Lake Tahoe Community College Math Department SLOs and MASLOs

Developmental

1. Identify and know when to apply basic arithmetic concepts.
2. Solve problems by application of arithmetic principles.
3. Represent problems in written language, in symbolic form, and in graphical form.
4. Select and apply appropriate formulas.
5. Organize work in a logical, clearly stated order, correctly using mathematical symbols and language.
6. Use calculators effectively and appropriately.
7. State solutions to application problems in the context of the problem and recognize inappropriate and/or impossible answers.
8. Follow and demonstrate understanding of mathematical exposition [text readings, handouts, and lecture.]
9. Recognize the usefulness of elementary mathematics

General

1. Follow mathematical exposition, including descriptions of algorithms and derivations of formulas, presented either orally or in writing.
2. Determine whether a theorem or definition applies in a given situation, and use it appropriately if it applies.
3. Use the language and notation of differential and integral calculus correctly, and use appropriate style and format in written work.
4. Demonstrate good problem-solving habits, including:
        a.       estimating solutions and    recognizing unreasonable results.
        b.       considering a variety of approaches to a given problem, and selecting one that is appropriate.
        c.       rejecting the temptation to rely on mechanical techniques (either pencil-and-paper or electronic) that they do not understand.
        d.       interpreting solutions correctly, and answering the questions that were actually asked.
5. Use technology (especially calculators) effectively and appropriately.

Las Positas College

1. Students will demonstrate the ability to use symbolic, graphical, numerical, and written representations of mathematical ideas.

2. Students will read, write, listen to, and speak mathematics with understanding.
3. Students will use mathematical reasoning to solve problems and a generalized problem solving process to work word problems.
4. Students will learn mathematics through modeling real-world situations.
5. Students will use appropriate technology to enhance their mathematical thinking and understanding, solve mathematical problems, and judge the reasonableness of their results.

1. Produce, interpret, and analyze data and graphs
2. Solve mathematical equations
3. Construct, manipulate, and utilize mathematical functions
4. Engage in logical and critical thinking
5. Apply mathematical techniques to solve problems that arise in the real world

Los Medanos Community College

Developmental Math Program

Students completing the Developmental Math Program will demonstrate:

1. Problem-solving abilities: Students will use mathematical reasoning to solve problems and a generalized problem solving process to work word problems.
    a. The student can apply standard problem-solving methods and use relevant concepts to solve problems.
    b. The student uses a generalized problem-solving rubric if such a rubric is used in the class.
    c. The student’s written work demonstrates a conceptual understanding of course concepts.
    d. The student’s written work supports his/her solution.
    e. The student evaluates the reasonableness of his/her answer.

2. Mathematical versatility: Students will use verbal, graphical, numerical, and symbolic representations of mathematical ideas to solve problems.
    a. Students will use a variety of representations to demonstrate their understanding of mathematical concepts.
    b. Students will use a multi-prong approach to problem solving.
    c. Students will use appropriate technology to solve mathematical problems and judge the reasonableness of their results.

3. Communication skills: Students will read, write, listen to, and speak mathematics with understanding.
    a. Students will read and listen to mathematical presentations and arguments with understanding.
    b. Students will communicate both in speaking and in writing their understanding of mathematical ideas and procedures using appropriate mathematical vocabulary and notation.
    c. Students will coherently communicate their own mathematical thinking to others.

4. Preparation: Students will recognize and apply math concepts in a variety of relevant settings and demonstrate the math skills and knowledge necessary to succeed in subsequent courses.

5. Effective Learning Attributes: Students will demonstrate the characteristics of an effective learner.
    a. Student has the will to succeed and demonstrates the characteristics of a successful student: motivation, responsibility, focus, perseverance, the ability to cope with anxiety, a good attitude toward learning, and time management skills.
    b. Student has the skills to succeed. (S)he uses appropriate resources to improve learning and reach goals.
    c. Student self-monitors and self-regulates. (S)he assesses personal strengths and weaknesses in his/her learning process and then seeks and implements a strategy for improving learning.

Transfer Math Program

Students completing transfer-level math courses at LMC will demonstrate:

1. Preparation and Mathematical Maturity: Be prepared for the mathematical reasoning required in upper division work in their major, including the ability to generalize mathematical concepts and comprehend increasing levels of mathematical abstraction.

2. Mathematical Literacy: Communicate using mathematics:
a. Read with comprehension documents having mathematical content and participate cogently in discussions involving mathematics;
b. Clearly articulate mathematical information accurately and effectively, using a form, structure and style that suit the purpose (including written and face-to-face presentation).

