functions
Set builder and interval notation
Union and intersection of intervals
Reading a point in the coordinate plane
Plotting a point in the coordinate plane
Introduction to functions: Notation and graphs
Sum, difference, and product of two functions
Domain and range: Problem type 1
Domain and range: Problem type 2
Domain and range: Problem type 3
Range of a real−valued function
Vertical translation of the graph of a function
Vertical and horizontal translations of the graph of a function
Classifying the graph of a function
Computing outputs for piecewise− defined functions
Graphing piecewise−defined functions
¨
Linear Functions
Solutions to a linear equation in two variables: Problem type 1
Solutions to a linear equation in two variables: Problem type 2
Y−intercept of a line
X− and y−intercepts of a line given the equation in standard form
Finding the slope of a line given its equation
Determining the slope of a line given its graph
Graphing a line given the x− and y−intercepts
Graphing a line given its equation in slope−intercept form
Graphing a line given its equation in standard form
Graphing a vertical or horizontal line
Graphing a line through a given point with a given slope
Graphing an equation involving absolute value in the plane
Writing an equation of a line given the y−intercept and a point
Writing the equation of a line given the slope and a point on the line
Writing the equations of vertical and horizontal lines through a given point
Writing the equation of the line through two given points
Slopes of parallel and perpendicular lines: Problem type 1
Slopes of parallel and perpendicular lines: Problem type 2
Writing equations and drawing graphs to fit a narrative
Application problem with a linear function: Problem type 1
Application problem with a linear function: Problem type 2
Interpreting the graphs of two functions

Parabolas
Finding the x−intercept(s) and the vertex of a parabola
Graphing a parabola: Problem type 1
Graphing a parabola: Problem type 2
Graphing a parabola: Problem type 3
¨
•Exponents and Polynomials
Integer Exponents
Product rule of exponents
Product rule of exponents in a multivariate monomial
Quotients of expressions involving exponents
Multiplying monomials
Power rule: Positive exponents
Ordering numbers with positive exponents
Writing a positive number without a negative exponent
Writing a negative number without a negative exponent
Power rule: Negative exponents
Ordering numbers with negative exponents
¨
Polynomial Arithmetic
Evaluation of a polynomial in one variable
Simplifying a polynomial expression
Degree of a multivariate polynomial
Multiplying two binomials
Squaring a binomial
Multiplying polynomials
Polynomial long division: Problem type 1
¨
Factoring
Greatest common factor of two monomials
Least common multiple of two monomials
Factoring a difference of squares
Factoring with repeated use of the difference of squares formula
Factoring a sum or difference of two cubes
Factoring a product of a quadratic trinomial and a monomial
Completing the square

Polynomial Equations and Functions
Graphing a simple cubic function
Inferring properties of a polynomial function from its graph
Solving a word problem involving a polynomial of degree 3
Solving a word problem by finding a local extremum of a polynomial function
¨
•Rational Expressions
Rational Expressions
Ordering fractions with variables
Ratio of multivariate polynomials
Simplifying a ratio of polynomials: Problem type 1
Simplifying a ratio of polynomials: Problem type 2
Multiplying rational expressions: Problem type 1
Multiplying rational expressions: Problem type 2
Dividing rational expressions
Complex fraction: Problem type 1
Adding rational expressions with common denominator
Adding rational expressions with different denominators
Adding and subtracting rational expressions: Problem type 1
Partial fraction decomposition
¨
Rational Equations
Solving a rational equation that simplifies to a linear equation: Problem type 1
Solving a rational equation that simplifies to a linear equation: Problem type 2
Solving a rational equation that simplifies to a linear equation: Problem type 3
Solving a rational equation that simplifies to a quadratic equation: Problem type 1
Solving a rational equation that simplifies to a quadratic equation: Problem type 2
Solving a rational equation that simplifies to a quadratic equation: Problem type 3
¨
Applications of Rational Expressions
Algebraic symbol manipulation
Word problem on direct variation
Word problem on inverse variation
Word problem on inverse proportions
Word problem involving multiple rates
¨
Rational Functions
Quotient of two functions
Sketching the graph of a rational function: Problem type 1
Sketching the graph of a rational function: Problem type 2
¨
·
Square root of a perfect square
Square root of a rational perfect square
Square root simplification
Square root multiplication
Simplifying a radical expression: Problem type 1
Simplifying a radical expression: Problem type 2
Simplifying a product of radical expressions
Simplifying a product of radical expressions using the distributive property
Rationalizing the denominator of a radical expression
Rationalizing the denominator of a radical expression using conjugates
Solving an equation with radicals: Problem type 1
Solving an equation with radicals: Problem type 2
Domain of a square root function

