Single Variable Calculus – First Semester
A. Description
Differential and integral calculus of a single variable. Functions; limits and continuity; differentiation, integration
B. Recommended Preparation
Three years of high school mathematics (or equivalent); college level
courses in the study of functions to include polynomial, rational,
algebraic, trigonometric, exponential, logarithmic, and other
transcendental functions
C. Prerequisites
Precalculus, or college algebra and trigonometry
D. Minimum Unit Requirement
Four (4) semester units
E. Student Learning Outcomes
Upon successful completion of the course, students will be able to:
• Compute the limit of a function at a real number
• Determine if a function is continuous at a real number
• Find the derivative of a function as a limit
• Find the equation of a tangent line to a function
• Compute derivatives using differentiation formulas
• Apply differentiation to solve related rate problems, optimization problems
• Use implicit differentiation
• Graph functions using methods of calculus
• Evaluate a definite integral as a limit
F. Topics
All the topics from the single variable calculus descriptor relating to
differentiation and some topics from integration. A suggested
list of topics could include
• Limits, lefthand and righthand limits
• Computing limits using numerical, graphical, and algebraic approaches
• Continuity; continuity at a real number, discontinuity at a real number, removable discontinuity
• Tangent lines
• Derivative as a limit
• Interpretation of the derivative as: slope of tangent line, a rate of change
• Derivative as a function
• Differentiation formulas; constants, power rule, product rule, quotient rule
• Rates of change
• Derivatives of trigonometric functions
• Chain rule
• Implicit differentiation
• Higherorder derivatives
• Related rates
• Maximum and minimum values (absolute and local)
• Critical numbers
• Mean Value Theorem
• Graphing functions using first and second derivatives
• Limits at infinity, horizontal asymptotes
• Infinite limits
• Optimization
• Antiderivatives
• Area under a curve
• Definite integral; Riemann sum
• Properties of the integral
• Fundamental Theorem of Calculus
• Indefinite integrals
• Integration by substitution
• Areas between curves
• Volume, volume of a solid of revolution

Single Variable Calculus – Second Semester
A. Description
Differential and integral calculus of a single variable.
Integration, techniques of integration; infinite sequences and series;
applications of differentiation and integration
B. Recommended Preparation
Three years of high school mathematics (or equivalent); college level
courses in the study of functions to include polynomial, rational,
algebraic, trigonometric, exponential, logarithmic, and other
transcendental functions
C. Prerequisites
Single variable calculus – first semester
D. Minimum Unit Requirement
Four (4) semester units
E. Student Learning Outcomes
Upon successful completion of the course, students will be able to:
• Evaluate a definite integral as a limit
• Evaluate a definite integral using integration formulas
• Find indefinite integrals
• Apply integration methods
• Find areas and volumes by integration
• Determine the limit of convergent sequences
• Determine the limit of convergent series
• Represent functions as power series
F. Topics
Remaining topics on differentiation and integration from the single
variable calculus descriptor not covered in the first semester,
parametric equations, polar curves, conic sections, sequences, series,
power series, Taylor and Maclaurin series, binomial series. A
suggested list of topics could include:
• Derivatives of inverse functions
• Logarithmic and exponential functions and their derivatives
• Logarithmic differentiation
• Inverse trigonometric functions and their derivatives
• Indeterminate forms and L'Hopital's Rule
• Techniques of integration; trigonometric integrals, trigonometric substitution, partial fractions
• Numerical integration; trapezoidal and Simpson's rule
• Improper integrals
• Applications; arc length, area of a surface of revolution, moments and centers of mass
• Separable first order differential equations
• Exponential growth and decay
• Parametric equations, calculus with parametric curves
• Polar curves, calculus in polar coordinates
• Conic sections
• Sequences; convergence, divergence
• Series; convergence and divergence, alternating series
• Tests for convergence of series (including integral
test, comparison tests, ratio test, and root test), divergence test
• Estimating the sum of a series
• Power series, radius of convergence, interval of convergence
• Differentiation and integration of power series
• Taylor and Maclaurin series; Taylor’s Inequality
• Binomial series

Multivariable Calculus
A. Description
Vector valued functions, calculus of functions of more than one
