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CALCULUS 1
An Introductory–Calculus Tutorial
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By Pheng Kim Ving, BA&Sc, MSc
Email: phengkimving@yahoo.ca
Toronto - Ontario - Canada
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Welcome To CALCULUS 1 PROBLEMS & SOLUTIONS!
This website posts level-1
calculus problems & solutions tutorial, free! If you're taking an introductory or a first
one-semester calculus course, you may find this tutorial helpful. It provides a
complete tutorial on beginning calculus of
functions of 1 real variable. It's organized to accompany a 1-semester 1st
calculus course. Virtually it treats all or even
more material that's usually covered over such a period. You'll hopefully find
here any topic that you need help with that's
taught in your introductory calculus course.
Each chapter is divided into
sections. Each section consists of 2 parts: Review and Problems &
Solutions. The Review part
reviews the topics that are subject of the section. The presentation of the
topics is fairly comprehensive and detailed,
almost as found in textbooks, not just a summary of the topics. The Problems
& Solutions part presents problems each
followed by their solution. The problems supply drills on the basic techniques
for the topics discussed in the section and
some are theoretical and/or difficult
If you need clarification or
explanation on any part of any review or on any problem & solution, or have
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Reference To A
Function
Splitting
Of The Topic Of The Applications Of The Derivative
Notations
And Abbreviations
1. Limits And Continuity
1.1 Limits
1.1.1
Limits
1.1.2
Properties Of Limits
1.1.3
The Indeterminate Form Of Type 0/0
1.1.4
One-Sided Limits
1.1.5 Limits At Infinity And Infinite Limits
1.1.5.1
Limits At Infinity And Infinite Limits
1.2 Continuity
1.2.1
Continuity
1.2.2
Extrema
1.2.3
The Intermediate-Value Theorem
2. The Derivative
2.1 The Derivative
2.1.1
Rates Of Change
2.1.2
Tangent Lines And Their Slopes
2.1.3
The Derivative
2.2
Differentiability Vs Continuity
2.3 Rules Of Differentiation
2.3.1
Differentiation Of Sums, Differences, And Polynomials
2.3.2
Differentiation Of Products And Quotients
2.3.3
Differentiation Of Compositions Of Functions - The Chain Rule
2.3.4
Differentiation Of Inverse Functions
2.4
Higher-Order Derivatives
2.5
Implicit Differentiation
2.6
The Differentials
3. Applications Of The Derivative – Part 1
3.1
The Mean-Value Theorem
3.2
Critical Points And Extrema
3.3
The First-Derivative Test
3.4
Concavity And Inflection
3.5
The Second-Derivative Test
3.6
Sketching Graphs Of Functions
3.7
Antiderivatives And Indefinite Integrals
3.8
Motion
3.9
Differential Equations
4. The Elementary Transcendental Functions
4.1 The Trigonometric Functions And Their Inverses
4.1.1 The Trigonometric Functions
4.1.1.1
The Trigonometric Functions
4.1.1.2
Trigonometric Identities
4.1.1.3
Limits Of Trigonometric Functions
4.1.1.4
Differentiation Of Trigonometric Functions
4.1.1.5
Graphs Of Trigonometric Functions
4.1.1.6
The Projectile Motion
4.1.1.7
The Simple Harmonic Motion
4.1.2 The Inverse Trigonometric Functions
4.1.2.1
The Inverse Trigonometric Functions
4.1.2.2
Differentiation Of The Inverse Trigonometric Functions
4.2 The Logarithmic And Exponential
Functions
4.2.1
The Natural Logarithm Function
4.2.2
The Natural Exponential Function
4.2.3
General Exponential And Logarithmic Functions
4.2.4
Logarithmic Differentiation
4.2.5
Growth And Decay
4.3 The Hyperbolic Functions And Their
Inverses
4.3.1
The Hyperbolic Functions
4.3.2
The Inverse Hyperbolic Functions
5. Applications Of The Derivative – Part 2
5.1
Optimization
5.2
Related Rates
5.3 Linear Approximations
5.3.1
Tangent-Line Approximations
5.3.2
Approximations Of Errors In Measurement
5.3.3
Approximations Of Roots Of Functions – Newton's Method
5.4.2
More Indeterminate Forms
6. The Integral
6.1 Areas And Riemann Sums
6.1.1
Summation Notation And Formulas
6.1.2
Areas And Riemann Sums
6.2
The Definite Integral
6.3
The Fundamental Theorem Of Calculus
6.4
Integration By Inspection
6.5 Techniques Of Integration
6.5.1 The Method Of Substitution
6.5.1.1
The Method Of Substitution
6.5.1.2
Integration Of Trigonometric Functions
6.5.1.3
Integration Of Powers Of Trigonometric Functions
6.5.1.4
The Inverse Trigonometric Substitution
6.5.1.5
Other Substitutions
6.5.2
The Method Of Partial Fractions
6.5.3
The Method Of Integration By Parts
6.6
Approximate Numerical Integration
6.7 Improper Integrals
6.7.1
Improper Integrals
6.7.2
Tests For Convergence Of Improper Integrals
7. Applications Of The Integral
7.1
The Mean-Value Theorem For Integrals
7.2
Areas Of Plane Regions
7.3 Volumes
7.3.1
Finding Volumes By Slicing
7.3.2
Finding Volumes By Using Cylindrical Shells
7.4
Distance
7.5
Arc Length
7.6
Areas Of Surfaces Of Revolution
7.7 Applications To Physics
7.7.1
Work
7.7.2
Force Exerted By A Fluid
7.8
Differential Equations – Variables Separable
Additional Topics
8. Plane Curves
8.1 Parametric Curves
8.1.1
Parametric Curves
8.1.2
Tangent And Sketching Of Parametric Curves
8.1.3
Arc Length And Area Of Surface Of Revolution Of Parametric Curves
8.1.4
Vector Study Of Motion In The Plane
8.2 The Polar Coordinate System
8.2.1
The Polar Coordinate System
8.2.2
Sketching Polar Curves
8.2.3
Area By Polar Curves
8.2.4
Arc Length And Area Of Surface Of Revolution Of Polar Curves
9.1
Infinite Sequences
9.2
infinite Series
9.3
The Comparison Tests
9.4
The Root And Ratio Tests
9.5
The Integral Test
9.6
The Alternating-Series And Absolute-Convergence Tests
9.7.
Approximations Of Sums Of Series
Last Updated: 14 Mar 2009