### Unit 1: Preview and Review

While a first course in calculus can strike you as something new to learn, it is not comparable to learning a foreign language where everything seems different. Calculus still depends on most of the things you learned in algebra, and the true genius of Isaac Newton was to realize that he could get answers for this something new by relying on simple and known things like graphs, geometry, and algebra. There is a need to review those concepts in this unit, where a graph can reinforce the adage that a picture is worth one thousand words. This unit starts right off with one of the most important steps in mastering problem solving: Have a clear and precise statement of what the problem really is about.

**Completing this unit should take you approximately 7 hours.**

Upon successful completion of this unit, you will be able to:

- approximate a slope of a tangent line from a function given as a graph;
- approximate the area of an irregular figure by counting inside squares;
- calculate the slope of the line through two points;
- write the equation of the line through two points using both slope-intercept and point-slope forms;
- write the equation of a circle with a given center and radius;
- evaluate a function at a point, given by a formula, graph, table, or words;
- evaluate a combination, or a composition, of functions when indicated by the symbols +, - , *, /;
- evaluate and graph the elementary functions as well as and ;
- state whether a given "if-then" statement is true or false, and justify the answer;
- state which parts of a mathematical statement are assumptions, or hypotheses, and which are conclusions; and
- state the contrapositive form of an "if-then” statement.

### 1.1: Preview of Calculus

Read this section for an introduction to calculus.

### 1.1.1: Practice Problems

Work through the odd-numbered problems 1 - 7 on page 5 and page 6. Once you have completed the problem set, check your answers for the odd-numbered questions here.

### 1.1.2: Review

Before moving on, you should be comfortable with each of these topics:

- The Slope of a Tangent Line; pages 1-2.
- The Area of a Shape; pages 3-4.
- Limits; page 4.
- Differentiation and Integration; page 4.

- The Slope of a Tangent Line; pages 1-2.

### 1.2: Lines in the Plane

Read this section and work through practice problems 1-9. For solutions to the practice problems, see page 15.

### 1.2.1: Practice Problems

Work through the odd-numbered problems 1-29 on pages 10-14. Once you have completed the problem set, check your answers here.

### 1.2.2: Review

Before moving on, you should be comfortable with each of these topics:

- The Real Number Line; page 1.
- The Cartesian Plane; page 2.
- Increments and Distance between Points in the Plane; pages 2-3.
- Slope between Points in the Plane; pages 4-6.
- Equations of Lines; page 6.
- Two-Point and Slope-Intercept Equations; pages 6-7.
- Angles between Lines; page 8.
- Parallel and Perpendicular Lines; pages 8-9.
- Angles and Intersecting Lines; page 10.

- The Real Number Line; page 1.

### 1.3: Functions and Their Graphs

Read this section for an introduction to functions and their graphs. Work through practice problems 1-5. For solutions to the practice problems, see pages 13-14.

### 1.3.1: Practice Problems

Work through the odd-numbered problems 1-23 on pages 8-13. Once you have completed the problem set, check your answers here.

### 1.3.2: Review

Before moving on, you should be comfortable with each of these topics:

- Definition of a Function; page 1.
- Function Machines; page 2.
- Functions Defined by Equations; pages 2-3.
- Functions Defined by Graphs and Tables of Values; pages 3-4.
- Creating Graphs of Functions; pages 4-5.
- Reading Graphs; pages 6-8.

- Definition of a Function; page 1.

### 1.4: Combinations of Functions

Read this section on pages 1-11 for an introduction to combinations of functions, then work through practice problems 1-9. For solutions to the practice problems, see pages 18-20.

### 1.4.1: Practice Problems

Work through the odd-numbered problems 1-31 on pages 11-16. Once you have completed the problem set, check your answers here.

### 1.4.2: Review

Before moving on, you should be comfortable with each of these topics:

- Multiline Definition of Functions; page 1.
- Wind Chill Index Sample; pages 1-3.
- Composition of Functions - Functions of Functions; pages 3-4.
- Shifting and Stretching Graphs; pages 5-6.
- Iteration of Functions; pages 6-7.
- Absolute Value and Greatest Integer; pages 7-9.
- Broken Graphs and Graphs with Holes; pages 10-11.

- Multiline Definition of Functions; page 1.

### 1.5: Mathematical Language

Read this section on pages 1-5 for an introduction to mathematical language, then work through practice problems 1-4. For the solutions to the practice problems, see pages 7-8.

### 1.5.1: Practice Problems

Work through the odd-numbered problems 1-25 on pages 5-7. Once you have completed the problem set, check your answers for the odd-numbered questions here.

### 1.5.2: Review

Before moving on, you should be comfortable with each of these topics:

- Equivalent Statements; page 1.
- The Logic of "And" and "Or"; page 1.
- Negation of a Statement; page 2.
- "If-Then" Statements; pages 2-3.
- Contrapositive of "If-Then" Statements; page 4.
- Converse of "If-Then" Statements; pages 4-5.

- Equivalent Statements; page 1.

### Problem Set 1

Take this assessment to see how well you understood these concepts.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment