# Course

Course Summary
Credit Type:
Course
ACE ID:
CSRA-0002
Organization:
Location:
Online
Length:
12 weeks (96 hours)
Dates Offered:
Credit Recommendation & Competencies
Level Credits (SH) Subject
Lower-Division Baccalaureate 3 Mathematics
Description

## Objective:

The course objective is to provide a comprehensive conceptual understanding of the differential and integral calculus in the context of applications to the physical, engineering, and social sciences. Special emphasis is placed on the role of local and global properties of functions, with continuous or discrete inputs, together with a consistent recourse to approximation and series expansions.

## Learning Outcomes:

• Understand single-variable functions in terms of inputs and outputs, and work with functions in a variety of contexts
• Use asymptotic or 'big O' notation to describe growth and decay of functions
• Conceptualize derivatives and differentials in terms of rates of change
• Use indefinite integrals to solve separable and autonomous linear ordinary differential equations
• Use integrals to compute physical quantities, such as mass, centroids, moments, work, and force
• Compute and use Taylor expansions to understand limits, approximations, and local analysis of functions
• Define definite integrals and compute them via indefinite integrals
• Use integrals to compute geometric quantities, such as areas, volumes, lengths, and surface areas
• Use integrals to compute simple probabilities and means
• Conceptualize infinite sequences and series as discrete-input functions
• Determine convergence of infinite series and power series
• Appreciate the use and impact of calculus in the physical, engineering, and social sciences

## General Topics:

• Taylor series and polynomial approximations of functions
• Standard Taylor series and computations using substitution
• Limits and L'Hopital's rule via Taylor expansion
• Orders of growth and asymptotic notation
• Derivatives: definitions, interpretations, and applications
• Implicit and logarithmic differentiation
• Differentials and linearization
• Optimization of single-variable functions
• Indefinite integrals: definition and examples
• Separable and linear differential equations, with applications
• Linearization of equilibria in autonomous differential equations
• Integration by substitution
• Integration by parts
• Integration by partial fractions and algebraic methods
• Definite integrals, definitions and computation
• The fundamental theorem of integral calculus, proof and applications
• Improper integrals
• Use of computers in integration
• Using integrals to compute areas
• Using integrals to compute volumes
• Using integrals to compute high-dimensional hyper-volumes
• Using integrals to compute arc length
• Using integrals to compute surface area
• Using integrals to compute work, force, and pressure
• Using integrals to compute averages
• Using integrals to compute mass, center of mass, and centroids
• Using integrals to compute moments and radius of gyration
• Probability and uniform distributions
• Probability density functions, mean, variance, and standard deviation
• Sequences as discrete functions
• Finite differences as discrete derivatives
• Recurrence relations as discrete differential equations
• Numerical integration as integrals of discrete functions
• Infinite series as discrete improper integrals
• Convergence types and tests for infinite series
• Power series and convergence
• The Taylor remainder theorem and mean value theorem
Instruction & Assessment

## Instructional Strategies:

• Audio Visual Materials
• Lectures
• Practical Exercises
Supplemental Materials