variable, partial derivatives, multiple integration, Green’s Theorem,
Stoke’s Theorem, divergence theorem
B. Recommended Preparation
One year of single variable calculus
C. Prerequisites
Calculus of a single variable (to be named)
D. Minimum Unit Requirement
Four (4) semester units
E. Student Learning Outcomes
Upon successful completion of the course, students will be able to:
• Compute vector operations
• Determine equations of lines and planes
• Compute the limit of a function at a point
• Compute derivatives
• Compute the equation of a tangent plane at a point
• Determine differentiability
• Determine and test for local extrema, saddle points
• Solve constraint problems using Lagrange multipliers
• Compute arc length
• Compute the divergence and curl of a vector field
• Evaluate two and three dimensional integrals
• Apply Green’s Theorem
• Apply Stoke’s Theorem
• Apply the divergence theorem
F. Topics
• Vectors in two and three dimensions
• Vector addition, scalar multiplication, standard basis vectors
• Vector equation of a line, parametric equation of a line
• Dot (inner) product, CauchySchwarz Inequality, projection
• Cross product, matrices, determinant (2 x 2, 3 x 3), triple product
• Vector equation of a plane, rectangular equation of a plane
• Functions of several variables, realvalued functions
• Level sets, curves, and surfaces
• Limit, properties of limits
• Continuity, properties of continuous functions
• Partial derivatives
• Differentiability
• Gradient (grad f)
• Curves, tangent vector
• Properties of derivatives
• Chain rule
• Gradients, directional derivatives, gradient vector field
• Higherorder derivatives
• Local maxima and minima, saddle point
• Global maxima and minima
• Lagrange multipliers
• Vectorvalued functions
• Arc length
• Vector fields
• Divergence and Curl
• Double and triple integrals
• Change of variables theorem, Jacobian
• Integrals in polar, cylindrical, and spherical coordinates
• Integrals over paths and surfaces
• Line integrals
• Integrals of realvalued functions over surfaces
• Green’s Theorem
• Stoke’s Theorem
• Conservative fields
• Divergence theorem

Ordinary Differential Equations
A. Description
The course is an introduction to ordinary differential equations
including both quantitative and qualitative methods as well as
applications from a variety of disciplines. Introduces the theoretical
aspects of differential equations, including establishing when
solution(s) exist and techniques for obtaining solutions, including,
series solutions, and singular points, Laplace transforms and linear
systems.
B. Recommended Preparation
Three semesters of calculus for science, mathematics and engineering.
C. Prerequisites
Multivariable Calculus
D. Minimum Unit Requirement
Three (3) semester units
E. Student Learning Outcomes
On completion of the course the student will be able to:
• Create and analyze mathematical models based on ordinary differential equations
• Determine the type of a given differential equation, determine the
existence of a solution and if a solution can be obtained, select the
appropriate analytical technique for finding the solution
• Utilize technology tools to find geometric, graphical and numeric techniques for the analysis of solutions
• Solve Linear Systems of equations using eigenvalues and eigenvectors
F. Topics
• Solutions of ordinary differential equations
• Separation of variables
• Equations with homogeneous coefficients
• Nonlinear differential equations
• Exact equations, Euler's method
• Existence and uniqueness
• Applications
• Second order linear differential equations
• Fundamental solutions, independence, Wronskian
• Complex and repeated eigenvalues
• Nonhomogeneous equations
• Application: The harmonic oscillator
• Variation of parameters
• Higher order linear equations
• Systems of Ordinary differential equations
• Matrices
• Solving linear systems of ordinary differential equations by diagonalization
• Complex eigenvalues and fundamental matrices
• Phase plane

Single Variable Calculus
A. Description
Differential and integral calculus of a single variable. Functions;
limits and continuity; differentiation, integration, techniques of
integration; infinite sequences and series; applications of
differentiation and integration
B. Recommended Preparation
Three years of high school mathematics (or equivalent); college level
courses in the study of functions to include polynomial, rational,
algebraic, trigonometric, exponential, logarithmic, and other
transcendental functions
C. Prerequisites
Precalculus, or college algebra and trigonometry
D. Minimum Unit Requirement
Eight (8) semester units (suggested two foursemester unit courses)
E. Student Learning Outcomes
Upon successful completion of the course, students will be able to:
• Compute the limit of a function at a real number
• Determine if a function is continuous at a real number
