Calculus
Calculus
About the Book
This book is an introduction to calculus, mainly concerned with real-valued functions of a single variable. It is based on several articles I have previously published on graphicmaths.com, plus a lot of new material.
The book assumes high school maths (UK KS4), and is suitable as an introduction to calculus at undergraduate level or for anyone with a general interest in mathematics.
It covers:
- Functions and limits, including domains and codomains, concepts of smoothness and continuity, tangents and slope, and squeeze theorem.
- Differentiation, including rates of change, differentiation from first principles, standard derivatives, stationary points, and L'Hopital's rule.
- Integration, including indefinite and definite integrals, standard integrals, Riemann sums, the fundamental theorem of calculus, and improper integrals.
- Differentiation techniques, including the product rule, chain rule, quotient rule, reciprocal rule, inverse function rule, implicit differentiation and logarithmic differentiation.
- Integration techniques, including integration by substitution, integration by parts and the LIATE rule, trig identities, trig substitution, and partial fractions.
The book takes the approach of providing intuitive explanations and examples followed by formal proofs that build on an intuitive understanding.
About the Author
Table of Contents
- 1 - Introduction
- 1.1 - Content summary
- 2 - Functions and limits
- 2.1 - Functions
- 2.2 - Limits
- 2.3 - Big O notation
- 2.4 - Squeeze theorem
- 2.5 - Tangents to a curve
- 2.6 - More about functions
- 2.7 - Summary
- 3 - Differentiation
- 3.1 - Rate of change
- 3.2 - Differentiation
- 3.3 - Derivative notation
- 3.4 - Differentiability
- 3.4.1 - Continuous functions and differentiability
- 3.5 - Standard derivatives
- 3.6 - Rules for differentiating powers
- 3.7 - Odd and even functions
- 3.8 - Differentiating exponential functions
- 3.9 - Differentiating trig functions
- 3.10 - Differentiating hyperbolic functions
- 3.11 - The link between e^x, sinh x and sin x
- 3.12 - Fermat's theorem of stationary points
- 3.13 - L'Hopital's rule
- 3.14 - Summary
- 4 - Integration
- 4.1 - Antiderivatives
- 4.2 - Indefinite integrals
- 4.3 - Standard integrals
- 4.4 - Integrals and the area under a curve
- 4.5 - Definite integrals
- 4.6 - Some useful theorems
- 4.7 - Mean value theorem for integrals
- 4.8 - The Riemann sum
- 4.9 - Fundamental theorem of calculus
- 4.10 - Improper integrals
- 4.11 - Summary
- 5 - Differentiation techniques
- 5.1 - The product rule
- 5.2 - The chain rule
- 5.3 - The quotient rule
- 5.4 - The reciprocal rule
- 5.5 - The inverse function rule
- 5.6 - Implicit differentiation
- 5.7 - Logarithmic differentiation
- 5.8 - The Weierstrass function
- 5.9 - Summary
- 6 - Integration techniques
- 6.1 - Integration by substitution
- 6.2 - Integration by parts
- 6.3 - LIATE rule for integration by parts
- 6.4 - Using trigonometric identities
- 6.5 - Trigonometric substitutions
- 6.6 - Partial fractions
- 6.7 - Summary
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