MATH 2413 - Calculus I Syllabus and Course Structure Study Guide

MATH 2413 - Calculus I: Course Overview and Structure

Course Description and Rationale

This course is the first in the calculus sequence for mathematics, science, and engineering students. It covers foundational topics such as functions, limits, continuity, derivatives, and integrals, emphasizing conceptual understanding through multiple representations and the use of technology.

• Credit Hours: 4

• Topics: Inequalities, functions, limits, continuity, derivatives, differentiation of elementary functions, applications of derivatives, integrals, integration of algebraic and trigonometric functions, numerical integration, and basic applications of the integral.

• Approach: Geometric, numerical, algebraic, and verbal presentations (the "rule of four").

• Technology: Graphing calculators (TI recommended), MyLab software, and online textbook access.

Course Calendar: Weekly Topics

The course is structured by weekly sections, each focusing on specific calculus concepts. Below is a summary of the main topics covered each week:

Week	Sections	Main Topics
1	Intro, 2.1	Policies, Prerequisite Review, Idea of Limits
2	2.2, 2.3, 2.4	Definitions of Limits, Techniques for Computing Limits, Infinite Limits
3	2.5, 2.6	Limits at Infinity, Continuity
4	3.1, 3.2	Introducing the Derivative, Derivative as a Function
5	3.3, 3.4	Rules of Differentiation, Product and Quotient Rules
6	3.5, 3.6, 3.7	Derivatives of Trigonometric Functions, Derivatives as Rates of Change, The Chain Rule
7	3.8, 3.9	Implicit Differentiation, Derivatives of Logarithmic and Exponential Functions
8	3.10	Inverse Trigonometric Functions
9	4.1, 4.2, 4.3	Maxima and Minima, Mean Value Theorem, What Derivatives Tell Us
10	4.3, 4.4, 4.5	Graphing Functions, Optimization Problems
11	4.6, 4.7	Linear Approximation and Differentials, L'Hôpital's Rule
12	4.8, 4.9	Newton's Method, Antiderivatives
13	5.1, 5.2	Approximating Areas Under Curves, Definite Integrals
14	5.3, 6.1	Fundamental Theorem of Calculus, Velocity and Net Change
15	5.4, 5.5	Working with Integrals, Substitution Rule
16	Review	Final Exam Preparation

Key Calculus Topics and Concepts

• Functions: Understanding algebraic and transcendental functions, their properties, and graphical representations.

• Limits and Continuity: Definitions, computation techniques, infinite limits, and limits at infinity. Continuity at a point and over intervals.

• Derivatives: Definition, interpretation as rates of change, differentiation rules (product, quotient, chain), and derivatives of trigonometric, exponential, and logarithmic functions.

• Applications of Derivatives: Maxima and minima, optimization, Mean Value Theorem, linear approximation, and Newton's Method.

• Integrals: Antiderivatives, definite integrals, Fundamental Theorem of Calculus, numerical integration, and applications such as area under curves and velocity.

Student Learning Outcomes

Upon successful completion, students will be able to:

• Solve tangent and area problems using limits, derivatives, and integrals.

• Draw graphs of functions considering limits, continuity, and differentiability.

• Determine continuity and differentiability at a point using limits.

• Use differentiation rules for algebraic and transcendental functions.

• Model real-world situations with calculus concepts and solve applied problems.

• Evaluate definite integrals using the Fundamental Theorem of Calculus.

• Understand the relationship between derivatives and integrals.

Mathematics Program Student Learning Outcomes

• Multiple Representations: Proficiency with verbal, numerical, graphical, and symbolic representations of functions.

• Applications: Ability to apply calculus techniques to real-world problems.

• Technology: Appropriate use of technology to supplement symbolic processes.

General Education Competencies

• Critical Thinking: Gathering, analyzing, synthesizing, evaluating, and applying information.

• Quantitative and Empirical Reasoning: Applying mathematical, logical, and scientific principles.

• Technology Skills: Using technology to retrieve, manage, analyze, and present information.

• Communication: Effective written, oral, and visual communication.

Assessment and Grading

Grades are determined by a weighted combination of homework, online homework, tests, and the final exam.

Component	Weight
Homework	10%
Online Homework	10%
Tests (4 exams)	60%
Final Exam	20%

• Grading Scale: A: 90-100, B: 80-89, C: 70-79, D: 60-69, F: <60

• Homework: Written and online assignments, due weekly. Two lowest homework grades dropped.

• Exams: Four in-class exams and one final exam. Lowest exam score (not including final) down-weighted to 5%.

• Late Work: 10% deduction per week late; deadline is first class after relevant exam.

• Missed Exams: Make-up allowed with valid reason; unexcused absences incur 25% deduction.

Course Resources and Support

• Office Hours: Instructor available for questions during posted times.

• Supplemental Instruction: SI Leader supports collaborative study sessions.

• Learning Labs: Free tutoring in math and other subjects.

• Academic Coaching: Support for study strategies and learning processes.

• Student Services: Academic, financial, personal, and technology support.

Homework Assignments

Homework is assigned by section, with both textbook and MyLab problems. Below is a sample of the assignment structure:

Section	Textbook Problems	MyLab Problems
2.1	5, 6, 10, 11, 23	1, 13, 17, 31
2.2	10, 19, 28, 33, 40, 47	3, 7, 15, 17, 41
3.1	7, 9, 23, 41, 53	27, 35, 41
4.1	3, 5, 7, 20, 21, 29, 37, 53, 71, 76	13, 15, 17, 27, 33, 35, 49, 55, 65
5.1	3, 5, 43	9, 11, 19, 23, 45, 71

Important Course Policies

• Attendance: Regular and punctual attendance required; instructor may withdraw students for unsatisfactory attendance.

• Withdrawal: Students responsible for withdrawal; instructor may also withdraw as necessary.

• Incomplete Grades: Only given in rare circumstances; must be up to date and have legitimate reason.

• AI Policy: Generative AI permitted for unproctored work; not allowed on major assessments unless explicitly permitted.

Technology and Materials

• Textbook: Calculus: Early Transcendentals, 3rd Edition by Briggs, Cochran, Gillette, & Schulz.

• MyLab Software: Required for online homework and course materials.

• Graphing Calculator: TI family recommended; must be able to graph functions, find zeros, and perform numerical integration.

Example: Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, providing a method to evaluate definite integrals.

• Statement: If f is continuous on [a, b] and F is an antiderivative of f, then:

$F'(x) = f(x)$

$\int_a^b f(x)\,dx = F(b) - F(a)$

• Application: Used to compute the area under a curve and solve problems involving net change.

Example: Derivative Rules

Common differentiation rules include the product rule, quotient rule, and chain rule.

• Product Rule: $(fg)' = f'g + fg'$

• Quotient Rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

• Chain Rule: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$

Example: Limit Definition of the Derivative

The derivative of a function at a point is defined as:

$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Interpretation: Represents the instantaneous rate of change or slope of the tangent line at a.

Additional info:

• All major Calculus I topics are covered, including graphical and applied aspects.

• Homework assignments are structured to reinforce each week's concepts.

• Technology use is integrated for visualization and computation.