A Modern Introduction to Calculus
About this Book
Acknowledgements
Prerequisites
Help! I can’t see the diagrams!!
Oh no! I destroyed a plot by mistake!!
1
Preliminaries and Precalculus
1.1
Motivating Calculus
1.1.1
Understanding the Universe
1.1.2
Art and Mathematics
1.2
Function Basics
1.2.1
Function Definition
1.2.2
Polynomial Functions
1.2.3
Rational Functions
1.2.4
Even and Odd Functions
1.3
Circle Theorems and Formulas
1.3.1
Radius and Tangent are Perpendicular
1.3.2
Arc Length of a Circle and Radians
1.3.3
Area of a Sector
1.4
Trigonometric Functions
1.4.1
Basic Trigonometry
1.4.2
Trigonometric Functions and The Unit Circle
1.4.3
Sine, Cosine, and Tangent Plots
1.4.4
Sum and Difference Identities
1.4.5
More Trigonometric Identities
1.4.6
Inverse Trigonometric Functions
1.4.7
Law of Sines
1.5
Introduction to Vectors
1.5.1
Definition
1.5.2
Vector Components
1.5.3
Negative of a Vector
1.5.4
Vector Addition
1.5.5
Vector Subtraction
1.5.6
The Dot Product
1.5.7
Law of Cosines
1.5.8
The Triangle Inequality
1.6
Exponents and Logarithms
1.6.1
Exponent Basics
1.6.2
Graphs of Exponential Functions
1.6.3
Examples Involving Exponents
1.6.4
The Number e
1.6.5
The Logarithm
1.7
Function Transformations
1.7.1
Vertical Translations
1.7.2
Horizontal Translations
1.7.3
Vertical Compression/Stretching
1.7.4
Horizontal Compression/Stretching
1.8
Complex Numbers
2
Limits
2.1
Discussion of Limits
2.2
Definition of Limits
2.3
Limits and Functions
2.3.1
Pointwise Limit
2.3.2
Holes
2.3.3
Limit at Infinity and Asymptotes
2.3.4
Breaking Limits
2.4
Limit Rules
2.4.1
Basic Rules
2.4.2
The Sandwich Theorem
2.5
Limit Examples and Problems
2.6
Applications of Sandwich Theorem
2.6.1
Limit of sin(h)/h
2.6.2
Limit of (1 - cos(h))/h
3
Introduction to the Derivative
3.1
Definition of a Derivative
3.2
Examples Applying the Limit Definition
3.2.1
Derivative of Constants
3.2.2
Derivative of Lines
3.2.3
Derivative of Monomials
3.2.4
Derivative of Square Root
3.3
Derivatives of log(x), exp(x), sin(x), and cos(x)
3.3.1
Log(x)
3.3.2
Exp(x)
3.3.3
Sin(x)
3.3.4
Cos(x)
3.4
Derivative Rules
3.4.1
Constant Rule
3.4.2
Product Rule
3.4.3
Chain Rule
3.4.4
Quotient Rule
3.4.5
Tan(x)
4
Applications of the Derivative
4.1
Related Rates and Implicit Differentiation
4.2
The Second Derivative
4.3
Finding Maxima and Minima
4.4
The Newton-Raphson Method and Finding Roots
4.5
The Mean Value Theorem and L’Hopital’s Rule
4.5.1
The Mean Value Theorem
4.5.2
L’Hopital’s Rule
5
Introduction to Integrals
5.1
Motivation - Arbitrary Areas and Distance Traveled
5.2
The Basic Idea
5.3
Limits of Riemann Sums
5.3.1
Leftpoint Sum
Rightpoint Sum
Midpoint Sum
Taking Limits of Sums
5.4
The Fundamental Theorem of Calculus, Part 1
5.5
Learning to Use Integrals
5.6
Basic Integral Formulas
5.6.1
Sum and Difference Rule
5.6.2
Constant Rule
5.6.3
Breaking the Interval Rule
5.6.4
Zero Rule for Integrals
5.6.5
Switching Limits of Integration
5.7
Indefinite Integrals
The Constant of Integration
5.7.1
Indefinite Integrals of Common Functions
5.8
Fundamental Theorem of Calculus, Part 2
6
Techniques of Integration
6.1
Expanding our Toolbox
6.2
u-substitution (Integration Chain Rule)
6.3
Visualizing u substitutions
6.4
Integration by Parts (Integration Product Rule)
6.5
Visualizing Integration by Parts
6.6
Trigonometric Integrals
6.7
Trigonometric Substitutions
6.8
Improper Integrals
6.9
Partial Fraction Decomposition
7
Integral Applications
7.1
Computing Arc Lengths
7.2
Probability and Calculus
Introduction to Probability
7.2.1
Continuous Probabilities
The Exponential Distribution
The Normal Distribution
Computing Probabilities
Introduction to Hypothesis Testing
7.3
Computing Centers of Masses
8
Sums and Infinite Series
8.1
Derivation of the Maclaurin Series
8.2
Examples of the Maclaurin Series
Sine Function
Cosine Function
Exponential Function
Euler’s Formula
Taylor Series
Derivation of Stirling’s Approximation
Derivation of Jensen’s Inequality
9
Polar Coordinates and Calculus
9.1
Play Me!
9.2
Changing coordinate systems
9.3
Plotting Functions in Polar Coordinates
Lines
Conic Sections
Cardioids
Limaçons (Snails)
Lemniscates
Rose Curves
Producing These Plots Yourselves
9.4
Derivatives of Polar Plots
9.5
Computing Areas of Polar Functions
9.6
Computing Arc Lengths of Polar Functions
10
Introduction to Differential Equations
10.1
What is a Differential Equation?
11
Calculus and Physics
11.1
Kinematic Equations and Newton’s Laws
11.1.1
Speed and Acceleration in Physics
11.1.2
x and y components
11.2
Range of Projectiles
11.3
Period of a Pendulum
11.4
Principle of Least Time and Snell’s Law
11.5
Deriving Bernoulli’s Equation
11.6
RLC Circuit Analysis
11.6.1
Kirchoff’s Current Law
11.6.2
Kirchoff’s Voltage Law
11.6.3
The Resistor
11.6.4
The Capacitor
11.6.5
The Inductor
11.6.6
The RLC Circuit
11.7
The End!
References
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An Interactive Calculus Textbook
References