• Find the derivative of a function as a limit
• Find the equation of a tangent line to a function
• Compute derivatives using differentiation formulas
• Apply differentiation to solve related rate problems, optimization problems
• Use implicit differentiation
• Graph functions using methods of calculus
• Evaluate a definite integral as a limit
• Evaluate a definite integral using integration formulas
• Find indefinite integrals
• Apply integration methods
• Find areas and volumes by integration
• Determine the limit of convergent sequences
• Determine the limit of convergent series
• Represent functions as power series
F. Topics
• Limits, lefthand and righthand limits
• Computing limits using numerical, graphical, and algebraic approaches
• Continuity; continuity at a real number, discontinuity at a real number, removable discontinuity
• Tangent lines
• Derivative as a limit
• Interpretation of the derivative as: slope of tangent line, a rate of change
• Derivative as a function
• Differentiation formulas; constants, power rule, product rule, quotient rule
• Rates of change
• Derivatives of trigonometric functions
• Chain rule
• Implicit differentiation
• Higherorder derivatives
• Related rates
• Maximum and minimum values (absolute and local)
• Critical numbers
• Mean Value Theorem
• Graphing functions using first and second derivatives
• Limits at infinity, horizontal asymptotes
• Infinite limits
• Optimization
• Antiderivatives
• Area under a curve
• Definite integral; Riemann sum
• Properties of the integral
• Fundamental Theorem of Calculus
• Indefinite integrals
• Integration by substitution
• Areas between curves
• Volume, volume of a solid of revolution
• Derivatives of inverse functions
• Logarithmic and exponential functions and their derivatives
• Logarithmic differentiation
• Inverse trigonometric functions and their derivatives
• Indeterminate forms and L'Hopital's Rule
• Techniques of integration; trigonometric integrals, trigonometric substitution, partial fractions
• Numerical integration; trapezoidal and Simpson's rule
• Improper integrals
• Applications; arc length, area of a surface of revolution, moments and centers of mass
• Separable first order differential equations
• Exponential growth and decay
• Parametric equations, calculus with parametric curves
• Polar curves, calculus in polar coordinates
• Conic sections
• Sequences; convergence, divergence
• Series; convergence and divergence, alternating series
• Tests for convergence of series (including integral
test, comparison tests, ratio test, and root test), divergence test
• Estimating the sum of a series
• Power series, radius of convergence, interval of convergence
• Differentiation and integration of power series
• Taylor and Maclaurin series; Taylor’s Inequality
• Binomial series

Introduction to Linear Algebra
A. Description
This course provides a careful development of the techniques and theory
needed to solve and classify systems of linear equations.
Solution techniques include row operations, Gaussian elimination, and
matrix algebra. Also covered is a thorough investigation of the
properties of vectors in two and three dimensions, leading to the
generalized notion of an abstract vector space. A complete
treatment of vector space theory is presented including topics such as
inner products, norms, orthogonality, eigenvalues, eigenspaces, and
linear transformations. Selected applications of linear algebra are
included.
B Recommended Preparation
A year of college calculus. Prior or concurrent course work with vector calculus or vectorintensive physics would be helpful.
C Prerequisites
Calculus II
D Minimum Unit Requirement
Three (3) semester units.
E Student Learning Outcomes
Upon successful completion of the course, students will be able to:
• Solve systems of linear equations by reducing an
augmented matrix to rowechelon or reduced rowechelon form.
• Determine whether a linear system is consistent or
inconsistent, and for consistent systems, characterize solutions as
unique or infinitely many.
• Simplify matrix expressions using properties of matrix algebra.
• Compute the transpose, determinant, and inverse of matrices if defined for a given matrix.
• Define vector space, subspace, linear independence, spanning set and basis
• Define an inner product.
• Determine if a function that maps two vectors from
a vector space to a scalar is an inner product on that vector space.
• Construct orthogonal and orthonormal bases using the GramSchmidt Process for a given basis.
• Construct the orthogonal diagonalization of a symmetric matrix.
• Define matrix transformations, linear
transformations, onetoone, onto, kernel, range or image, rank,
nullity and isomorphism.
• Compute the characteristic polynomial, eigenvalues,
eigenvectors and eigenspaces for both matrices and linear
transformations.
