Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Limits
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Derivatives
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Differentials
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
Graphing
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Optimization
Integrals
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Work
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
p-Series
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Sequences
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametrics
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Area
Arc Length
Surface Area
Volume
Polar Coordinates
Converting
Slope & Tangent Lines
Area
Arc Length
Surface Area
Conics
Parabolas
Ellipses
Hyperbolas
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Vectors
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Work
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Curvature
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Curl
Divergence
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
More Math Help
Tutoring
Tools and Resources
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books

You CAN Ace Calculus

17calculus > logic

### Calculus Main Topics

Single Variable Calculus

Multi-Variable Calculus

### Tools

math tools

general learning tools

Mathematical Logic

Although you may not have had a class in mathematical logic yet, there are a few concepts that we will cover here that should help you understand calculus.
First, let us recommend that you read the proofs in your textbook when you come across a new theorem. Even if you don't understand each step, starting to read them now will help you understand them much more easily later on, when you DO need to understand them. We recommend this even if you are not planning on taking any more math in the future. Just the effort of trying to understand the proofs will change the chemistry of your brain now and will help you in your current course.

Okay, now for a few logic concepts. We hope you have noticed in the statement of some theorems that the language has an if-then structure (even if they don't explictly state the words 'if' and 'then'). Other theorems use an 'if and only if' phrase. It is important that you understand the difference and how the wording affects your use of the theorem.

One Way Implication; if A, then B

Let's start with 'if-then'. This is called an implication. It may be written as, 'if A, then B' and symbolically we usually write it as A → B. This means that if we know that A is true, then, based on that information alone, we can correctly conclude that B is also true. But you need to be careful here. The other direction cannot be assumed, i.e. you cannot automatically say B → A.

The character '~' is called a tilde. You may have seen it on your keyboard and used it in an emoticon or something. In mathematics, we use it to mean 'not' or 'negation'. So if I write ~A, I mean 'not A'. It turns out that if you know that A → B, then, based on that information alone, can also correctly say ~B → ~A. Notice the direction of the arrow. It may be easier for you to see if you write ~A ← ~B. Basically the idea is that if we have A → B and we know A is true, then B is also true. You can also be guaranteed that by negating both and changing the direction of the arrow, you get ~A ← ~B, which is also true. However, mathematicians don't really like to have arrows going backwards like this, so they usually leave the arrow and switch A and B to get ~B → ~A.

Now, I know this is a lot to take in and when you get into a mathematical logic course, you will learn why these work this way. For now, let me just give you an example.

A: I am a college math teacher.
B: I have a degree in mathematics.
Now, since most schools, including where I have taught calculus, require college math teachers to have a degree in mathematics, we know that A → B, i.e. if I am a college math teacher, then I have a degree in mathematics. However, you cannot assume B → A, if I have a degree in mathematics, then I am a college math teacher. There are a lot of people with math degrees who are not college math teachers. Some of them engineers. So based on the information that I am a college math teacher, which is true, then you know I have a degree in mathematics.

Now, let's look at ~B → ~A ( called the contrapositive of A → B ). Using this same example, if I do NOT have a degree in mathematics, then I am NOT a college math teacher. This is also true because, as I said above, the schools where I have taught calculus all require a math degree. Do you see how this works? In mathematical logic, if the implication A → B is true, then the contrapositive ~B → ~A is also true. This can be shown using truth tables.

Two Way Implication; A if and only if B

You can say that 'if-then' is a part or a subset of 'if and only if'. Because what 'if and only if' (sometimes shortened to iff) says is that the other direction is true also. What I mean is if you have 'A iff B', then you know A → B and B → A. Is that cool or what?!

Why is this important? Because starting with calculus, theorems become increasing important as you go on in mathematics. And you need to be able to use theorems correctly, even at the beginning calculus level. If you are studying infinite series, you have to understand these ideas in order to know how to determine convergence or divergence of an infinite series.