3. Problem-solving ability:
    a. Reason with and apply mathematical concepts, principles and methods to solve problems or analyze scenarios in real-world contexts relevant to their major;
    b. Use technology effectively to analyze situations and solve problems;
    c. Estimate and check answers to mathematical problems in order to determine reasonableness, identify alternatives, and select optimal results.

4. Modeling ability:
a. Construct and interpret mathematical models using numerical, graphical, symbolic and verbal representations with the help of technology where appropriate in order to draw conclusions or make predictions;
b. Recognize and describe the limits of mathematical and statistical methods.

5. Effective Learning skills:
a. Independently acquire further mathematical knowledge without guidance, take responsibility for their own learning, and function effectively in different learning environments.
b. Succeed in different learning environments, particularly in a group setting of working collaboratively with others.

1. Students should improve their attitude towards math.
2. Students should have problem solving skills at an appropriate level.
3. Students should retain information from course to course.
4. Students should be completing the Math certificates and degree.

Three Categories:

1. Students completing advanced math core courses.
2. Students meeting the minimum requirements to transfer.
3. Students meeting the minimum requirements to graduate with Associates Degree.

Student Learning Outcomes

Be able to:

Solve a problem applying appropriate math concepts and ideas.
Effectively communicate solution(s).

Pre-Algebra

1. Strengthen core entry skills, which are to perform
        a. Operations with whole numbers
        b. Operations with fractions
        c. Operations with decimals
        d. Operations with percentages
2. Perform operations on integers.
3. Simplify and evaluate variable expressions.
4. Solve a one variable first degree linear equation that models situation.
5.   Construct linear graphs.
6. Convert units of measure (includes American and Metric systems).
7. Perform operations with polynomial

Part I

1. Perform arithmetic operations with whole numbers, fractions, and decimals.
2. Translate written language into mathematical statements.
3. Apply the concepts in the course to real-life situations.

Part II

1. Solve problems involving decimals, percents and beginning algebra
2. Translate written statements in to mathematical statements.
3. Apply the topics of Basic Arithmetic (Part II) to real life situations.

The student will:

1. Solve numerous problems in order to gain mastery of the arithmetic skills needed for everyday situations.
2. Demonstrate a systematic and logical approach to solving arithmetic problems.
3. Demonstrate the knowledge and skills required to select the correct introductory formulas, procedures, and concepts from algebra and geometry and use them to calculate and problem solve.
4. Solve word problems that require concept of central tendency and interpretation of statistical graphs.

By completing MATH 811 course students will be able to:

Correctly choose and apply the four basic arithmetic operations with whole numbers, decimals, fractions and sign numbers to estimate and solve application problems that are part of their daily lives. (Number Sense)
Apply “Proportional Reasoning” to solve related problems including ratios, rates, proportion, percent and conversions of units.
Compute the area and perimeter of geometric figures.
Learn to use the services available to improve study skills, test taking skills, problem solving skills and attitude toward learning mathematics

1. Perform arithmetic operations on signed values including integers, fractions, and decimals.
2. Simplify algebraic expressions, evaluate formulas, and solve basic linear equations and application problems.
3. Students obtain sufficient math proficiency to be successful in a subsequent elementary algebra course.

Beginning Algebra

1. Identify and apply basic algebraic concepts including slope, absolute value, scientific notation, equivalent equations, laws of exponents, intercepts, horizontal lines, and vertical lines.
2. Solve systems of linear equations in two unknowns using graphing, elimination, and substitution.
3. Solve equations and inequalities in one variable.
4. Solve quadratic equations by factoring and by using the quadratic formula.
5. Solve elementary radical equations.
6. Graph linear equations.
7. Solve problems by application of linear functions.
8. Apply the properties of and perform operations with radicals.
9. Apply the properties of and perform operations with integer exponents.

First Quarter

1. Solve linear equations and inequalities.
2. Define and employ terminology and arithmetic relating to polynomials in one variable.
3. Determine the equation and graph a line given information about the line.
4. Manipulate expressions with integral exponents.
5. Apply course topics to real-world situations.

Second Quarter

1. Factor a polynomial.
2. Apply the four basic operations to rational and radical expressions.
3. Solve equations with rational and radical expressions.
4. Solve a 2 x 2 system of linear equations.
5. Solve quadratic equations.
6. Apply course topics to real world situations.