• Prove basic results in linear algebra using accepted proofwriting conventions.
• Evaluate linear algebra proofs for accuracy and completeness.
F Topics
• Systems of linear equations: basic terminology and notation
• Gaussian elimination: row operations, rowechelon
form, reduced rowechelon form, Gaussian elimination algorithm,
GaussJordan elimination algorithm, back substitution
• Matrix algebra: operations, properties
• Inverse of matrix: definition, method of computing the inverse of a matrix, invertibility
• Relationship between coefficient matrix invertibility and solutions to a system of linear equations
• Transpose of matrix
• Special matrices: diagonal, triangular, and symmetric
• Determinants: definition, methods of computing
• Properties of the determinant function
• Vector algebra for Rn.
• Dot product, norm of a vector, angle between vectors, orthogonality of two vectors in Rn.
• Real vector space: definition, properties
• Subspaces of a real vector space
• Linear independence and dependence
• Basis and dimension of a vector space
• Matrixgenerated spaces: row space, column space, null space, rank, nullity
• Inner products on a real vector space
• Angle and orthogonality in inner product spaces
• Orthogonal and orthonormal bases: GramSchmidt process
• Best approximation: least squares technique
• Change of basis
• Eigenvalues, eigenvectors, eigenspace
• Diagonalization
• Orthogonal diagonalization of a symmetric matrix
• Linear Transformations: definitions, examples
• Kernel and range
• Inverse linear transformation
• Matrices of general linear transformations
• Isomorphism

Differential Equations and Linear Algebra
A. Description
First course in differential equations and linear algebra.
B. Recommended Preparation
Three semesters of calculus for science, mathematics, and engineering.
C. Prerequisites
Multivariable calculus.
D. Minimum Unit Requirement
Four (4) semester units
E. Student Learning Outcomes
Upon successful completion of the course, students will be able to:
• Solve first order differential equations
• Solve applications of first order differential equations in the sciences
• Solve higher order linear differential equations
• Solve homogeneous differential equations with constant coefficients
• Solve systems of differential equations
F. Topics
• Classification of differential equations as linear,
nonlinear, ordinary, partial, first order, second order, constant
coefficient
• First order differential equations and slope fields
• Qualitative behavior of solutions without explicit solving the equation
• Existence and uniqueness of solutions of first order equations
• Separable first order equations and their solutions
• Linear first order equations and their solutions
• Euler’s method for approximating solutions
• Applications of first order equations such as the
logistic equation, bodies moving in a resistant medium, mixing problems
• Linear differential equations of higher order
• Nature of the solution space
• Homogeneous equations with constant coefficients and their solutions
• Method of undetermined coefficients
• Unforced harmonic oscillator, undamped, underdamped, critically damped, overdamped
• Forced harmonic oscillator, damped and undamped
• Resonance
• CauchyEuler equation (second order, homogeneous)
• Vector spaces
• Lines and planes in 3space
• Vector spaces of functions
• Matrix algebra
• Systems of linear algebraic equations in matrix form
• Nature of the solution space
• Gaussian elimination and row echelon form
• Rank of a matrix
• Column space and row space of a matrix
• Determinants
• Eigenvalues
• Diagonalization
• Systems of linear differential equations in matrix form
• Solution by diagonalization including complex eigenvalues
• Matrix exponential

Calculus with Applications
A. Description
Functions and graphs; limits; derivatives, antiderivatives, and
differential equations with emphasis on applications in the life
sciences
B. Recommended Preparation
College algebra or precalculus
C. Prerequisites
Completion of General Education quantitative reasoning
D. Minimum Unit Requirement
Three (3) semester units
E. Student Learning Outcomes
Upon successful completion of the course, students will be able to:
• Compute first and second derivatives of a function
• Graph functions using the methods of calculus
• Apply differentiation to life science areas
• Compute antiderivatives
• Evaluate a definite integral as a limit
• Evaluate a definite integral using integration formulas
• Apply integration to life science areas
• Solve differential equations that arise in life sciences
F. Topics
• Limits, lefthand and righthand limits
