Calculus III Syllabus and Study Guide: Infinite Sequences, Series, and Vector Calculus

Course Overview

This course, Calculus III with Laboratory (MATH 243-01), covers advanced topics in calculus including infinite sequences and series, vector-valued functions, partial derivatives, and their applications. The course is designed for students in the College of Science, Engineering, and Technology at Jackson State University.

Instructor and Course Information

• Instructor: Dr. Cakara Chem

• Office Hours: MWF 8:00am-9:00am, 12:00pm-1:00pm or by appointment

• Lecture Location: JSH 244

• Lab Location: JSH 103

• Email: [e-mail]

Course Description

This course introduces students to the following main topics:

• Infinite Sequences and Series: Understanding convergence and divergence, power series, and related tests.

• Vector-Valued Functions: Operations, derivatives, and applications to motion in space.

• Partial Derivatives: Chain rule, directional derivatives, gradients, and applications.

• Technology Integration: Use of graphing utilities and mathematical software for problem-solving and visualization.

Prerequisites

• Completion of MATH 242 (Calculus II with Laboratory) or equivalent with a grade of C or better.

Learning Outcomes

Upon completion of the course, students will be able to:

• Use various tests and criteria to discuss the convergence or divergence of sequences and series.

• Determine the convergence and divergence of series.

• Determine and sketch the domain and range of functions of two variables.

• Compute limits, continuity, partial and total derivatives of functions of several variables.

• Interpret and analyze real functions of several variables and their directional derivatives and gradients.

• Be proficient in mathematical computation and graphing utilities.

Course Topics and Schedule

The course is organized into weekly modules, each focusing on a specific topic. Laboratory activities are integrated throughout.

Week	Topics
1	Infinite Sequences and Series: Sequences, Infinite Series
2	Integral Test, Comparison Tests
3	Alternating Series, Absolute and Conditional Convergence
4	Power Series, Taylor and Maclaurin Series
5	Convergence of Taylor Series, Parametrizations of Plane Curves
6	Calculus with Parametric Curves, Polar Coordinates
7	Graphing Polar Coordinate Equations
8	Vectors in Space, The Dot Product
9	The Cross Product
10	Vector-Valued Functions, Derivatives and Integrals of Vector Functions
11	Velocity and Acceleration, Tangent and Normal Vectors
12	Partial Derivatives, Limits and Continuity in Higher Dimensions
13	Final Exam Review

Key Concepts and Definitions

Infinite Sequences and Series

• Sequence: An ordered list of numbers, typically defined by a function $a_n$ where $n$ is a positive integer.

• Series: The sum of the terms of a sequence, written as $\sum_{n=1}^{\infty} a_n$.

• Convergence: A series $\sum a_n$ converges if the sequence of partial sums $S_N = \sum_{n=1}^N a_n$ approaches a finite limit as $N \to \infty$.

• Divergence: If the sequence of partial sums does not approach a finite limit, the series diverges.

Example: The geometric series $\sum_{n=0}^{\infty} ar^n$ converges if $|r| < 1$ and diverges otherwise.

Tests for Convergence

• Integral Test: If $f(n) = a_n$ is positive, continuous, and decreasing for $n \geq N$, then $\sum a_n$ and $\int_N^{\infty} f(x) dx$ both converge or both diverge.

• Comparison Test: Compare $a_n$ to a known convergent or divergent series $b_n$.

• Alternating Series Test: If $a_n$ decreases to zero, then $\sum (-1)^{n} a_n$ converges.

• Ratio Test: If $\lim_{n \to \infty} |a_{n+1}/a_n| = L$, then the series converges if $L < 1$ and diverges if $L > 1$.

Power Series and Taylor Series

• Power Series: A series of the form $\sum_{n=0}^{\infty} c_n (x - a)^n$.

• Taylor Series: The expansion of a function $f(x)$ about $x=a$: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$

• Maclaurin Series: A Taylor series centered at $a=0$.

Example: The Maclaurin series for $e^x$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$.

Parametric and Polar Curves

• Parametric Equations: Curves defined by $x = f(t)$, $y = g(t)$ for $t$ in an interval.

• Polar Coordinates: Points defined by $(r, \theta)$, where $x = r \cos \theta$, $y = r \sin \theta$.

Example: The circle of radius $a$ can be written in polar coordinates as $r = a$.

Vectors and Vector-Valued Functions

• Vector: A quantity with both magnitude and direction, often written as $\vec{v} = \langle v_1, v_2, v_3 \rangle$.

• Dot Product: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$

• Cross Product: $\vec{a} \times \vec{b}$ gives a vector perpendicular to both $\vec{a}$ and $\vec{b}$.

• Vector-Valued Function: A function whose output is a vector, e.g., $\vec{r}(t) = \langle x(t), y(t), z(t) \rangle$.

Example: The position of a particle in space as a function of time can be described by $\vec{r}(t)$.

Partial Derivatives and Multivariable Functions

• Partial Derivative: The derivative of a function of several variables with respect to one variable, holding the others constant. Notation: $\frac{\partial f}{\partial x}$.

• Gradient: The vector of all partial derivatives: $\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle$.

• Directional Derivative: The rate at which $f$ changes in the direction of a vector $\vec{u}$: $D_{\vec{u}} f = \nabla f \cdot \vec{u}$.

Example: For $f(x, y) = x^2 + y^2$, $\frac{\partial f}{\partial x} = 2x$ and $\frac{\partial f}{\partial y} = 2y$.

Assessment and Grading

Component	Number	Percent
Midterm Exam	1	30
Attendance		10
Homework Assignments	3	10
Quizzes	3	10
Laboratory Activities	10	10
Final Examination	1	30

Grading Scale:

Score (%)	Grade
85-100	A
75-84	B
65-74	C
55-64	D
0-54	F

Academic Support and Policies

• Attendance: Required for all sessions.

• Academic Integrity: Collaboration without consent and plagiarism are prohibited.

• Support Services: Library, writing center, and technical support are available.

Textbook and Resources

• Textbook: "Thomas' Calculus" by Joel Hass, Christopher Heil, et al. (14th Edition, Pearson)

• Online Resources: MyMathLab, JSU V.I.B.E. program for digital course materials

Additional Info

• Students are expected to use graphing utilities and mathematical software for assignments and laboratory activities.

• Course policies include academic engagement, communication, and compliance with university regulations.