Ohlone College

The student will:
1. Demonstrate basic skills in algebra up through quadratics and problem solving.
2. Set up stated problems algebraically and solve the resulting equations.
3. Solve problems presented via formulas or procedures.
4. Graph linear equations.
5. Solve systems of linear equations using graphing, substitution, and elimination methods.
6. Simplify exponential expressions with integer exponents.
7. Identify polynomials and perform operations with polynomials.
8. Factor polynomials using grouping, FOIL, special products formulas, and trial and error methods.
9. Solve quadratic equations using factoring and their applications.
10. Simplify rational expressions and complex fractions and solve applications of rational equations.

Upon completion of the course:
• Through real world applications, students will be able to create, manipulate, and interpret mathematical models of relationships defined by either a constant rate of change or a constant relative rate of change.
• Students will recognize, apply, and interpret multiple representations (graphic, symbolic, numerical/data, verbal/applied) of linear functions and their applications.
• Students will develop skills and attitudes for effectively solving problems at an introductory algebra level.

1. Distinguish between and give examples of equations, solutions to equations, and algebraic expressions.
2. Solve mathematical equations appropriate to the elementary algebra curriculum.
3. Formulate real-world problems quantitatively and interpret the results.

Intermediate Algebra

1. Identify and apply basic algebraic concepts including domain, range, slope, absolute value, scientific notation, equivalent equations, laws of exponents, intercepts, parallel lines, perpendicular lines, horizontal lines, and vertical lines;
2. Solve systems of linear equations in three unknowns using elimination and substitution
3. Solve equations and inequalities in one or two variables and involving absolute values
4. Solve quadratic equations by factoring, completing the square, and quadratic formula;
5. Solve exponential and logarithmic equations
6. Solve equations involving radicals
7. Perform basic operations on complex numbers
8. Find complex roots of a quadratic equation;
9. Sketch the graphs of functions and relations:
        a. algebraic, including polynomial and rational
        b. logarithmic
        c. exponential
        d. circles;
10. Find and sketch inverse functions;
11. Problem solve by application of linear and quadratic functions
12. Apply the concepts of logarithmic and exponential functions;
13. Apply the properties of and perform operations with radicals;
14. Apply the properties of and perform operations with rational exponents;
15. Apply linear and quadratic functions
16. Graph linear inequalities in two variables;
17. Find the distance between two points;
18. Find the midpoint of a line segment.

1. Apply the course topics to real-world situations.
2. Sketch and interpret the graphs of functions and relations introduced in intermediate algebra.
3. Simplify mathematical expressions into forms more amenable to analysis.
4. Provide solutions to equations using methods from intermediate algebra.

The student will:
1. Solve problems involving the mathematical concepts of function and functional inverse.
2. Show increased skill in setting up and solving applications.
3. Solve mathematical problems using concepts that may be useful for learning statistics: logarithms, sigma notation, and the binomial theorem.
4. Solve mathematical problems in topics useful for trigonometry: functions and inverses and their graphs, quadratic equations, and conic sections.

Upon completion of the course:
• Through real world applications, students will be able to create, manipulate, and interpret mathematical models of relationships involving linear, exponential, polynomial, radical, and rational functions.
• Students will recognize, apply, and interpret multiple representations (graphic, symbolic, numerical/data, verbal/applied) of functions and their applications.
• Students will develop skills and attitudes for effectively solving problems at an intermediate algebra level.

1. Evaluate functions and utilize functions and their graphs appropriate to the intermediate algebra curriculum to solve and interpret the solutions to real-world problems.
2. Recognize exponential relationships and apply logarithmic principles to solve and interpret the solutions to real-world problems.

Geometry

1. Demonstrate familiarity with geometric vocabulary
        a. Classify angles as acute, obtuse, straight, or right angles.
        b. Identify and name specific polygons
        c. Identify and name simple conic sections
        d. Identify alternate interior angles and alternate exterior angles
        e. Identify the four parts of an axiomatic system.
        f. Identify the hypothesis and conclusion in and the converse and contrapositive of a conditional statement.
        g. State and apply the triangular inequality
2. Apply the parallel postulate.
3. Identify congruent figures and apply theorems relative to congruent figures.
4. Perform basic geometric constructions using a compass and straight edge.
5. Prove geometric theorems
        a. Use reflexive, symmetric, and transitive properties
        b. Prove theorems with respect to lines are parallel.
        c. Prove two triangles are congruent or not congruent
        d. Construct indirect proofs
6. Perform calculations requiring geometric knowledge and formula
        a. Apply formulas and theorem to find area and perimeter.
        b. Find volume and surface area of prisms and pyramids.
        c. Apply the Pythagorean Theorem.
        d. Use ratio and proportion to solve similar figures.
        e. Solve for perimeter, area, and volume of similar figures
7. Apply theorems about the geometry of a circle.
8. Graph figures.
9. Recognize transformations.

