You CAN Ace Calculus

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > mathematical logic

 One Way Implication Two Way Implication

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, we 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.

Before we get started, here is a great video.

### PatrickJMT - Introduction to Fundamental Math Proof Techniques [14mins]

video by PatrickJMT

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.