Higher Roots and Rational Exponents
Cube root of an integer
Converting between radical form and exponent form
Rational exponents: Basic
Rational exponents: Negative exponents and fractional bases
Rational exponents: Powers of powers
Even root property
Odd root property
¨
•Exponentials and Logarithms
Function Compositions and Inverse Functions
Composition of two functions: Basic
Inverse functions: Basic
¨
Properties of Logarithms
Exponential and logarithmic equations
Evaluating a logarithmic expression
Basic properties of logarithms
Change of base for logarithms: Problem type 1
Change of base for logarithms: Problem type 2
¨
Exponential and Logarithmic Equations
Solving an exponential equation: Problem type 1
Solving an exponential equation: Problem type 2
Solving a logarithmic equation: Problem type 1
Solving a logarithmic equation: Problem type 2
Solving a word problem using an exponential equation: Problem type 1
Solving a word problem using an exponential equation: Problem type 2
Solving a word problem using an exponential equation: Problem type 3
Solving a word problem using an exponential equation: Problem type 4
¨
Exponential and Logarithmic Functions
Sketching the graph of an exponential function: Basic
Sketching the graph of an exponential function: Advanced
Sketching the graph of a logarithmic function
Translating the graph of a logarithmic or exponential function
¨
·
•Geometry and Trigonometry
Perimeter, Area, and Volume
Perimeter of a square or a rectangle
Area of a square or a rectangle
Area of a piecewise rectangular figure
Finding the side length of a rectangle given its perimeter or area
Area and perimeter of a rectangle
Circumference and area of a circle
Perimeter involving rectangles and circles
Circumference ratios
Area between two concentric circles
Arc length and area of a sector of a circle
Area involving rectangles and circles: Advanced problem
Volume of a cube or a rectangular prism
Volume of a cylinder
Volume of a cone
Volume of a sphere
Rate of filling of a solid
Ratio of volumes
Surface area of a cube or a rectangular prism
Surface area of a cylinder

Coordinate Geometry
Pythagorean Theorem
Distance between two points in the plane
Midpoint of a line segment in the plane
Graphing a circle given its equation in standard form
Graphing a circle given its equation in general form
¨
Right Angle Trigonometry
Converting between degree and radian measure
Coterminal angles
Sketching an angle in standard position
Sine, cosine, and tangent ratios
Using a trigonometric ratio to find a side length in a right triangle
Using a trigonometric ratio to find an angle measure in a right triangle
Finding trigonometric ratios given a right triangle
Common angles and trigonometric functions
Finding values of trigonometric functions given information about an angle: Problem type 1
Finding values of trigonometric functions given information about an angle: Problem type 2
Finding values of trigonometric functions given information about an angle: Problem type 3
¨
Trigonometric Functions
Sketching the graph of a sine or cosine function: Problem type 1
Sketching the graph of a sine or cosine function: Problem type 2
Values of inverse trigonometric functions
Composition of a trigonometric function and an inverse trigonometric function: Problem type 1
Composition of a trigonometric function and an inverse trigonometric function: Problem type 2
Composition of a trigonometric function and an inverse trigonometric function: Problem type 3
¨
Trigonometric Identities and Equations
Cofunction identities
Double− angle identities
Product−to−sum and sum−to−product identities
Solving a basic trigonometric equation involving sine or cosine
Solving a basic trigonometric equation involving tangent, cotangent, secant, or cosecant
¨
Polar Coordinates
Plotting a point in polar coordinates
Converting rectangular coordinates to polar coordinates
Converting polar coordinates to rectangular coordinates
Converting an equation written in rectangular coordinates to one written in polar form
Converting an equation written in polar form to one written in rectangular coordinates
¨
•Limits and Continuity
Introduction to Limits
Estimating a limit numerically
Finding limits from a graph
¨
Computing Limits Algebraically
Finding limits for a piecewise−defined function
Finding a limit by using the limit laws: Problem type 1
Finding a limit by using the limit laws: Problem type 2
Finding a limit by using the limit laws: Problem type 3
Squeeze Theorem

Continuity
Determining points of discontinuity from a graph
Determining a parameter to make a function continuous
¨
Limits Involving Infinity
Limits at infinity and graphs
Limits at infinity and rational functions
Infinite limits and graphs
Infinite limits and rational functions
¨
Limits of Trigonometric Functions
Finding a limit of a trigonometric function by using continuity
Finding a limit by using special trigonometric limits

# Applications of Linear Algebra & Dierential Equations

Course Description:

Linear algebra is the branch of mathematics concerned with the study of linear transformations and
vector spaces
and lies at the heart of all numerical mathematics. In recent times linear algebra research
has flourished with new ideas about how to solve linear equations , carry out least squares procedures, and
find eigenvalues of matrices. In this course, we will learn to solve systems of linear equations and fully
characterize when solutions exist beginning with the case of n equations in n unknowns. We will also begin
to conceptualize vector spaces through an examination of Euclidean space.

