Calculus I Syllabus and Study Guide: Limits, Derivatives, and Applications

Study Guide - Smart Notes

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Course Overview

Introduction

This syllabus outlines the structure, objectives, policies, and weekly topics for a college-level Calculus I course. The course covers foundational concepts such as limits, derivatives, and their applications, with a focus on exponential and logarithmic functions, optimization, and the Fundamental Theorem of Calculus.

Course Objectives

Learning Goals

Visualize and determine whether a function is one-to-one: Understand invertible functions and how to find their inverses, especially for trigonometric, exponential, and logarithmic functions.
Limits of functions: Investigate the concept of a limit and determine continuity at a point. Apply the Intermediate Value Theorem to solve problems.
Basic differentiation: Understand the definition of the derivative and use basic rules to compute derivatives of functions.
Applications of derivatives: Use derivatives to solve problems including tangent lines, maxima and minima, and rates of change. Apply L'Hospital's Rule to evaluate limits.
Definite integrals and the Fundamental Theorem of Calculus: Calculate areas under curves and understand the relationship between differentiation and integration.
Substitution and integration techniques: Use substitution to integrate functions and solve applied problems.

Student Learning Outcomes

Upon completion, students will be able to:

Define, differentiate, and integrate elementary functions.
Apply derivatives and integrals to solve real-world problems.
Prepare for subsequent courses in engineering, mathematics, and related fields.

Prerequisite Courses

Required Background

Completion of MTH 116 or equivalent is required.

Textbook and Required Sections

Textbook

Calculus Volume I, OpenStax (free online PDF available).

Required Sections

1.4-5, 2.2-5, 3.1-4, 4.3-8, 4.10, 5.1-7, 6.1

Grading Breakdown

Assessment Components

Component	Weight
Skills Tests	10%
Quizzes	10%
Online Homework	15%
Tests	35%
Final Exam	30%

Assessment Details

Skills Tests

Three computer-based tests, each 60 minutes, covering basic skills and concepts.
Immediate feedback; retakes allowed until deadline.

Quizzes

Weekly or bi-weekly, covering recent material.
Lowest two scores dropped.

Homework

Online assignments throughout the semester.
Lowest three scores dropped.

Tests

Three in-class tests; no makeups except for extraordinary circumstances.

Final Exam

Cumulative, scheduled at end of semester.

Grade Scale

Grade Letter	Percentage
A	93-100
A-	90-92
B+	87-89
B	83-86
B-	80-82
C+	77-79
C	73-76
C-	70-72
D	60-69
F	0-59

Weekly Topics and Schedule

Topic Progression

Week	Date	Topic	Section
1	Aug 25, 26, 27, 29	Overview, inverse functions, exponential and logarithmic functions	2.1, 1.4, 1.5
2	Sep 2, 3, 5	The limit of a function, limit laws	2.2, 2.3
3	Sep 8, 9, 10, 12	Continuity, the precise definition of limit	2.4, 2.5
4	Sep 15, 16, 17, 19	Derivative, differentiation rules	3.1-3.4, 3.3
5	Sep 22, 23, 24, 26	Rates of change, derivatives of trig functions, the chain rule	3.4, 3.5
6	Sep 30	Test 1 review	1.4 to 3.5
7	Oct 1, 3	The chain rule (continued), derivatives of inverse functions	3.6, 3.7
8	Oct 6, 8, 10	Implicit differentiation, derivatives of exponential and logarithmic functions, related rates	3.8, 3.9, 4.1
9	Oct 13, 15, 17	Related rates (continued), maxima, minima, the mean value theorem	4.2
10	Oct 20	Test 2 review	3.6 to 4.3, except 4.2
11	Oct 21	Derivative and the shape of a graph, limits at infinity and asymptotes	4.3, 4.4, except 4.2
12	Oct 22, 29, 31	Applied optimization, L'Hospital's Rule	4.5, 4.8
13	Nov 3, 4, 7	Antiderivatives, definite integral	4.10, 5.1-2
14	Nov 11	Test 3 review	4.4 to 5.2, except 4.9
15	Nov 12, 14	Fundamental theorems of calculus, net change theorem, substitution	5.3, 5.4, 5.5, 5.7
16	Nov 17, 18, 19	Substitution (continued), integrals involving exponential and logarithmic functions, trigonometric integrals	5.3, 5.4, 5.5, 5.7
17	Nov 24, 25	Areas between curves	6.1
18	Dec 1, 2, 3	Area between curves (continued), final review	6.1

Key Calculus Concepts

Limits and Continuity

Limit of a function: The value that a function approaches as the input approaches a certain point.
Continuity: A function is continuous at a point if the limit exists and equals the function value at that point.
Limit Laws: Rules for combining limits, such as sum, product, and quotient laws.

Formula:

Derivatives

Definition: The derivative of a function measures the rate at which the function value changes as its input changes.
Differentiation Rules: Power rule, product rule, quotient rule, and chain rule.
Applications: Tangent lines, rates of change, optimization problems.

Formula:

Integrals

Definite Integral: Represents the area under a curve between two points.
Fundamental Theorem of Calculus: Connects differentiation and integration.

Formula:

Course Policies

Attendance

Attendance is required for all class sessions and recitations.
Absences must be excused for extraordinary circumstances.

Makeup Policy

No makeups for assignments or tests except for documented emergencies.

Electronic Devices & Calculators

Graphing calculators are not permitted on tests or skills tests.
Only basic scientific calculators allowed.

Academic Integrity

All assignments must be completed individually unless otherwise specified.
Plagiarism and cheating are strictly prohibited.

Accommodations

Students with disabilities or religious needs should contact the instructor or relevant university office for accommodations.

Additional Resources

Math Help Center: Free drop-in tutoring available.
Online Learning Resources: Supplemental materials and tutoring.
Academic Calendar: Important dates and deadlines.

Example Applications

Optimization: Use derivatives to find maximum profit or minimum cost in business applications.
Area under curves: Use definite integrals to calculate physical quantities such as distance or total accumulated change.

Important Formulas

Chain Rule:
Product Rule:
Quotient Rule:
L'Hospital's Rule: (when or )

Summary

This syllabus provides a comprehensive guide to the topics, policies, and resources for Calculus I. Students are expected to master limits, derivatives, and integrals, and apply these concepts to solve real-world problems. Regular attendance, academic integrity, and utilization of available resources are essential for success in this course.