# Study Guide

Here is a list of topics/ideas you need to be familiar with for tests. The list may be incomplete; refer to lecture notes for everything.

You can find example problems to work on in the textbook for this course. Highly recommended!

## Chapter 1: Preliminaries

1. Exponent laws
2. How to add fractions (LCD)
3. How to solve quadratics $ax^2 + bx + c = 0$
4. When you are allowed to cancel common factors (all $+$ and $-$ encapsulated in parentheses)
5. Coordinate plane
6. Set and interval notation
1. union, intersection, number line techinque to find these
7. Polynomial/rational functions
8. How to isolate a variable
9. the list goes on...

## Chapter 2 : Functions, Limits, and the Derivative

### 2.1: Functions and Their Graphs

1. Definition of a function
2. How to evaluate a function, i.e. $f(a + h)$
3. How to find domain of a function (exclude two problems)
4. Definition of the graph of a function
5. How to graph functions by hand (make a table)
1. $ax + b$ has shape of a line
2. $ax^2 + bx + c$ has shape of a parabola
6. Piecewise functions and their graphs
7. Vertical line test
8. Word problems of the above

### 2.2: Algebra of Functions

1. How to $+, -, \times, \div$ functions and their domains
2. Physical meaning of adding two functions together
3. fixed/variable/total cost, total profit and revenue
4. Definition of composition of functions
5. How to decompose a composition into two functions
6. Word problems (including in a business context) of the above

### 2.4: Limits

1. Definition of a limit
2. When does a limit exist (cannot be not a number or left and right approach disagree)
3. Table + graph method to find limit
4. Find limit given a graph
5. 10 limit laws and how to apply them correctly
6. How to deal with indeterminate forms
7. How to find limits at infinity for any function, such as rational function
8. Word problems (including in a business context) of the above

### 2.5: One-sided limits and continuity

1. Left- and right-handed limits and the fact that limit laws can be used on them
2. How to use them to prove a limit exists
3. Intuitive definition of continuity
4. Mathematical defintiion of continuity (three conditions)
5. Properties of continuous functions
6. Any polynomial and rational function are continuous on their domains
7. Intermediate value theorem and how it can be used to prove a zero (x-intercept) exists
8. Left- and right-handed limits

### 2.6: The Derivative

1. Mathematical definition of a derivative
2. Two meanings of the derivative
3. Find derivative of a function with the definition of the derivative
4. When a derivative fails to exist

## Chapter 3: Differentiation

### 3.1: Basic Rules of Differentiation

1. Derivative of a constant
2. Power Rule
3. Pulling constant out
4. Derivative distributues across addition and subtraction

### 3.2-3.3: Product, Quotient, and Chain Rule

1. When to use each one
2. General Power Rule as a shortcut to the chain rule
3. How to apply multiple rules in the same problems

### 3.4: Marginal Functions in Economics

1. Actual cost vs. marginal cost
2. Average cost functions
3. Marginal cost and marginal average cost function
4. Demand curve and demand equation $p = f(x)$
5. Marginal revenue and profit function
6. Finding the revenue function given a demand equation
7. English interpretations of each marginal
8. Elasticity of demand: calculating + interpretation
9. Relationship between elasticity and revenue

### 3.5: Higher-Order Derivatives

1. How to calculate higher-order derivatives
2. position, velocity and acceleration
3. Consumer Price Index: meaning of first and second derivative

### 3.6: Implicit Differentiation and Related Rates

1. How to implicity differentiate
2. When to use product, quotient, and chain rule in implicit differentiation
4. Related rates problems

## Chapter 4: Applications of the Derivative

### 4.1: Applications of the First Derivative

1. What an increasing/decreasing function is
2. Relationship between $f'(x)$ and whether $f(x)$ is increasing/decreasing
3. Be able to find increasing/decreasing intervals
4. Definition of relative extrema
5. First Derivative Test for Relative Extrema

### 4.2: Applications of the Second Derivative

1. What concavity means
2. Determining intervals of concavity
3. Second Derivative Test for Relative Extrema
4. Weakness of Second Derivative Test vs. First (for example, at sharp corners second derivative does not exist)
5. Signs of $f'(x)$ and $f''(x)$ force a certain graph shape

### 4.4: Optimization I

1. What an absolute extrema is and how it is different than relative extrema
2. Continuous function + closed interval forces absolute extrema to exist
3. Closed interval test for finding absolute extrema
4. Second derivative test + concavity for finding absolute extrema

#### 4.5: Optimization II

1. Be able to translate English into equations and diagrams
2. Solve 4.4 type world problems except the equation to find absolute min/max needs to be parsed from the sentence