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Calculus Concepts and Modelling:
An introduction to the concepts of calculus
Jeremy Sylvestre
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Front Matter
Colophon
Author Biography
Preface
1
Variables and variation
1.1
Variables, constants, and parameters
1.2
Accumulation
1.3
Rate of variation
1.4
Constant variation
2
Functions and graphs
2.1
Basics
2.2
Graph of a function
2.3
Piecewise functions
2.4
Continuous functions
2.4.1
Definition
2.4.2
Common types of discontinuity
2.4.3
The Intermediate Value Theorem
2.5
Graph transformations
2.5.1
Vertical transformations
Vertical shift
Vertical scaling
Vertical reflection
2.5.2
Horizontal transformations
Horizontal shift
Horizontal scaling
Horizontal reflection
3
Long-term and singular behaviour
3.1
Long-term behaviour
3.1.1
Definition and first examples
3.1.2
Fundamental long-term behaviour patterns
3.1.3
Analyzing more complex examples
3.1.4
Ancient behaviour
3.1.5
Slant asymptotes
3.2
Comparing growth of two functions
3.3
Singular behaviour
3.3.1
Concept, definition, and basic examples
3.3.2
One-sided singularities
Vertical asymptotes
4
Models
4.1
Functions modelling quantity and rate
4.2
Rate equations
4.3
Direction fields
5
Step functions
5.1
Concept
5.2
Accumulation for a stepped rate function
5.2.1
Calculation pattern
5.2.2
Geometric interpretation
5.2.3
Oriented area
5.3
Sigma notation
5.3.1
Summing patterns
5.3.2
Properties of Sigma notation
5.3.3
Summing in Sage
5.4
Riemann sums
6
Accumulation from rate data
6.1
Approximating accumulation using step functions
6.1.1
Simplifying models
6.1.2
Obtaining better approximations
6.1.3
General form of approximations
6.1.4
Systematic approximation
6.1.5
Using Sage to calculate Riemann sums
6.2
Computing an exact accumulation value using step functions
6.3
The definite integral
6.3.1
Definition
6.3.2
The definite integral as net oriented area
6.3.3
Using Sage to compute definite integrals
6.3.4
Properties of the definite integral
6.3.5
Re-basing an integral
6.4
Recovering a quantity function from a rate function
6.4.1
The definite integral as an accumulation function
6.4.2
Accumulation versus quantity
7
Polynomial growth
7.1
Power and polynomial functions
7.1.1
Definitions
7.1.2
Polynomials of small degree
Constant polynomials
Linear polynomials
Quadratic polynomials
Cubic polynomials
7.1.3
Roots and factoring
7.2
Constant rate of variation
7.3
Linear rate of variation
7.4
Polynomial rate of variation
8
Logarithmic growth
8.1
The natural logarithm
8.2
Properties of the natural logarithm
9
Exponential growth
9.1
Exponential functions
9.2
Basic properties of exponential functions
9.3
The natural exponential function
9.3.1
Euler’s number
9.3.2
The natural exponential function as a reflection of the natural logarithm
9.3.3
Computing values of exponential functions
9.3.4
Undoing the natural exponential and logarithm functions
9.4
Logarithms in other bases
9.5
More properties of exponential functions
9.6
Exponential rate
10
Trigonometric functions
10.1
Periodic behaviour
10.2
Radian measure
10.3
The six trigonometric functions
10.4
Trigonometry in triangles
10.5
Trigonometric rate functions
11
Average and instantaneous rates
11.1
Averaging variation over an extended time period
11.2
Graphical interpretations of average rate of accumulation
11.3
Approximating instantaneous rate of accumulation
11.4
Calculating instantaneous rate of accumulation
11.5
The derivative
11.6
Graphical interpretation of the derivative
11.6.1
Tangent lines
11.6.2
Local linearity
11.7
Some non-differentiable examples
11.7.1
Cusps
11.7.2
Jump discontinuities
11.7.3
Singularities
11.7.4
Vertical tangents
12
The derivative function
12.1
Definition
12.2
Computing algebraically
12.3
Fundamental differentiation formulas and principles
12.3.1
Two fundamental formulas
12.3.2
Scaling and adding quantities does the same to the derivative
12.4
Derivatives of polynomials
12.5
Derivatives of the natural logarithm and exponential functions
