Calculus I Brain Summary

Malcolm E. Hays                  28 October 2002


This is a list of all the thoughts located in the Calculus I Brain. Each thought is followed by a statement indicating the content associated by that thought.  Italicized thoughts are all example problems. Thoughts followed by (jump) are jump thoughts, which are located on the left hand side of the central thought in the Brain matrix.  Jump thoughts often take the user to entirely different sections of the Brain or provide reference information about the central thought.


Review of Basic Concepts  -  A short refresher course of concepts covered in Algebra, Trig and Geometry

      Cartesian Plane  -  A brief discussion about the Cartesian plan and how it works

             Definition of Circle in Plane  -  Gives the standard equation of a circle

             Distance Formula  -  Just what it says

             Midpoint Formula  -  Just what it says

             Standard Form of Circle -  Same as "Definition of Circle in Plane"

      Function Review  -  Explanation of function notation and how a function is defined

             Function Representation -  Equivalent means of representation of functions

                   Examples -  An example of how we utilize function representation

             Translation/Combination  -  Shifting functions around the Cartesian plane

                   Composition Examples  - Examples of composing two functions with each other

Even More Examples  -  2 examples of real world applications

Trans/Combo Examples  -  Yet more examples involving shifting graphs of functions around the   Cartesian Plane

Trig Example  -  A graph showing how to scale trig functions based on amplitude and frequency

Types of Functions  -  A brief summary of the different kinds of functions and properties of said functions

Geometry Review  -  A brief reminder of the various formulas involving geometric shapes

             Related Rates (jump)  -  This link takes you to the area of Related Rates in Differentiation,

             Circles  -  Area, circumference of different shapes based on circles, such as ellipses, sectors, and rings

             Triangles  -  Includes equilateral, right, and general triangle formulas

             Other  -  Other common geometric shapes such as trapezoids and parallelograms

             Volumes  -  Formulas of 3-dimensional shapes such as cones, spheres, and wedges

Real Numbers and Number Line  -  Review of key concepts about real numbers, important to understanding calculus

             Absolute Value  -  Definition and properties of the absolute value operator

             Definition of Order  -  Basically a review of inequalities and intervals

             Intervals  -  Intervals and their notation as used in mathematics

             Properties of Inequalities  -  Just what it sounds like


Limits  -  An introduction to limits with an informal definition

      Example Problems (jump)  -  Several examples of limits using the informal definition

             Example 1  -  Uses a technique to factor a denominator and cancel like terms

             Example 2  -  A special trick is introduced to eliminate a hostile term to make the limit easier to evaluate

             Example 3  -  Rationalization of the expression to get rid of radical signs

      Limits at Infinity (jump)  -  More fully covered under Differentiation | Advanced Principles

      Basic Limits  -  3 simple formulas for evaluating limits of a particular form

             Functions vs. Limits  -  A brief discussion of the difference between a function and a limit

      Continuity and One-Sided Limits  -  Definition of continuity (and discontinuity)

             Continuity on Closed Interval  -  Definition of what it means to be continuous on a closed interval

             Existence of a Limit  -  Definition of when a (two-sided) limit exists

                   Example  -  Shows an example of when a (two-sided) limit does not exist

             Intermediate Value Theorem  -  A brief discussion of the IVT and how it works graphically

             One-Sided Limits        -  Explanation of what they are, including a Flash Video Example

             Properties of Continuity  -  Several common properties exploited by various functions

      Delta-Epsilon Definition  -  The formal definition of what a limit actually is, explained in graphical terms

             Example 1  -  Shows how to use the D-E Definition, using a specified value of

Example 2  -  Shows how to choose a value for  to use the D-E definition in order to prove that a given limit is true

             Example 3  -  Uses a graph to find a particular value for

      Infinite Limits  -  Explanation of what happens when a given limit equals

             Properties  -  Several characteristic properties of what happens when limits equal

             Vertical Asymptotes  -  The graphical explanation of infinite limits

      Properties of Limits  -  Just what it sounds like

             Example 1  -  Finding the limit using a trick of factoring, then canceling factors

             Example 2  -  Finding the limit by rationalizing the numerator

             Example 3  -  Finding the limit using a trick of finding the least common denominator

             Example 4  -  Finding the limit of a function involving absolute values

      Special Limits  -  A list of limits of different types of functions

             Composite Functions  - Limit of two functions composed with each other

             Polynomial Functions  -  Limit of a polynomial function

             Radical Functions  -  Limit of a function involving a  symbol

             Rational Functions  -  Limit of one polynomial function divided by another

             Trig Functions  -  Limit of sine, cosine, tangent and their associated cofunctions

      Squeeze Theorem       -  How to find the limit of a function through indirect means

             Example  -  An application of the Squeeze Theorem

      Tangent Line Problem  -  The problem which gave rise to the differentiation side of calculus

Derivative at a Point  -  Basic definition of what the derivative is in graphical terms including a Flash Video Example