1. Prove geometric statements using classical axioms and theorems.
2. Perform ruler and compass constructions.
3. Make deductions using the rules of logic.
4. Solve problems involving parallel lines, triangles, and angles.

The student will:
1. Master the terms, postulates, and theorems of geometry.
2. Construct both direct and indirect proofs in geometric problems.
3. Const ruct and rmal logical arguments and give counterexamples to disprove statements.
4. Prove basic theorems involving congruence and similarity.
5. Solve practical problems such as finding lengths, areas, volumes and angles of geometric figures.

1. Apply deductive reasoning to construct formal proofs of geometric theorems.
2. Apply relevant theorems to solve problems related to geometric figures.

Statistics

1. See that statistical analyses are useful for "real world" questions and problems.
2. Understand the kinds of problems for which statistical analysis can be of help.
3. Become knowledgeable and critical users of descriptive and inferential statistical analyses; specifically:
a. Assess whether a descriptive analysis is appropriate for the data and problem posed.
b. Explain what a descriptive analysis says about the data it is describing.
c. Demonstrate understanding of the concepts of variable, variability, the distribution of a variable, the center, spread and shape of a distribution of a variable, and how these features of a distribution are appropriately measured and modeled.
d. Describe in words the relationship between two variables from graphical or numerical analyses in the context of the data, whether the variables be quantitative or categorical.
e. Identify appropriate methods and models of analyzing the relationship between variables whether the variables be quantitative or categorical.
f. Construct graphical presentations displaying data, measure features of distributions, and interpret the results in writing for data that the student helped to collect.
g. Explain, with a mixture of prose and mathematics, why a formula does what it is said to do.
4. Demonstrate awareness that the methods used to produce data can be as important as what is done with the data.
5. Apply the language and the logic of probability, not only in “probability problems” but also in the context of inference.
6. Show familiarity with the idea of a theoretical (or mathematically) defined probability distribution, as illustrated by uniform distribution, normal distributions or binomial distributions.

1. Design and implement an unbiased study that will produce sound statistical results.
2. Generate and interpret statistics graphs from data that arise from surveys and experiments.
3. Implement the rules of probability.
4. Apply confidence intervals and test hypotheses to make conclusions about data that come from practical applications.
5. Perform regression analysis to make informed predictions about relationships between quantitative variables.

The student will:
1. Understand basic statistical concepts and vocabulary.
2. Understand basic probability concepts and vocabulary.
3. Use statistical formulas.
4. Choose correct statistical tool for analysis of word problems.
5. Use technology for statistical applications.

Upon completion of this course the student will be able to:
• Organize, analyze, and utilize appropriate methods to draw conclusions based on sample data by using tables, graphs, measures of central tendency, and measures of dispersion.
• Have sufficient command of the concepts and terminology of probability and statistics.
• Collect data, interpret and communicate the results using statistical analyses such as confidence intervals, hypothesis tests, and regression analysis.
• Successfully apply statistics to ones selected major at a transfer institution

1. Critically analyze statistical information presented in media, journals, etc."
2. Convert data to statistical evidence and interpret the evidence.

Liberal Arts Mathematics

1. apply principles from algebra and elementary number theory to current technology, i.e. encryption techniques.
2. identify similarities and differences between Euclidean and non-Euclidean Geometries
3. explain certain relationships between geometry and topology
4. demonstrate familiarity with the Platonic Solids
5. apply problem solving techniques learned in one area to another area
6. measure uncertainty using basic probability techniques
7. count using permutations and combinations
8. identify contributions of selected mathematicians

1. Apply combinatorics and the rules of probability to real life situations.
2. Analyze statistical information and the 'Normal' distribution to make conclusions based on data.
3. Incorporate the mathematics of finance to be consumer-wise.
4. Utilize trigonometric formulas to solve problems involving triangles.
5. Develop exponential growth and decay models.

The student will:
1. Apply mathematical principles and techniques to solve problems in areas such as systems of numeration, algebraic modeling, basic trigonometry, intuitive calculus, and math of finance.
2. Use critical thinking to arrive at conclusions from Venn Diagrams, syllogistic forms, and truth tables.
3. Demonstrate a knowledge of probability and statistics by solving a variety of counting problems, by calculating the probability of games of chance, and by analyzing statistical data.
4. Relate a knowledge of the people, history and uses of mathematics through research papers, projects, presentations, and class discussions.