12.5.1
The natural logarithm
12.5.2
The natural exponential
12.6
Derivatives of the sine and cosine functions
12.7
Calculating derivatives with Sage
13
Average value and the Fundamental Theorem
13.1
Average value of a function
13.2
The Mean Value Theorem
13.3
The Fundamental Theorem
13.4
Antiderivatives
13.5
Indefinite integrals
13.6
Using antiderivatives to compute definite integrals
14
Related rates
14.1
Quantities varying in tandem
14.2
The Chain Rule
14.3
Implicit differentiation
14.4
Related rate problems
14.5
Logarithmic differentiation
15
Reversing the Chain Rule
15.1
Integrating a derivative
15.2
Method of substitution
15.3
Separable rate equations
16
Composite and inverse functions
16.1
Composite functions
16.2
Inverse functions
16.3
One-to-one functions
16.4
Graph of an inverse function
16.5
Derivative of an inverse function
17
Product rule and integrating factors
17.1
Derivatives of products
17.2
Integrating factors
18
Accumulation trends
18.1
Domains of increase and decrease
18.2
Critical points and local extremes
18.3
The second derivative
18.4
Graph trends
19
Extreme values and optimization
19.1
Absolute extreme values
19.2
The Extreme Value Theorem
19.3
Searching for extremes
19.4
The Second Derivative Test
19.5
Optimization problems
20
Autonomous systems
20.1
Introduction and method of analysis
20.2
Exponential growth
20.3
Logistic growth
20.4
Growth with threshold
20.5
Logistic growth with threshold
21
Polynomial approximations
21.1
Linear approximations
21.1.1
Linearization at a point
21.1.2
Error in a linear approximation
21.1.3
Differentials
Another look at linearization
Relative error in a linear approximation
Approximating relative variation
21.2
Quadratic approximations
21.3
Higher-degree approximations: Maclaurin and Taylor polynomials
21.3.1
Higher derivatives
21.3.2
Matching higher derivatives
22
Sequences
22.1
Motivation
22.2
Definition, properties, and examples
22.2.1
Defining sequences
22.2.2
Bounded and monotonic sequences
22.3
Sequence limits
22.3.1
Definitions and examples
22.3.2
Limits of subsequences
22.3.3
Convergence versus boundedness
22.3.4
Sequence limit laws
23
Function limits
23.1
Concept and definition
23.1.1
Motivation
23.1.2
Limit points
23.1.3
Definition and examples
23.2
Function limit laws
23.3
One-sided limits
23.4
An alternative characterization
23.5
Previous concepts as function limits
23.5.1
Continuity
Continuity in terms of limits
Continuity of combinations of functions
One-sided continuity
23.5.2
Long-term and singular behaviour
23.5.3
Definite integrals
23.5.4
The derivative
23.6
L’Hopital’s rule
23.6.1
\(0/0\)
forms
23.6.2
Other indeterminate forms
\(\infty/\infty\)
form
\(0 \cdot \infty\)
form
\(\infty - \infty\)
form
Indeterminate forms involving exponents
23.6.3
Justification
24
Improper integrals
24.1
Integrals on unbounded domains
24.2
Properties
24.3
Comparison tests
24.3.1
Comparing areas
24.3.2
Comparing rates
24.4
Integrals near a vertical asymptote
25
Series
25.1
Definitions and examples
25.2
Properties
25.3
Comparison tests
25.3.1
Integral comparison test
25.3.2
Comparing two series
25.3.3
Comparing rates
25.4
More convergence tests
25.4.1
Root test
25.4.2
Ratio test
25.5
Absolute convergence
25.5.1
Absolute versus conditional convergence
25.5.2
Alternating series
25.5.3
Estimating alternating sums
26
Power and Taylor series
26.1
Motivation
26.2
Power series
26.2.1
Basics
26.2.2
Radius of convergence
26.2.3
Shifted power series
26.3
Taylor series
26.3.1
Creating Taylor series
26.3.2
Limits of Taylor series
26.4
Functions defined by power series
26.4.1
Term-by-term integration/differentiation
26.4.2
Convergence of integrated/differentiated power series
26.4.3
Using differentiation/integration to compute power series
26.4.4
Convergence at endpoints: Abel’s Theorem
26.5
Power series versus Taylor series
Back Matter
A
GNU Free Documentation License
Bibliography
Index
Colophon
Colophon
Colophon
This book was authored in PreTeXt.