                   Derivative as a Function  -  How to find the slope of a function at any point on the curve

                         Example  -  Uses the definition of the derivative to find the slope of a function

                   Derivative Notation  -  Various ways in which we can express the notion of a derivative

                   Graphing the Derivative  -  How to graph the derivative based on key points

             Differentiability and Continuity  -  Any function that is differentiable is by definition also continuous

             Tangent Line Examples                    

                   Example 1  -  How to find the tangent line to a graph using a process of estimation

                   Example 2  -  Uses the definition of the derivative at a point to find the tangent line to a curve

                   Example 3  - Finding the slope of a tangent line and a secant line connecting two points on a curve


Differentiation  - One of the two elemental concepts in calculus, concerning the slope of a function at any point

      Definitions (jump)  -  Basic concepts, mostly covered under the Tangent Line Problem (see above)

      Basic Principles  -  Elementary rules and applications of differentiation

             Applications  -  The two most basic applications are Related Rates and Rates of Change

                   Examples (jump)  -  3 common examples of differentiation as used in physics and geometry

                         Circle  -  The rate of change of the area of an expanding circle

                         Optics  -  The rate of change of the focal length of a double-convex lens

                         Pressure-Volume  -  The rate of change of volume with respect to time as pressure increases

                   Rates of Change  -  One of the fundamental definitions of the derivative

                         Acceleration  -  The rate of change of velocity with respect to time

                         Velocity  -  The rate of change of position with respect to time

                                Average Velocity  -  The sort of velocity we usually deal with

                                Instantaneous Velocity  -  The calculus brand of velocity

Related Rates  -  Changing the rate of one variable with respect to another rate of change (Flash Video Example)

                         Chain Rule (jump)  -  A link to the Chain Rule, which is instrumental in solving this type of problem

Geometry Review (jump)  -  Another key element to solving Related Rates problems is understanding geometry

Method – Solving Related Rates (jump)  -  A brief summary of how to solve this type of problem

                         Example 1  -  Find the change in height given a particular change in volume

                         Example 2  -  Ideal Gas Law example finding change in volume given a change in pressure

                         Example 3  -  Generic related rates example

             Higher Order Derivatives  -  How to find the second, third and so on derivatives of a given function

                   More on Higher Order Derivatives  -  Further explanation on Higher Order Derivatives

             Implicit Differentiation  -  A method of differentiating a function indirectly

                   Method of Implicit Diff (jump)  -  A summary of the technique of Implicit Differentiation

                   Example 1  -  Polynomial function example    

                   Example 2  -  Trigonometric function example

             Linear Approximations  -  Definition of the linearization of a function

                   Examples (jump)                                            

Example 1  -  Generic polynomial function example

Example 2       -  Example involving the accuracy of the approximation

                   Differentials  -  Definition of the differential and its graphical representation

                         General Formulas (jump)  -  Basic formulas involving differentials similar to those for derivatives

                         Example 1  -  Generic examples on how to find the differential of a given function

                         Example 2  -  Slightly more complicated example using differentials

                         Relative Size  -  Shows how big y is compared to dy as x  0

             Rules  -  The various elementary rules governing the derivative operator

                   Rules Summary (jump)  -  A list of all the rules for easy reference

                         Examples  -  Several examples using the rules of differentiation

                                Ex 1 – 4   -  Examples 1 through 4

                                      Example 1

                                      Example 2

                                      Example 3

                                      Example 4

                                Ex 5 – 7  - Examples 5 through 7

                                      Example 5

                                      Example 6

                                      Example 7

                   Chain Rule  -  The cornerstone of differentiation and without which calculus would be impossible

                         Related Rates (jump)  - Related Rates uses the Chain Rule extensively

                         Example 1  -  Simple example of the Chain Rule

Example 2  -  Much more complicated example involving the graphical shape known as the Witch of Maria Agnesi

                   Constant Multiple Rule  -  The constant can be factored out before differentiating the function

                   Power Rules  -  How to differentiate a function raised to a power

                         General Power Rule  -  Power Rule involving use of the Chain Rule

                         Power Rule  -  Simplest form of the General Power Rule, where u(x) = x

                                Example 4  -  Several examples showing how to use various rules of differentiation

                   Product Rule  -  How to differentiate two functions multiplied together

                         Product Rule Example  -  Several examples of how to apply the Product Rule

             Quotient Rule  -        How to differentiate a rational function

                   Example 2  -  Same as Example 2 under Chain Rule

             Sum/Difference Rules  -  How to differentiate the sum of two functions

                         Example 4  -  Same as Example 4 under Power Rule

                   Trig Function Rules  -  How to differentiate any of the six trig functions

                         Sin/Cos  - Differentiation of sine and cosine

                         Tan/Cot/Sec/Csc  -  Differentiation of tangent, cotangent, secant, and cosecant

      Advanced Principles  -  More advanced ideas about differentiation such as Extrema and Rolle's Theorem

