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Prerequisites for Calculus
Chapter Overview
Chapter Overview
Connections
Connections
Willard Libby won the 1960 Nobel Prize in Chemistry for his 1949 discovery of the method of radiocarbon dating for estimating the age of organic substances that were once living tissue. The method relies on the slow decay of carbon-14, a radioactive isotope of carbon with a half-life of approximately 5730 years. Using exponential functions and the properties reviewed in section 1.3, researchers have used radiocarbon dating to study the carefully preserved remains of early Egyptian dynasties and thus reconstruct the history of an important ancient civilization.
Chapter 1 Overview
Chapter 1 Overview
The main prerequisite for a student who wants to undertake the study of calculus is an understanding of functions. It is the context of functions that brings coherence to the study of algebra and provides a connection between algebra and geometry, especially through the graphical representations of algebraic expressions. Students who have taken a modern precalculus course (emphasizing algebraic, numerical, and graphical representations of functions) will probably have already seen everything in his first chapter, but we offer it here mainly in the spirit of review. Since calculus is basically a tool for understanding how functions of various kinds model real-world behavior, a solid understanding of the basic functions makes the applications of calculus considerably easier. Any time you spend in this chapter strengthening understanding of functions and graphs will pay off later in the course. We begin by reviewing the easiest type of function, surely the most familiar to students. Hopefully, it is also the most important ( by far) for understanding how calculus works.
Section 1.1: Linear Functions
Section 1.1: Linear Functions
Lesson Objectives
Create linear equations given information about points, slope, and intercepts.
Solve problems by writing two-variable linear equations.
Section 1.2: Functions and Their Graphs
Section 1.2: Functions and Their Graphs
Lesson Objectives
Use the language, notation, and graphical representation of functions to express relationships between variable quantities.
Section 1.3: Exponential Functions
Section 1.3: Exponential Functions
Lesson Objectives
Identify exponential functions.
Determine the domain and range of exponential functions.
Graph exponential functions.
Quick Quiz: Sections 1.1-1.3
Quick Quiz: Sections 1.1-1.3
Section 1.4: Parametric Equations
Section 1.4: Parametric Equations
Lesson Objectives
Define curves parametrically.
Graph parametric equations.
Determine the Cartesian equation that contains a given parametric equation.
Section 1.5: Inverse Functions and Logarithms
Section 1.5: Inverse Functions and Logarithms
Lesson Objectives
Find the inverse of a function.
Use composition to verify that functions are inverses.
Identify logarithmic functions.
Determine the domain and range of logarithmic functions.
Identify and analyze the graphs of logarithmic functions.
Evaluate, expand, and simplify logarithmic expressions using properties of logarithms.
Section 1.6: Trigonometric Functions
Section 1.6: Trigonometric Functions
Lesson Objectives
Convert between degree and radian measure.
Use the definition of radian measure to calculate arc lengths, radii, and angle measures.
Analyze key features of inverse trigonometric functions from equations and graphs.
Evaluate inverse trigonometric functions over a specified domain.
Solve trigonometric equations over a specified domain.
Complete the following exercises, beginning on page 50:
1–5 (odd problems only), 9–19 (odd problems only), 23–27 (odd problems only), 31, 33, 37, 43
Quick Quiz: Sections 1.4-1.6
Quick Quiz: Sections 1.4-1.6
Unit 1 Test
Unit 1 Test
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