1. Apply problem solving skills to solve unfamiliar problems related to the topics studied.

Pre-Calculus / College Algebra

1. Solve and apply equations and inequalities including linear, quadratic, absolute value, polynomial, rational, radical, exponential, logarithmic, and trigonometric equations.
2. Graph linear, quadratic, absolute value, polynomial, rational, radical, exponential, logarithmic and trigonometric functions, and parametric equations.
3. Perform function operations including composition, transposition, and finding inverse functions.
4. Apply techniques for finding zeros of polynomial functions.
5. Solve systems of equations by application of algebraic techniques and/or matrix techniques.
6. Apply formulas from analytic geometry.
7. Define, recognize, and solve for terms of arithmetic and geometric series.

First Quarter

1. Produce and interpret graphs of functions and relations.
2. Apply techniques to solve polynomial and rational equations and inequalities.
3. Model real life situations using algebraic methods.
4. Simplify algebraic expressions using skills obtained in the course.

Second Quarter

1. Prove and derive mathematical statements using various methods including induction.
2. Employ matrices and their properties to solve systems of equations.
3. Construct and interpret graphs of conic sections and transcendental functions.
4. Apply the topics of the course to real world situations.

The student will:
1. Solve equations involving algebraic and transcendental functions, and systems of equations.2. Solve real-world applications of the above.
3. Graph algebraic and transcendental functions.
4. Apply a graphing calculator to all of the above.
5. Solve introductory sequence and series problems.

• Apply Algebraic concepts to equation and problem solving.
• Apply modeling techniques to solve real world problems.
• Apply graphing methods to be able to synthesize graphical concepts to check algebraic solutions, as well as, to find solutions where algebraic ones are not possible.
• Apply their functional awareness to be successful in Calculus

1. Demonstrate the ability to use functions as a mathematical tool to model the conceptual ideas of algebra and trigonometry.
2. Construct, derive, and graph basic trigonometric relationships.

Trigonometry

1.       State and apply correctly the definitions (unit circle, right triangle, and x-y-r), values for key angles, properties (e.g. periodicity and domain and range), and basic identities, for the six trig functions.
2.       Work with and apply the algebraic relationships among the six trig functions: use algebra and identities to derive other identities, verify identities, simplify expressions, and solve trigonometric equations.
3.       Solve right triangles using right triangle definitions of trig functions, and oblique triangles using the laws of sines and cosines
4.       Solve applied trigonometry problems involving triangles or periodic behaviors.
5.       Produce and interpret graphs of sine and cosine functions, with correct amplitude, period, phase shift, and vertical shift.
6.       Demonstrate understanding of inverse trig functions and their applications.
7.       Model periodic phenomena using sine and cosine functions.
8.       State solutions to application problems in context and recognize inappropriate or impossible answers.
9.       Follow and demonstrate understanding of mathematical exposition [e.g. text readings, handouts, and on-line resources].
10.   Organize work in a logical, clearly stated order, correctly using mathematical symbols and language.

1. Provide and analyze graphs of trigonometric functions.
2. Apply trigonometric techniques to solve problems in real world contexts.
3. Derive and prove trigonometric properties and identities.
4. Produce solutions to equations using skills developed in trigonometry.

The student will:
1. Identify six trigonometric functions and express them as the ratio of the sides of a right triangle.
2. Solve right triangle problems.
3. Convert angles from degrees to radians and from radians to degrees.
5. Define the domain and range of trigonometric functions.
6. Graph trigonometric functions using amplitude, period, phase shift and/or vertical translation.
7. Define domain and range of inverse trig functions.
8. Verify trigonometric identities using fundamental identities, sum and difference formulas, double angle formulas and half angle formulas.
9. Solve trigonometric equations on either a restricted domain or a general domain.
10. Applications.
11. Use Law of Sines and Law of Cosines to solve problems using oblique triangles.

Upon completion of the course:
• Through real world applications, students will be able to create, manipulate, and interpret mathematical models of periodic relationships.
• Students will recognize, apply, and interpret multiple representations (graphic, symbolic, numerical/data, verbal/applied) of periodic functions and their applications.
• Students will recognize, apply, and interpret proportional reasoning in the context of right triangles and circles by analyzing a variety of problems, then applying the trigonometric definitions, choosing appropriate ideas, and applying these ideas to the solution of the problems.
• Students will develop skills and attitudes for effectively solving problems at a transfer math course level.

