Single Variable Calculus | Spring Semester 2019
Course Description
Description: The course offers a general view to some important ideas and techniques of differentiation and integration, and reveals the relationship between them. The fundamental objects that we deal with in calculus are functions. We discus the basic ideas concerning functions, their graphs, and ways of transforming and combining them.
We will see how to interpret derivatives as slopes and rates of change, and also develop rules for finding derivatives. These differentiation rules enable us to calculate with relative ease the derivatives of polynomials, rational functions, algebraic functions, exponential and logarithmic functions, and trigonometric and inverse trigonometric functions.
The course shows how to use the integral to solve problems concerning volumes, lengths of curves, population predictions, work, consumer surplus and many others. There is a connection between integral calculus and differential calculus. The Fundamental Theorem of Calculus relates the integral to the derivative, and we will see in this course that it greatly simplifies the solution of many problems.
Prerequisite(s): PreCalculus (Optional)
Attendance: It is mandatory to attend at least 20 lectures. The attendance lists are updated manually every week.
Course Format
Study materials: Students are recommended to read the Lecture Notes and Study Guides before and after the lectures. It is necessary to test their knowledge using the online quizzes designed at this web-page.
Homework and Quiz: There are also 2 (two) Mandatory Homework assignments during the semester. We assign 10 (ten) short-quizzes during the lectures with similar questions from homework assignments and online quizzes. Students performed the homework and the online quizzes should be able to succeed the quizzes.
2 (two) worst quiz and homework results will be dropped to allow students to be absent.
Software projects: During the semester, there are three software projects assigned which illustrate the derivatives of curves and numerical integration. Students need to submit them in a group of two or three students during the assigned slots. The list of software projects:
Lecture 1.2: Introduction to functions.
- Functions transformations.
- Trigonometric functions.
Lecture 2.1: Introduction to derivatives.
- Average velocity vs Instantaneous velocity.
- Definition of derivatives.
- Rates of change in our daily life
Lecture 2.2: Derivative rules.
- Binomial formula.
- Derivatives of power functions and polynomials.
- Derivatives of trigonometric functions.
Lecture 3.1: Differentiation techniques.
- The product and quotient rules.
- Composite functions.
- Chain rule.
Lecture 3.2: Implicit differentiation and its applications
- Implicit Differentiation.
- Linear Approximation.
Lecture 4.1: Max & Min Values
- Finding extremum points.
- Finding max/min of functions.
Lecture 4.2: Shapes of graphs
- Increasing / decreasing regions
- Identifying local max/min
- Concavity
Lecture 5.1: Indetermined limits. Optimization.
- L'Hopital's rule
- Optimization problems
Lecture 5.2: Search for roots of functions
- Finding roots using binary search
- Newton's iteration
Lecture 6.1: Introduction to integration
- Antiderivatives
- Reimann's sum
- Definition of definite integral
Lecture 6.2: Fundamental Theorems of Calculus
- Fundamental theorems of Calculus
- Newton-Leibnitz theorem
Lecture 7.1: Substitution rule for integration
- Substitution for indefinite integrals
- Substitution for definite integrals
Lecture 7.2: Areas using Integration
- Areas under curves
- Finding the areas using vertical rectangles
- Finding the areas using horizontal rectangles
Lecture 8.1: Intro to volumes using Integration
- Finding volumes using disks
Lecture 9.1: Volumes using disks
Lecture 9.2: Volumes using Cylindrical Shells
Lecture 10: Integration techniques
- Substitution rule
- Trigonometric integration
- Integration by parts
- Integration of rational functions
- Integration of radicals
Lecture 11.1: Average value. Arc length.
- Average value of a function
- Mean value theorem
- Finding the Length of curves
Lecture 11.2: Surface Area of Revolution.
- Surface areas of cylinders, cones
- Approximation of the surface area with bands
- Example: surface area of a sphere
Lecture 12: Calculus with Parametric equations
- curves defined by parametric equations
- the area under a parametric curves
- derivatives of a parametric equation
- guides of constructing a tangent lines
- arc length of a parametric curve
- surface area of revolution of a parametric curve
Lecture 13: Curves in Polar Coordinates
- Points in Polar Coordinates
- Change of coordinates
- Curves in Polar Coordinates
- Derivatives of Polar Equations
- Tangent lines to Polar Curves
Lecture 13.2: Calculus with Polar Curves
- Areas of Polar Curves
- Arc Lengths of Polar Curves
Lecture 14: Final Review