Calculus 1A (Math 1776) Syllabus and Study Guide

Course Overview

Introduction to Calculus 1A

This course introduces students to the foundational concepts of calculus, focusing on limits, derivatives, and their applications. It emphasizes both conceptual understanding and computational skills, preparing students for further study in mathematics and related fields.

• Course Name and Number: Math 1776 – Calculus 1A

• Instructor: Dr. Ana Grossi

• Schedule: Seven-week course with multiple sections

• Textbook: Briggs, W., Cochran, L., Gillett, B., Schulz, C. CALCULUS Early Transcendentals, Third Edition

Course Topics and Structure

Limits

Limits are a fundamental concept in calculus, describing the behavior of functions as inputs approach a particular value. Understanding limits is essential for defining derivatives and integrals.

• Definition: The limit of a function $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ gets closer to as $x$ gets closer to $a$.

• Notation: $\lim_{x \to a} f(x)$

• Key Properties:

  • Limits can be evaluated graphically, numerically, and algebraically.

  • One-sided limits consider approach from the left ($x \to a^-$) or right ($x \to a^+$).

• Example: $\lim_{x \to 2} (3x + 1) = 7$

Continuity

A function is continuous at a point if its limit at that point equals its value. Continuity is crucial for many calculus theorems and applications.

• Definition: $f(x)$ is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$

• Types of Discontinuity: Removable, jump, and infinite discontinuities

• Example: The function $f(x) = x^2$ is continuous everywhere.

Derivatives

The derivative measures the rate at which a function changes. It is defined as the limit of the average rate of change as the interval approaches zero.

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

• Interpretation: The derivative represents the slope of the tangent line to the function at a point.

• Notation: $f'(x)$, $\frac{df}{dx}$

• Example: If $f(x) = x^2$, then $f'(x) = 2x$

Differentiation Rules

Several rules simplify the process of finding derivatives for various types of functions.

• Product Rule: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

• Quotient Rule: $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

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

• Example: For $f(x) = (3x^2 + 2x)(x^3)$, use the product rule to find $f'(x)$.

Applications of Derivatives

Derivatives are used to solve real-world problems, such as finding rates of change, optimizing functions, and modeling physical phenomena.

• Rate of Change: Velocity, acceleration, and other physical rates are derivatives of position, velocity, etc.

• Optimization: Finding maximum and minimum values of functions

• Example: The maximum profit for a business can be found by setting the derivative of the profit function to zero and solving for the critical points.

Course Learning Outcomes

• Calculate limits graphically, numerically, and algebraically.

• Evaluate derivatives graphically, using the limit definition, and with derivative rules.

• Interpret the derivative as the slope of the tangent line to a function.

• Apply derivatives to solve real-world problems.

Grading Criteria

Grades are determined by homework, in-class participation, quizzes, and a cumulative final exam. The grading system uses both numerical and letter definitions.

Grade	Weight	Numerical Definition
A	4.00	93-100
A-	3.67	90-92
B+	3.33	87-89
B	3.00	83-86
B-	2.67	80-82
C+	2.33	77-79
C	2.00	73-76
C-	1.67	70-72
D	1.00	67-69
F	0.00	0-66

Component	Percentage
Homework	10%
In-class participation	10%
Quizzes	45%
Cumulative final exam	35%

Course Schedule (Sample)

Week	Monday	Wednesday	Friday
1	Assessment Function Review	2.1 (Idea of limits)	2.1 (Idea of limits)
2	2.2 (Definition of limits)	2.3 (Computing limits) 2.4 (Infinite limits)	2.2 (Definition of limits) Quiz 1
3	2.5 (Limits at infinity)	2.6 (Continuity)	2.5 (Limits at infinity) Quiz 2
4	3.1 (Derivatives as functions)	3.2 (Derivatives)	3.3 (Differentiation rules) Quiz 3
5	3.6 (Rates of change)	3.4 (Product rule)	3.5 (Quotient rule) Quiz 4
6	3.7 (Chain rule)	3.7 (Chain rule)	Final Exam

Course Policies and Support

• Attendance: Regular attendance is required; participation affects your grade.

• Homework: Must be submitted on time; late submissions are penalized.

• Quizzes: No make-up quizzes; lowest two scores may be replaced by the final exam score.

• Final Exam: Mandatory; covers the entire course.

• Academic Integrity: Honesty and proper citation are required.

• Student Support: Access to tutoring, wellness, and library resources is available.

Additional info:

• Students are encouraged to use online graphing tools such as Desmos for visualizing functions and limits.

• Collaboration is encouraged for learning, but all submitted work must be individual unless otherwise specified.

• Accessibility accommodations are available through the Success Studio and Student Accessibility Services.