1. Solve application problems involving trigonometric functions and graphs.

Calculus I

1. Calculate limits when they exist, and explain why when they do not.
2. Determine where a function is continuous and/or differentiable, and explain why.
3. Compute derivatives of polynomial, rational, algebraic, exponential, logarithmic, and trigonometric functions.
4. Use techniques of differentiation, including the product, quotient, and chain rules, and implicit differentiation.
5. Apply differentiation to the study of functions and their graphs, to optimization and related rate problems, and to applications from science and economics.
6. Compute antiderivatives of polynomial, rational, algebraic, exponential, logarithmic, and trigonometric functions.
7. Interpret Riemann sums as definite integrals, relate definite integrals to areas, and evaluate definite integrals using the fundamental theorem of calculus.

First Quarter

1. Differentiate functions of a single variable using the basic rules of differentiation.
2. Apply the derivative to describe phenomena arising from real-life situations.
3. Sketch and analyze graphs using the first and second derivatives.
4. Prove corollaries and derive equations using the theorems that relate to differential calculus.
5. Determine limits and continuity using graphical, analytical, and tabular techniques.

Second Quarter

1. Employ integrals to applications from physics.
2. Apply the Fundamental Theorem of Calculus in determining indefinite integrals.
3. Compute geometric quantities using integrals.
4. Solve differential equations.
5. Determine integrals and derivatives of transcendental functions.

The student will:
1. Compute limits using numerical, graphical, and algebraic methods.
2. Differentiate algebraic, trigonometric, logarithmic, exponential, and inverse trig functions.
3. Apply differentiation to problems in the areas of geometry, physics, engineering, and business, including slopes of tangent lines and rates of change.
4. Integrate algebraic, trigonometric, and exponential functions using introductory techniques.
5. Apply integration to finding the area under a curve.
6. Demonstrate logical thinking, correct use of notation, and mathematical precision in formulating and solving problems in the above areas.
7. Apply the appropriate use of a graphing calculator to each of the above areas.

• Students will be able to explain and apply the techniques of differential calculus to construct derivatives graphically, numerically and analytically.

• Students will be able to translate problems from the physical, life and social sciences into workable mathematical form, suitable for graphical, numerical, analytical or verbal solutions.

• Students will be able to use technology where appropriate, in particular, the use of a graphing calculator to
a. produce the graph of any function in an arbitrary viewing window
b. estimate the value of a numerical derivative of a function at a point and
c. estimate the value of a definite integral to a given degree of accuracy

1. Define and apply the concepts of limits, continuity, derivatives and antiderivatives to solve a variety of word problems (both familiar and unfamiliar) and corroborate their solutions with practical reasoning.
2. Demonstrate understanding of the geometric relationship between a function, its first and second derivatives and its antiderivatives.
3. Interpret and analyze information to develop strategies for solving problems involving related rates, optimization, and approximation by linear models.

Calculus II

1. Evaluate definite integrals using the fundamental theorem of calculus.
2. Analyze geometric and physical situations to obtain Riemann sums, and interpret and evaluate them as definite integrals.
3. Use numerical methods to estimate the value of definite integrals.
4. Use techniques of integration, including algebraic and trig substitutions, integration by parts, and partial fractions, to evaluate definite and indefinite integrals.
5. Find limits of sequences, or show that the limit does not exist
6. Determine whether series diverge, converge conditionally, or converge absolutely, and find or estimate sums of series.
7. Find intervals of convergence of power series.
8. Find Taylor and Maclaurin series of functions.
9. Interpret and solve certain types of differential equations, including separable and first order linear.

1. Test series for convergence.
2. Relate analytic functions to their power series.
3. Apply calculus to functions of several variables.
4. Model real life applications using three-dimensional constructs.
5. Perform arithmetic on vectors using both component and geometric forms.

The student will:
1. Apply the use of integrals to problems involving volumes of solids, arc length, surface area, and physics applications.
3. Solve basic differential equations.
4. Determine the convergence or divergence of infinite sequences and series by using appropriate tests.
5. Determine polynomial representations of mathematical functions by using power series.
6. Analyze mathematical relationships given in parametric and polar forms.
7. Graph conic sections and determine information about the conic from its algebraic equation.
8. Demonstrate logical thinking, correct use of notation, and mathematical precision in formulating and solving problems in the above areas.
9. Apply the use of a graphing calculator to each of the above areas.

• Students will be able to explain and apply the techniques of integral calculus to construct anti-derivatives graphically, numerically and analytically.

• Students will be able to translate problems from the physical, life and social sciences into workable mathematical form, suitable for graphical, numerical, analytical or verbal solutions.

• Students will be able to use technology where appropriate, in particular, the use of a graphing calculator to
    o produce the graph of any function in an arbitrary viewing window
    o estimate the value of a numerical derivative of a function at a point and
    o estimate the value of a definite integral to a given degree of accuracy

• Students will be able to solve differential equations with an emphasis on qualitative solutions, modeling and interpretation.