• Computing limits using numerical, graphical, and algebraic approaches
• Continuity; continuity at a real number, discontinuity at a real number, removable discontinuity
• Tangent lines
• Derivative as a limit
• Interpretation of the derivative as: slope of tangent line, a rate of change
• Derivative as a function
• Differentiation formulas; constants, power rule, product rule, quotient rule
• Rates of change
• Second derivative, concavity, inflection points, and acceleration
• Derivatives of polynomial, rational, exponential, and logarithmic functions
• Chain rule
• Maxima and minima, global extrema, second derivative test
• Rolle's Theorem and Mean Value Theorem
• Limits at infinity, horizontal asymptotes
• Leading behavior, indeterminate forms and L'Hopital's Rule
• Polynomial approximation, Taylor Polynomials
• Differential equations, autonomous differential
equations, y' = f(y), puretime differential equations, y' = f(t)
• Antiderivatives, indefinite integrals
• Rules for antiderivatives, power rule, constant product rule, sum rule
• Antiderivatives of polynomial, rational, and exponential functions
• Integration by substitution
• Integration by parts
• Riemann sums, left and righthand sums
• Definite integral
• Properties of the integral
• Fundamental Theorem of Calculus
• Applications of the integral, area, average value
• Improper integrals
• Solving differential equations, Euler's method, separable equations
• Applications of differential equations such as
Newton's law of cooling, bacterial population growth, predatorprey
dynamics

Statistics for Psychology
a. Course Description
The theory of parametric and nonparametric statistical methods and
their application to psychology data. Topics include: descriptive
statistics; probability and sampling distributions; statistical
inference and power; linear correlation and regression; chisquare;
ttests; and oneway analysis of variance. Application of both
handcomputation and statistical software to data in a psychology
context, including the interpretation of the relevance of the
statistical findings
b. Recommended Preparation
A working knowledge of a spreadsheet program such as Microsoft Excel.
c. Prerequisites
Intermediate Algebra
d. Minimum units:
3 semester units
e. Required Topical Coverage
Scales of measurement
Summarizing data graphically and numerically
Introduction to probability
Assigning Probabilities to Simple Events
Conditional Probability
Probability Rules
Independent and Mutually Exclusive Events
Discrete Distribution – Binomial
Continuous Distributions – Normal
Measures of central tendency, dispersion, and correlation
Mean, median, and mode
Calculation of expected value
Measures of dispersion – range, interquartile range, mean
absolute deviation, variance, standard deviation, standard error
Calculation of sample and population variance and standard deviation
Expected values and variances of random variables
Methods of random sampling, biases in sampling, observational studies, design of experiments
Estimation based on Sampling
Sampling distributions
The central limit theorem
Determining point and confidence intervals for the population mean when the population variance is known
Determining point and confidence intervals for the
population mean when the population variance is unknown – the t
distribution
Determining point and confidence interval estimates for the population proportion
Determining point and confidence interval estimates for the population variance
Selecting the appropriate sample size
Hypothesis Testing
Type I and II errors
Selection of alpha and beta
Calculation of p values
Inference about population mean – variance known and unknown
Inference about population variance using the F distribution
Inference about population proportion
Using the ChiSquared Distribution for tests of independence and goodness of fit
Hypothesis testing for comparing two populations
Comparing population means
Comparing population variances
Comparing population proportions
Using One Way ANOVA to compare multiple populations
Simple linear regression analysis
Calculation of regression coefficients
Testing for independence
Correlation analysis
Nonparametric Statistics
Wilcoxon Rank Sum Test
Sign Test
MannWhitney U test
KruskalWallis
Freidman’s ANOVA
Exposure to doing statistical analysis using a software program such as Excel, SPSS, SAS, Minitab, etc.
f. Measurable Learning Outcomes
Upon successful completion of the course students will be able to:
• Distinguish among different scales of measurement and their implications
• Interpret data displayed in tables and graphically.
