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Differential and Integral Calculus
Calculus Group
Contents
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Contents
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Front Matter
1
Approximations and bounds
Where should I put my ladder?
Bounding
Concavity
2
Variables, functions and graphs
Notation and terminology
Some useful functions
Properties of functions
Graphing
3
Units, proportionality and mathematical modeling
Physical units and formulas
Modeling
Inverse functions
Exponential and logarithmic relationships
4
Limits
Definitions of limit
Variations
Continuity
Computing limits
5
Derivatives
Concept of the derivative
Definitions
First and second derivatives, and sketching
6
Computing derivatives
Tools for computing derivatives
Arguments and proofs
7
Asymptotic analysis and L'Hôpital's Rule
L'Hôpital's Rule
Orders of growth at infinity
Comparisons elsewhere and orders of closeness
8
Optimization
Definitions of Minima and Maxima, and their existence
The first derivative and extrema
Some example applications
9
Applying the optimization procedure
Optimization in geometry
Optimization in economics and business
10
Summation
Sequences
Finite series
Some series you can explicitly sum
Infinite series
Financial applications
11
Integrals
Area
Riemann sums and the definite integral
Interpretations of the integral
The fundamental theorem of calculus
Estimating sums via integrals
12
Computing integrals
Remembering and guessing
Integration by parts
Substitution
13
Integrals over the whole real line
Definitions
Convergence
14
Integration and Probability
Probability densities
Summary statistics
Some common probability densities
15
Taylor approximations
Taylor polynomials
Taylor and Maclaurin polynomials in graphing
Computing Taylor Polynomials
The Mean Value Theorem and Taylor's Theorem
Authored in PreTeXt
Differential and Integral Calculus
Calculus Group
Department of Mathematics
University of Pennsylvania
Philadelphia, USA
ancoop@math.upenn.edu