Differential equations arise in many areas of science and technology whenever a deterministic relation-
ship involving some continuously changing quantities (modeled by functions) and their rates of change
(expressed as derivatives) is known or postulated. This course is devoted to help build our familiarity with
many applications and mathematical models involving ordinary differential equations (equations that in-
volve derivatives of a function of a single variable { usually time). We will learn classical and computational
techniques for solving differential equations and/or analyzing solution behavior.

Throughout the course we will use Matlab as our computational software. Learning to use this software
for computations, simulations, and visualization is a primary course goal.

Course topics:

UNIT I: Complex Numbers and Matrix Algebra. (4 meetings)

1. Introduction to Complex Numbers

2. The Cartesian and Exponential Forms

3. Roots of Polynomial Equations and Numbers

4. Matrix Notation and Terminology

5. The Solution of Simultaneous Equations

6. The Algebra of Matrices

7. Matrix Multiplication

8. The Inverse of a Matrix

9. The Computation of the Inverse

10. Determinants

(a) Definitions and Fundamental Theorems
(b) Minors and Cofactors

11. Linear Independence

12. Vector Spaces

UNIT II: First Order Ordinary Differential Equations. (4 meetings)

1. The First-Order Linear Equation

(a) Homogeneous Equations
(b) Nonhomogeneous Equations

2. Applications

(a) Linear Rate Equations
(b) Fluid Flow
(d) Population Growth
(e) Compound Interest
(f) Newton's Law of Cooling

3. Nonlinear Equations of First Order

(a) Separable Equations
(b) Exact Equations
(c) Euler's Method
(d) Direction Fields

UNIT III: First Order Systems of Ordinary Differential Equations. (6 meetings)

1. Eigenvalues and Eigenvectors

2. First-Order Homogeneous Systems

(a) Complex Eigenvalues
(b) Repeated Eigenvalues
(c) Phase Planes

3. Linear Differential Operators

4. Linear Independence and the Wronskian

5. Second-Order Linear Equations with Constant Coefficients

(a) Real and Unequal Roots
(b) Real and Equal Roots
(c) Complex Roots

6. Spring-Mass Systems in Free Motion

(a) Undamped Motion
(b) Damped Motion

7. Solutions of Nonhomogeneous Systems

(a) Undetermined Coefficients
(b) Variation of Parameters

8. Nonhomogeneous Initial Value Problems

9. Higher Order Equations

10. Phase Plane Analysis

Resources:

 Course information will be posted on Blackboard Vista.

Notes:
Notes for each class meeting will be posted on Blackboard the night before class. Download, print, and
bring these notes with you to each meeting.

Homework:
Homework exercises will be assigned during each class meeting for you to complete prior to our next class.
You are encouraged to post questions, results, or revelations stimulated by the homework on the discussion
board of Blackboard. Nontrivial comments authored by you about the homework or lectures will be tallied

Tests:
There will be three take-home tests corresponding to each of the three units. The Unit I Test will be
posted May 8 and due May 13, the Unit II Test will be posted May 22 and due May 27, and the Unit
III Test will be posted June 5 and is due June 10. Tests must be neatly presented in a single electronic
document posted to Blackboard. Almost always use complete sentences in your responses to questions and
exposition of problem solutions . Your solutions will likely include mathematical computations , formulae,
and plots that should be integrated into the write-up so the reader can easily follow and understand your
reasoning. Edit les that contain Matlab output so that only the relevant information is included.

Projects:

There will be two projects. Each requires you to research a topic and write a 2-page report of your results.
The first project is about population modeling with harvesting and will be due June 3. The second is
about the pendulum equation and is due June 17.

Each of the three tests, the two projects, and your homework participation will be equally weighted in the

Accommodations:

Students with physical, sensory, emotional or medical impairments may be eligible for reasonable accom-
modations in accordance with the Americans with Disabilities Act and Section 504 of the Rehabilitation
Act of 1973. All accommodations are coordinated through the Disability Resource Center (DRC) in Room
the DRC as early in the semester as possible. Alternate format materials (Braille, large print or digital)
making use of these resources and/or require accommodations for this course.