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Active Prelude to Calculus
Matthew Boelkins
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Front Matter
Colophon
Acknowledgements
Contributors
Our Goals
Features of the Text
Students! Read this!
Instructors! Read this!
1
Relating Changing Quantities
1.1
Changing in Tandem
1.1.1
Using Graphs to Represent Relationships
1.1.2
Using Algebra to Add Perspective
1.1.3
Summary
1.1.4
Exercises
1.2
Functions: Modeling Relationships
1.2.1
Functions
1.2.2
Comparing models and abstract functions
1.2.3
Determining whether a relationship is a function or not
1.2.4
Summary
1.2.5
Exercises
1.3
The Average Rate of Change of a Function
1.3.1
Defining and interpreting the average rate of change of a function
1.3.2
How average rate of change indicates function trends
1.3.3
Summary
1.3.4
Exercises
1.4
Linear Functions
1.4.1
Properties of linear functions
1.4.2
Interpreting linear functions in context
1.4.3
Summary
1.4.4
Exercises
1.5
Quadratic Functions
1.5.1
Properties of Quadratic Functions
1.5.2
Modeling falling objects
1.5.3
How quadratic functions change
1.5.4
Summary
1.5.5
Exercises
1.6
Composite Functions
1.6.1
Composing two functions
1.6.2
Composing functions in context
1.6.3
Function composition and average rate of change
1.6.4
Summary
1.6.5
Exercises
1.7
Inverse Functions
1.7.1
When a function has an inverse function
1.7.2
Determining whether a function has an inverse function
1.7.3
Properties of an inverse function
1.7.4
Summary
1.7.5
Exercises
1.8
Transformations of Functions
1.8.1
Translations of Functions
1.8.2
Vertical stretches and reflections
1.8.3
Combining shifts and stretches: why order sometimes matters
1.8.4
Summary
1.8.5
Exercises
1.9
Combining Functions
1.9.1
Arithmetic with functions
1.9.2
Combining functions in context
1.9.3
Piecewise functions
1.9.4
Summary
1.9.5
Exercises
2
Circular Functions
2.1
Traversing Circles
2.1.1
Circular Functions
2.1.2
Properties of Circular Functions
2.1.3
The average rate of change of a circular function
2.1.4
Summary
2.1.5
Exercises
2.2
The Unit Circle
2.2.1
Radians and degrees
2.2.2
Special points on the unit circle
2.2.3
Special points and arc length in non-unit circles
2.2.4
Summary
2.2.5
Exercises
2.3
The Sine and Cosine Functions
2.3.1
The definition of the sine function
2.3.2
The definition of the cosine function
2.3.3
Properties of the sine and cosine functions
2.3.4
Using computing technology
2.3.5
Summary
2.3.6
Exercises
2.4
Sinusoidal Functions
2.4.1
Shifts and vertical stretches of the sine and cosine functions
2.4.2
Horizontal scaling
2.4.3
Circular functions with different periods
2.4.4
Summary
2.4.5
Exercises
3
Exponential and Logarithmic Functions
3.1
Exponential Growth and Decay
3.1.1
Exponential functions of form
\(f(t) = ab^t\)
3.1.2
Determining formulas for exponential functions
3.1.3
Trends in the behavior of exponential functions
3.1.4
Summary
3.1.5
Exercises
3.2
Modeling with exponential functions
3.2.1
Long-term behavior of exponential functions
3.2.2
The role of
\(c\)
in
\(g(t) = ab^t + c\)
3.2.3
Modeling temperature data
3.2.4
Summary
3.2.5
Exercises
3.3
The special number
\(e\)
3.3.1
The natural base
\(e\)
3.3.2
Why any exponential function can be written in terms of
\(e\)
3.3.3
Summary
3.3.4
Exercises
3.4
What a logarithm is
3.4.1
The base-
\(10\)
logarithm
3.4.2
The natural logarithm
3.4.3
\(f(t) = b^t\)
revisited
3.4.4
Summary
3.4.5
Exercises
3.5
Properties and applications of logarithmic functions
3.5.1
Key properties of logarithms
3.5.2
The graph of the natural logarithm
3.5.3
Putting logarithms to work
3.5.4
Summary
3.5.5
Exercises
3.6
Modeling temperature and population
3.6.1
Newton’s Law of Cooling revisited
3.6.2
A more realistic model for population growth
3.6.3
Summary
3.6.4
Exercises
4
Trigonometry
4.1
Right triangles
4.1.1
The geometry of triangles
4.1.2
Ratios of sides in right triangles
4.1.3
Using a ratio involving sine and cosine
4.1.4
Summary
4.1.5
Exercises
4.2
The Tangent Function
4.2.1
Two perspectives on the tangent function
4.2.2
Properties of the tangent function
4.2.3
Using the tangent function in right triangles
4.2.4
Summary
4.2.5
Exercises
4.3
Inverses of trigonometric functions
4.3.1
The arccosine function
4.3.2
The arcsine function
4.3.3
The arctangent function
4.3.4
Summary
4.3.5
Exercises
4.4
Finding Angles
4.4.1
Evaluating inverse trigonometric functions
4.4.2
Finding angles in applied contexts
4.4.3
Summary
4.4.4
Exercises
4.5
Other Trigonometric Functions and Identities
4.5.1
Ratios in right triangles
4.5.2
Properties of the secant, cosecant, and cotangent functions
4.5.3
A few important identities
4.5.4
Summary
4.5.5
Exercises
5
Polynomial and Rational Functions
5.1
Infinity, limits, and power functions
5.1.1
Limit notation
5.1.2
Power functions
5.1.3
Summary
5.1.4
Exercises
5.2
Polynomials
5.2.1
Key results about polynomial functions
5.2.2
Using zeros and signs to understand polynomial behavior
5.2.3
Multiplicity of polynomial zeros
5.2.4
Summary
5.2.5
Exercises
5.3
Modeling with polynomial functions
5.3.1
Volume, surface area, and constraints
5.3.2
Other applications of polynomial functions
5.3.3
Summary
5.3.4
Exercises
5.4
Rational Functions
5.4.1
Long-range behavior of rational functions
5.4.2
The domain of a rational function
5.4.3
Applications of rational functions
5.4.4
Summary
5.4.5
Exercises
5.5
Key features of rational functions
5.5.1
When a rational function has a “hole”
5.5.2
Sign charts and finding formulas for rational functions
5.5.3
Summary
5.5.4
Exercises
Back Matter
Index
Colophon
Colophon
Colophon
This book was authored in PreTeXt.
