Calculus I: Course Syllabus and Key Concepts Overview

Calculus I Syllabus Overview

Course Information

• Course Code: MATH 112

• Course Title: Calculus

• Textbook: Thomas' Calculus, Early Transcendentals

• Author: Hass, Heil, Bogacki

• Publisher: Pearson

• Year: 2023

Assessment Details

Component	Percentage
Homework	10%
First Exam	25%
Second Exam	25%
Final Exam	40%

Main Topics and Subtopics

1. Functions and Their Graphs

• Definition of a Function: A function is a relation that assigns each element in the domain to exactly one element in the range.

• Domain and Range: The set of all possible input values (domain) and output values (range).

• Graph of a Function: Visual representation of the function on the coordinate plane.

• Vertical Line Test: Used to determine if a graph represents a function.

• Piecewise-Defined Functions: Functions defined by different expressions over different intervals.

• Transformations: Includes vertical and horizontal shifts, reflections, and stretching/compressing.

Example: The function $f(x) = x^2$ is a parabola opening upwards. Its domain is all real numbers, and its range is $[0, \\infty)$.

2. Combining Functions; Shifting and Scaling Graphs

• Sum, Difference, Product, and Quotient of Functions: Operations that combine two functions to produce a new function.

• Composition of Functions: $(f \\circ g)(x) = f(g(x))$

• Transformations: Shifting (vertical/horizontal), scaling (stretch/compress), and reflecting graphs.

Example: If $f(x) = x$ and $g(x) = x^2$, then $(f + g)(x) = x + x^2$.

3. Trigonometric Functions

• Definition: Functions based on angles, including sine, cosine, tangent, etc.

• Graphs: Periodic and oscillatory behavior.

• Domain and Range: For example, $\sin(x)$ has domain $(-\\infty, \\infty)$ and range $[-1, 1]$.

Example: The graph of $y = \sin(x)$ oscillates between $-1$ and $1$.

4. Exponential Functions

• Definition: Functions of the form $f(x) = a^x$, where $a > 0$ and $a \\neq 1$.

• Domain: All real numbers.

• Range: $(0, \\infty)$

• Properties: Rapid growth or decay depending on the base.

Example: $f(x) = 2^x$ doubles for each increase of $x$ by 1.

5. Inverse Functions and Logarithms

• Inverse Function: If $f(x)$ is a function, its inverse $f^{-1}(x)$ reverses the effect of $f$.

• Logarithmic Function: $f(x) = \log_a(x)$ is the inverse of $f(x) = a^x$.

• Properties: $\log_a(a^x) = x$ and $a^{\log_a(x)} = x$

Example: $f(x) = \log_{10}(x)$ is the common logarithm.

6. Limits and Continuity

• Limit of a Function: The value that $f(x)$ approaches as $x$ approaches a certain point.

• One-Sided Limits: Limits from the left ($x \\to a^-$) and right ($x \\to a^+$).

• Infinite Limits: Limits where $f(x)$ increases or decreases without bound.

• Continuity at a Point: A function is continuous at $x = a$ if $\\lim_{x \\to a} f(x) = f(a)$.

• Continuity Test: Used to determine if a function is continuous at a point or on an interval.

Example: $f(x) = 1/x$ is not continuous at $x = 0$ because the limit does not exist.

Summary Table of Sections

Section	Title	Main Concepts	Example
1.1	Functions and Their Graphs	Domain, Range, Graphs, Transformations	$f(x) = x^2$
1.2	Combining Functions; Shifting and Scaling Graphs	Operations, Composition, Transformations	$(f + g)(x) = x + x^2$
1.3	Trigonometric Functions	Sine, Cosine, Tangent, Graphs	$y = \sin(x)$
1.4	Exponential Functions	Definition, Domain, Range, Properties	$f(x) = 2^x$
1.5	Inverse Functions and Logarithms	Inverse, Logarithms, Properties	$f(x) = \log_{10}(x)$
2.1-2.6	Limits and Continuity	Limits, One-Sided Limits, Infinite Limits, Continuity	$f(x) = 1/x$ at $x = 0$

Additional info: The syllabus provides a structured outline of the foundational topics in Calculus I, including definitions, properties, and examples. Students are expected to master these concepts for success in subsequent calculus topics.