• Correctly apply the following concepts from sets and probability to
solve problems: Venn diagrams, samples spaces, tree diagrams,
probability distributions, complementary events, mutually exclusive
events, and the addition rule
• Determine measures of central tendency and variation for a given data set
• Discuss the standard methods of obtaining data and enunciate the advantages and disadvantages of each
• Calculate the mean and variance of a discrete distribution
• Use tables for normal and Student’s t distributions to calculate probabilities
• Explain the difference between sample and population distributions and the role played by the central limit theorem
• Construct and interpret confidence intervals
• Interpret levels of statistical significance including pvalues
• Interpret of the output of a computerbased statistical analysis
• Explain the basic concept of hypothesis testing including Type I and II errors
• Formulate a hypothesis test (i.e., choose the forms of null and
alternative hypotheses) involving samples from two or more populations
• Select the appropriate technique for testing a hypothesis and interpret the result
• Identify the dependent and independent variables in a regression analysis
• Use regression analysis for prediction and estimation and determine whether two variables are significantly correlated
• Conduct non parametrical statistical tests

Probability & Statistics
a. Course Description
The use of probability techniques, hypothesis testing, and predictive
techniques to facilitate decisionmaking. Topics include:
descriptive statistics; probability and sampling distributions;
statistical inference and power; linear correlation and regression;
chisquare; ttests; and oneway analysis of variance.
Application of statistical software to data, including the
interpretation of the relevance of the statistical findings.
NOTE: To be eligible to be included in the LDTP this course must be approved for General Education
b. Recommended Preparation
Computer Literacy – including a working knowledge of a spreadsheet program such as Microsoft Excel
c. Prerequisites
Intermediate Algebra.
d. Minimum units:
3 semester units
e. Required Topical Coverage
Scales of measurement
Summarizing data graphically and numerically
Introduction to probability
Assigning Probabilities to Events
Conditional Probability
Probability Rules
Independent and Mutually Exclusive Events
Discrete Distribution – Binomial and Poisson
Continuous Distributions – Normal and Exponential
Measures of central tendency, dispersion, and correlation
Mean, median, and mode
Calculation of expected value
Measures of dispersion – range, interquartile range, variance, standard deviation, standard error
Calculation of sample and population variance and standard deviation
Expected values and variances of random variables
Methods of random sampling, biases in sampling, observational studies, design of experiments
Estimation based on Sampling
Sampling distributions
The central limit theorem
Determining point and confidence intervals for the population mean when the population variance is known
Determining point and confidence intervals for the
population mean when the population variance is unknown – the t
distribution
Determining point and confidence interval estimates for the population proportion
Determining point and confidence interval estimates for the population variance
Selecting the appropriate sample size
Hypothesis Testing
Type I and II errors
Selection of alpha and beta
Calculation of p values
Inference about population mean – variance known and unknown
Inference about population variance using the F distribution
Inference about population proportion
Using the ChiSquared Distribution for tests of independence and goodness of fit
Hypothesis testing for comparing two populations
Comparing population means
Comparing population variances
Comparing population proportions
Using one way ANOVA to compare multiple populations
Simple linear regression analysis
Calculation of regression coefficients
Testing for independence
Correlation analysis
Nonparametric Statistics
Wilcoxon Rank Sum Test
Sign Test
Exposure to doing statistical analysis using a software program such as Excel, SPSS, SAS, Minitab, etc.
f. Measurable Learning Outcomes
Upon successful completion of the course students will be able to:
• Distinguish among different scales of measurement and their implications.
• Interpret data displayed in tables and graphically.
• Correctly apply the following concepts from sets
and probability to solve simple problems: Venn diagrams, sample spaces,
tree diagrams, samples spaces, probability distributions, complementary
events, mutually exclusive events, and the addition rule.
• Determine measures of central tendency and variation for a given data set.
• Discuss the standard methods of obtaining data and enunciate the advantages and disadvantages of each.
• Calculate the mean and variance of a discrete distribution.
• Calculate probabilities using normal and Student’s t distributions.
• Explain the difference between sample and
population distributions and the role played by the central limit
theorem.
• Construct and interpret confidence intervals.
• Interpret levels of statistical significance including pvalues.
• Interpret the output of a computerbased statistical analysis.
• Explain the basic concept of hypothesis testing including Type I and II errors.
• Formulate a hypothesis test (i.e., choose the forms
of null and alternative hypotheses) involving samples from two
populations.
• Select the appropriate technique for testing a hypothesis and interpret the result
• Use the Chi Square distribution to test independence and goodness of fit.
• Use simple regression analysis for estimation,
inference, and prediction for decisionmaking in business and economics
(e.g., cost estimation, evaluation of risk, forecasting) and interpret
the associated statistics.
• Use nonparametric statistical tests such as the Wilcoxon Rank Sum test and the Sign test
