Calculus I (MAT 131-770) Syllabus and Topical Outline – Study Guide

Course Overview

Introduction to Calculus I

Calculus I is the foundational course in a three-part calculus sequence, focusing on the concepts of limits, derivatives, and integrals. The course emphasizes the limit process, which is central to modern mathematics, and develops both differential and integral calculus for elementary functions. Applications to geometry, physics, economics, and other sciences are integrated throughout the curriculum.

• Course Code: MAT 131-770

• Credits: 5.00 credit hours

• Textbook: Thomas’ Calculus, 15th edition (online access via MyMathLab)

• Prerequisite: Pre-calculus (MAT 111) with grade 'C' or better, or appropriate placement

Course Learning Outcomes

• Define and identify when to apply the theory of limit, continuity, derivative, anti-derivative, and definite integral.

• Calculate derivatives and anti-derivatives.

• Apply calculus concepts and techniques to problems in geometry, physics, economics, and other sciences.

• Read and interpret mathematical proofs and appreciate the need for precise language and notation.

• Translate applied problems into mathematical language.

• Memorize and apply necessary calculus formulas to solve problems.

Topical Outline

The following outline summarizes the main topics and chapters covered in Calculus I, along with their key subtopics and applications.

Chapter 1: Functions and Preliminaries

• Functions: Definitions, types, and properties

• Equations and Graphs: Understanding and interpreting graphical representations

• Trigonometry Review: Essential trigonometric functions and identities

Example: Sketch the graph of $f(x) = \sin(x)$ and identify its period and amplitude.

Chapter 2: Limits and Continuity

• Rates of Change and Tangent Lines: Introduction to instantaneous rate of change

• Limits and Limit Laws: Evaluating limits using algebraic techniques

• Precise Definition of a Limit: $\lim_{x \to a} f(x) = L$

• Continuity: Definition and identification of continuous functions

Example: Evaluate $\lim_{x \to 2} (3x^2 - 4)$.

Solution: Substitute $x = 2$: $3(2)^2 - 4 = 12 - 4 = 8$.

Chapter 3: Derivatives

• Derivative as a Function: Definition and interpretation

• Tangents and Derivatives at a Point: Slope of the tangent line

• Differentiation Rules: Power, product, quotient, and chain rules

• Trigonometric Derivatives: Derivatives of $\sin(x)$, $\cos(x)$, etc.

• Implicit Differentiation: Differentiating equations not solved for $y$

• Related Rates: Applications to real-world problems

• Linearization and Differentials: Approximating function values

Key Formula: The derivative of $f(x)$ at $x = a$ is $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Example: Find the derivative of $f(x) = x^3$.

Solution: $f'(x) = 3x^2$

Chapter 4: Applications of Derivatives

• Extreme Values of Functions: Finding maxima and minima

• First and Second Derivative Tests: Determining concavity and inflection points

• Curve Sketching: Using derivatives to analyze graphs

• Applied Optimization: Solving real-world optimization problems

• Anti-derivatives: Introduction to integration as the reverse process of differentiation

Example: Find the local maximum of $f(x) = -x^2 + 4x + 1$.

Solution: Set $f'(x) = -2x + 4 = 0 \implies x = 2$. The local maximum occurs at $x = 2$.

Chapter 5: Integrals

• Area Estimation with Finite Sums: Approximating area under curves

• Sigma Notation: $\sum_{i=1}^n a_i$

• Limits of Finite Sums: Transition to definite integrals

• Definite Integrals: $\int_a^b f(x)\,dx$

• Fundamental Theorem of Calculus: Connecting differentiation and integration

• Indefinite Integrals: General antiderivatives

• Integration by Substitution: Change of variables technique

• Areas Between Curves: Calculating net area

Key Formula: $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F'(x) = f(x)$

Example: Compute $\int_0^2 x^2\,dx$.

Solution: $\left.\frac{1}{3}x^3\right|_0^2 = \frac{8}{3}$

Chapter 7: Transcendental Functions

• Derivatives of Inverse Functions: Including $\ln(x)$ and inverse trigonometric functions

• Derivatives and Integrals of Transcendental Functions: $e^x$, $\ln(x)$, $\sin(x)$, $\cos(x)$, etc.

• Indeterminate Forms and L’Hôpital’s Rule: Evaluating limits of the form $0/0$ or $\infty/\infty$

• Separable Differential Equations: Solving first-order ODEs by separation of variables

• Hyperbolic Functions: $\sinh(x)$, $\cosh(x)$, etc.

• Inverse Trigonometric Functions: $\arcsin(x)$, $\arccos(x)$, $\arctan(x)$

• Relative Rates of Growth: Comparing growth of functions as $x \to \infty$

Key Formula: $\frac{d}{dx}e^x = e^x$, $\frac{d}{dx}\ln(x) = \frac{1}{x}$

Example: Evaluate $\lim_{x \to 0} \frac{\sin(x)}{x}$ using L’Hôpital’s Rule.

Solution: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$

Course Structure and Assessment

Grading Components

Assessment	Points	Percent of Final Grade
Homework (39 sections)	195	18%
Syllabus Quiz	30	3%
Quizzes (5 out of 6)	150	14%
Learning Reflection	50	5%
Midterm Exam	250	23%
Final Exam	400	37%
Total	1075	100%

Grading Scale

Percentage	Total Points	Grade
90-100	967.5-1075	A
80-89	860-967	B
70-79	752.5-859	C
60-69	645-752	D
0-59	< 645	F

Additional Information

• Homework: Assigned for each section, completed online via MyMathLab. Unlimited attempts before deadline.

• Quizzes: Timed, online; lowest score dropped. Two attempts allowed if homework is completed with at least 70% in all sections.

• Exams: Midterm (Chapters 2 & 3) and cumulative Final (Chapters 2, 3, 4, 5, 7). Proctored online or in-person. Written work submission required.

• Learning Reflection: End-of-course survey for feedback and self-assessment.

• Academic Honesty: Strictly enforced; use of AI or unauthorized aids is prohibited.

• Support Services: Free tutoring, academic coaching, and other student resources are available.

Summary Table: Main Calculus I Topics

Chapter	Main Topics	Key Concepts
1	Functions, Equations, Graphs, Trigonometry	Domain, Range, Graphing, Trig Identities
2	Limits and Continuity	Limit Laws, Continuity, Tangent Lines
3	Derivatives	Differentiation Rules, Chain Rule, Implicit Diff.
4	Applications of Derivatives	Optimization, Curve Sketching, Related Rates
5	Integrals	Definite/Indefinite Integrals, FTC, Substitution
7	Transcendental Functions	Exponential, Logarithmic, Inverse Trig, L’Hôpital’s Rule

Study Tips

• Review each chapter’s homework and quizzes before exams.

• Practice showing all steps in written solutions, as required for exams.

• Utilize online resources, video lectures, and tutoring services for additional support.

• Memorize key formulas and understand their applications.

• Stay engaged with course communications and meet all deadlines.

Additional info: This guide is based on the official syllabus and topical outline for Calculus I at Triton College. For detailed examples, proofs, and further applications, refer to the online textbook and MyMathLab resources provided with your course access.