MATH 86857 Calculus of One Variable
Lecture 1 Notes
**Lecture 1: Introduction to Calculus of One Variable**
**Overview:**
- Calculus is the branch of mathematics that deals with rates of change and accumulation, and it
is essential for understanding many natural phenomena and mathematical concepts.
- In this course, we will focus on the calculus of one variable, which involves studying functions
of a single variable and their derivatives and integrals.
**Key Concepts:**
1. **Functions**:
- A function is a rule that assigns to each input value a unique output value.
- Functions can be represented by equations, graphs, or tables.
- We will work with various types of functions, such as polynomial functions, trigonometric
functions, exponential functions, and logarithmic functions.
2. **Limits**:
- The concept of a limit is fundamental in calculus as it allows us to understand the behavior of
functions as the input value approaches a certain value.
- We will learn how to calculate limits algebraically, graphically, and using limit laws.
3. **Derivatives**:
- The derivative of a function measures the rate at which the output value changes with respect
to the input value.
- It gives us information about the slope of the function at a particular point.
- We will study techniques for finding derivatives of various functions, including the power rule,
product rule, quotient rule, and chain rule.
4. **Applications of Derivatives**:
- Derivatives have many practical applications, such as determining maximum and minimum
values of functions, modeling rates of change, and solving optimization problems.
- We will explore these applications through examples and problems.
5. **Integration**:
- Integration is the reverse process of differentiation and involves finding the area under a
curve.
- We will learn how to calculate integrals using techniques like substitution, integration by
parts, and trigonometric integrals.
