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Calculus
Small Changes, Big Ideas
Bill Shillito
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Front Matter
Colophon
About this project
I
Describing Change
1
Differentiation
1.1
Sensitivity to change
Introduction to sensitivity
Sensitivity of nonlinear functions
Sensitivity at a point
Interpreting sensitivity
Exercises
1.2
The derivative
The derivative of a function
Differentiating linear functions
1.3
The Product Rule and the Chain Rule
1.4
Implicit differentiation
2
Applications of derivatives
2.1
Modeling and related rates
2.2
Higher order derivatives
2.3
Optimization
2.4
Linear approximation
2.5
Relative change and elasticity
3
Integration
3.1
The integral: accumulation of change
3.2
Properties of integrals
3.3
The Fundamental Theorem of Calculus
3.4
Average value and average rates
4
Differential equations
4.1
Differential equations
4.2
Exponential functions
4.3
Trigonometric functions
4.4
Logarithmic functions
4.5
Separation of variables
5
Limits
5.1
Limits and continuity
5.2
Analytic techniques for limits
5.3
Limits and differentiation
5.4
Limits and integration
5.5
L’Hôpital’s rule
5.6
Valuable theorems
5.7
CAPSTONE: Proving the fundamental theorem
II
Harnessing Infinity
6
Analytic geometry
6.1
Motion in the plane
6.2
Parametric curves
6.3
Polar coordinates
6.4
Arc length
6.5
Curvature
7
Area and volume
7.1
Area between curves
7.2
Volume by slicing
7.3
Volume by shells
7.4
Surfaces of revolution
8
Integration techniques
8.1
Integration by substitution
8.2
Integration by parts
8.3
Trigonometric substitution
8.4
Partial fractions
8.5
Advanced integration strategy
9
Special functions
9.1
Functions defined by integrals
9.2
Numerical integration
9.3
Improper integrals
9.4
Probability distributions
10
Infinite series
10.1
Taylor approximations and series
10.2
Error in Taylor series
10.3
Manipulating power series
10.4
Geometric series
10.5
Generating functions
10.6
Hyperbolic functions
10.7
CAPSTONE: Fourier series
III
New Dimensions
11
Higher dimensions
11.1
What are dimensions, really?
11.2
Extending to higher dimensions
11.3
Slicing through dimensions
11.4
Functions of multiple variables
12
Partial derivatives
12.1
Partial derivatives
12.2
Tangent plane approximations
12.3
Partial elasticity
12.4
Higher order partial derivatives
12.5
Multivariable optimization
13
Vectors
13.1
Vector-valued functions
13.2
Motion in space
13.3
The dot product and cross product
13.4
The Frenet-Serret frame
14
Gradients
14.1
The chain rule in higher dimensions
14.2
Directional derivatives and the gradient
14.3
Geometry of the gradient
14.4
Lagrange multipliers
14.5
Gradient descent
15
Multiple integrals
15.1
Double integrals and iterated integrals
15.2
Double integrals over general regions
15.3
Surface area
15.4
Triple integrals
15.5
Center of mass
15.6
Multivariate probability
15.7
CAPSTONE: Calculus of variations
IV
Transforming Perspectives
16
Complex functions
16.1
The geometry of complex numbers
16.2
Euler’s formula and its consequences
16.3
Visualizing complex functions
16.4
Multivalued function behavior
17
Vector fields
17.1
Vector fields
17.2
Dynamical systems
17.3
New derivatives and the del operator
17.4
The Cauchy-Riemann equations
17.5
Potential functions
18
Coordinate transformations
18.1
Coordinate transformations
18.2
The Jacobian matrix and its determinant
18.3
Double integrals over polar regions
18.4
Cylindrical and spherical coordinates
19
Line and surface integrals
19.1
Line integrals
19.2
Work, circulation, and flux
19.3
Contour integrals
19.4
Surface integrals
19.5
Integrals over oriented surfaces
20
Fundamental theorems
20.1
Conservative vector fields
20.2
Green’s theorems
20.3
Fundamental theorems in higher dimensions
20.4
Integrals of holomorphic functions
20.5
Singularities and Laurent series
20.6
The residue theorem
20.7
CAPSTONE: Differential forms and exterior calculus
V
Firm Foundations
21
Properties of the real numbers
21.1
The need for rigor
21.2
Ordered fields
21.3
The completeness property
21.4
Topology of the real numbers
22
Sequences and limits
22.1
Convergence of sequences
22.2
Proving limit laws
22.3
Limits of functions
22.4
Cauchy sequences
22.5
Constructing the real numbers
23
Differentiability
23.1
Differentiability classes
23.2
More on the Mean Value Theorem
23.3
Differentiability in higher dimensions
23.4
The inverse and implicit function theorems
24
Integrability
24.1
Compactness
24.2
Uniform continuity
24.3
Riemann and Darboux integrals
24.4
Integrability conditions and properties
24.5
Lebesgue measure
25
Series and limiting processes
25.1
Convergence of series
25.2
Absolute and conditional convergence
25.3
Pointwise and uniform convergence
25.4
Interchanging limit processes
25.5
Convergence of power series
25.6
Solutions to differential equations
25.7
CAPSTONE: Non-standard analysis
Back Matter
Bibliography
Calculus
Small Changes, Big Ideas
Bill Shillito
Mathematics Department
Oglethorpe University
Last Updated: May 11, 2026
Colophon
About this project
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