• Students will be able to analyze the convergence or divergence of sequences and series of constants with a variety of techniques.

1. Apply numerical methods to approximate definite integrals, improper integrals that converge, and infinite series that converge and bound the errors.
2. Find antiderivatives using a variety of techniques of integration.
3. Interpret analyze information to develop strategies for finding area, volume, and arc length.

Calculus III

1. Compute limits of functions of several variables, or show that the limits do not exist.
2. Determine where functions of several variables are continuous.
3. Compute partial derivatives and directional derivatives.
4. Solve extremum problems, using a) partial derivatives and b) Lagrange multipliers.
5. Set up and evaluate multiple integrals, and use them in geometric and physical applications.
6. Compute dot and cross products of vectors, and use them to find equations of lines and planes in R3.
7. Find arc length, curvature, tangent and normal vectors to space curves.
8. Compute line and surface integrals.
9. Use cylindrical and spherical coordinates, and parametric equations, to study functions of several variables and surfaces.
10. Compute and apply the gradient of scalar functions, and the divergence and curl of vector fields.

1. Perform calculus on vector value functions.
2. Evaluate double and triple integrals.
3. Integrate vector value functions.
4. Relate types of single and multiple integrals using the major theorems of vector calculus.
5. Apply multivariable calculus to problems arising from physics.

The student will:
1. Use vector methods to solve problems in three dimensional analytic geometry.
2. Analyze problems involving vector functions of a single variable. Topics include two dimensional normal and tangential acceleration and curvature.
3. Determine the extreme value(s) of a multi-dimensional function, the tangent plane to a three dimensional function, the directional derivative and gradient of a function by using partial derivatives.
4. Use double and triple integrals to determine the areas and volumes bounded by curves and surfaces, determine the surface area and center of mass of a solid. Use polar, cylindrical and spherical coordinates for solving these types of problems.
5. Evaluate line and surface integrals by using Green's Theorem, the Divergence Theorem, and Stokes' Theorem
6. Demonstrate using a graphing calculator.

• Students will be able to explain and apply the techniques of multivariable calculus to solve problems graphically, numerically and analytically.
• Students will be able to translate problems from the physical, life and social sciences into workable mathematical form, suitable for graphical, numerical, analytical or verbal solutions.
• Students will be able to calculate derivatives and integrals, using vectors and other tools fundamental to multivariable calculus.
• Students will be able to solve multivariable problems graphically and analytically, by examining level sets, partial derivatives, finding extrema, etc.

1. Define, understand and apply concepts of three dimensional analytical geometry, vector calculus, and functions of two or more variables.
2. Make connections between the theorems and applications in vector calculus.

Linear Algebra

1. Parametrically construct the solution space of a linear system using Gaussian elimination.
2. Use elementary row operations to reduce a matrix to row echelon form.
3. Successfully employ all of the standard operations with matrices and vectors.
4. Judge when a function is or is not a linear transformation.
5. Given a linear transformation,
        a. Recognize the standard matrix (if the domain and range are of the form Rn).
        b. Construct the null space as a span of vectors.
        c. Construct the range space as a span of vectors.
        d. Assess if it is invertible, and, if it is, construct the inverse.
        e. Assess if it is one to one and/or onto.
6. Construct elementary matrices corresponding to elementary row operations and can use both to construct the inverse of an invertible square matrix.
7. Judge if a set of vectors is linearly independent or dependent.
8. Construct a basis for a given vector space.
9. Calculate the dimension of a given vector space.
10. Judge if a subset of a given vector space is a subspace.
11. Assess whether a given set and field with addition and scalar multiplication is or is not a vector space.
12. Calculate the new representations of a vector or standard matrix under a change of basis.
13. Calculate the determinant of a square matrix and use it to judge the linear independence of row or columns and judge invertibility.
14. Apply the basic properties of the determinant.
15. Calculate a determinant by expansion by majors.
16. Apply the concept of matrix similarity.
17. Calculate the eigenvalues and construct a basis for the eigenspaces of a matrix or linear transformation.
18. Construct the diagonal decomposition of a square matrix or else explain why the matrix can not be diagonalized.
19. Judge when a given (vector, vector) to scalar operation is or is not an inner product.
20. Express vectors and the representation of a linear transformation in terms of a new basis.
21. Construct orthonormal bases of Rⁿ.

1. Apply the theory and techniques of linear algebra in applications from physics, operations research and other scientific disciplines.
2. Solve linear systems, including under- and over-determined systems.
3. Prove lemmas and corollaries in linear algebra.
4. Relate linear transformations to their matrices with respect to given bases.
5. Describe linear transformations as functions mapping an n-dimensional space to an m-dimensional space.

The student will:
1. Use elementary row operations to reduce a matrix to reduced row echelon form and apply this technique to solving systems of linear equations.
2. Perform the arithmetic of vectors and matrices.
3. Identify definitions and properties of the determinant of a matrix.
4. Identify definitions and properties of vector spaces, including subspaces, basis and dimension.
5. Perform linear transformations and find the range, kernel, nullity and rank of a transformation.
6. Expand their levels of thinking to more abstract mathematics and apply this abstract thinking to proofs.
7. Apply the above material to current applications.

Upon completion of the course:
• Students will be able to formulate linear systems as mathematical models.
• Students will be able to represent any linear system with a suitable matrix equation.
• Students will be able to compute the general solution to any linear system.
• Students will recognize inherent geometric and analytical properties of a given matrix.

1. Compute matrix row operations, determinants, and inverses by hand and using technology.
2. Define, understand and apply concepts of linear systems, invertible matrices, vector spaces, a basis, linear transformations, range and kernel change of basis, eigenvalues and eigenvectors, matrix, inner product and orthogonality.
3. Write simple proofs and give explanations that demonstrate understanding of the concepts of linear systems, invertible matrices, vector spaces, a basis, linear transformations, range and kernel change of basis, eigenvalues and eigenvectors, matrix, inner product and orthogonality.

Differential Equations

1. Recognize the types of first order and second order ODE's.
2. Appropriately employ qualitative methods to analyze first order ODE's.
3. Solve separable first order ODE's.
4. Solve first order ODE's with homogeneous coefficients.
5. Explain the meaning of existence and uniqueness.
6. Recognize when a first order ODE must or must not have a unique solution.
7. Calculate approximate solutions to first order ODE's using selected numerical methods.
8. Solve second order homogeneous linear ODE's with constant coefficients.
9. Solve second order linear ODE's with constant coefficients and a forcing function.
10. Explain the meaning of linear independence.
11. Evaluate when given functions are linearly independent or not.
12. Employ the Wronskian to assess the linear independence of given functions.
13. Employ reduction of order to find the general solution of a second order ODE given one solution.
14. Employ variation of parameters to solve second order ODE's.
15. Calculate the Laplace transform of a given function.
16. Calculate the inverse Laplace transform of a given function.
17. Calculate the Laplace transform for piecewise defined functions.
18. Employ convolution to calculate the Laplace transform of a given function.
19. Solve first and second order ODE's by employing Laplace transforms.
20. Solve first and second order ODE's using power series in the case of non-singular and singular points.
21. Construct the first order system equivalent to a given second order equation and vice versa.
22. Employ both solution trajectories and phase plane analysis to explain solutions to first order 2 x 2 systems of ODE's.
23. Solve first order, linear, autonomous first order systems of ODE's.
24. Categorize equilibrium points of systems of first order ODE's.
25. Calculate the Fourier series of given functions.
26. Solve given partial differential equations using separation of variables and Fourier series.

1. Apply ordinary differential equations to problems from physics, biology, and other scientific disciplines.
2. Employ the technique of transformations in finding solutions to ordinary differential equations.
3. Prove results from the field of differential equations.
4. Sketch direction fields for first-order ordinary differential equations.
5. Solve differential equations using sequences, series, and matrices.

The student will:
1. Solve first and second order linear differential equations using techniques including separation of variables, reduction of order, constant coefficients, and undetermined coefficients.
2. Solve an ordinary differential equation through the use of infinite series.
3. Solve an ordinary differential equation through the use of Laplace Transforms.
4. Convert a higher order ordinary differential equation into a system of first order differential equations.
5. Solve a partial differential equation by the method of separation of variables.
6. Apply the methods of solving differential equations to problems in spring-mass systems, electrical circuits, exponential growth and decay, mixing, and Newton's Law of Cooling.
7. Use computer software to draw direction fields and solutions to differential equations.

Upon completion of the course:
• Through real world applications, students will be able to create, manipulate, solve, and interpret mathematical models describing various physical phenomena using scalar or vector differential equations.
• Students will be able to apply analytical, numerical, and qualitative methods to achieve solutions to a wide class of differential equations.
• Students will develop skills for effectively solving problems at an introductory level of differential equations study.

1. Analyze a differential equation and select and implement an appropriate method to solve the equation.
2. Construct and solve a differential equation that models a given situation.
3. Use numerical methods to approximate the solution of a differential equation.

Others