             Business and Econ  -  Common formulas encountered in business and economic applications

             Concavity  -  Description of functions based on how much it curves

                   Example (jump)  -  A comprehensive example looking at the complete description of a function

Points of Inflection  -  The points on a curve at which the concavity changes from positive to negative and vice versa

                   Second Derivative Test  -  A test for finding points of relative extrema

                   Test for Concavity  -  How to determine if a function is concave up/down

             Limits at Infinity  -  What happens to the limit of a function as x  

                   Limits (jump)  -  A link to the basic concepts of limits

                   Formal Definitions  -  The Delta-Epsilon Definition as applied to limits at infinity

                   Horizontal Asymptotes  -  The graphical interpretation of limits at infinity (as opposed to infinite limits)

                   Rational Limit at Infinity                              

             Min/Max Values  -  The relative extrema of a functions, with Flash Video Example

                   Example 1 (jump)  -  How to use a graph to find relative extrema

                   Example 2 (jump)

             Extrema  -  The high and low points on the graph of a function

                         Examples (jump)                                            

                                Example 1  -  Finding the absolute extrema of a function

                                Example 2  -  Proving that a given function has no extrema

                         Guidelines (jump)  - A brief summary of how to find the extrema of a function

Critical Numbers  -  The values used to determine extrema, since extrema can only occur at critical numbers

                         Extreme Value Theorem  -  On any given closed interval, there is at least one maximum and minimum

                         First Derivative Test  -  How to determine if a critical number is a maxima or minima

                                Increasing/Decreasing Functions  -  Basic definition

                                      Guidelines (jump)  -  Summary of how to find intervals of increase and decrease

                                      Test for Finding Inc/Dec Functions  -  Just what it sounds like

                         Relative Extrema  -  Basic definition

             Newton's Method  -  How to approximate the roots of a function

                   Example 1  -  Find the seventh root of 3 using Newton's Method

                   Example 2  -  Find the indicated root of an equation to the desired accuracy

             Optimization Problems  -  Applications of extrema

                   Example 1  -  Fencing in a field using a fixed length of fence and maximizing the area enclosed

                   Example 2  -  Find the specified volume of a box using the minimum amount of material

             Rolle's Theorem  -  An existence theorem for critical points

                   Mean Value Theorem  -  Application of Rolle's Theorem

                         Cauchy MVT (jump)  -  Broader explanation of the Mean Value Theorem

                         Example  -  Application of the Mean Value Theorem involving the speed of a car


Integration  -  The other fundamental concept covered in calculus, concerning the area underneath a curve                                

      Basic Integration Formulas (jump)  -  A summary of the most commonly used integration formulas

      Guidelines for Integration (jump)  -  A set of steps for approaching problems involving integrals

      Procedures (jump)  -  How to transform certain integrands to fit a particular integration formula

      Basics  -  The basic concept of the integral and its associated properties

             Antiderivatives -  One interpretation of the indefinite integral

                   Basic Rules (jump)  -  A list of simple integration formulas


                         Example 1  -  Finding the antiderivative of a polynomial function

                         Example 2  -  Finding the linear density of a rod

                         Example 3  -  Finding the height of a cliff by dropping a stone off of it

                   Integral Notation  -  The most common means of representing integrals in mathematics

             Area  -  The problem for which the definite integral was developed to solve

                   Area of Region in Plane  -  How to find the area of a planar region

                   Limit of Lower/Upper Sums  -  A prelude to the infamous Riemann Sum

                         Example 1  -  Using lower/upper sums to approximate the area under curve                       

                   Sigma Notation  -  Definition of what sigma notation actually means

Riemann Sums (jump)  -  A link to the Riemann Sum, which sets the stage for the Definite Integral

                         Summation Formulas  -  Common summation formulas

                         Summation Properties  -  Properties of summation along with proofs of those properties

             Definite Integral  -  The area under a curve

                   Examples (jump)                                                  

                         Example 1  -  Sketch a defined region, then find its area

                         Example 2  -  Evaluate a Riemann Sum

                         Example 3  -  Evaluate a Definite Integral

                   Area of a Region  -  What the Definite Integral is used for


                         Additive Interval Property  -  Separating one integral into two integrals

                         Basic Properties  -  Fundamental properties associated with the Definite Integral

                         Midpoint Rule of Integrals  -  A rule used for approximation of an area

                         Preservation of Inequality  -  Definite Integrals obey same laws of inequality as any real number

                   Riemann Sums  -  Formal explanation of Definite Integral

                         Sigma Notation (jump)  -  Link back to sigma notation for more clarification of Riemann Sums

             Fundamental Theorem  -  The link between differentiation and integration

                   Examples (jump)                                       

                         Example 1  -  Evaluate several definite integrals

                         Example 2  -  Find the area of a region

                         Example 3  -  Find the area of a region

                   2nd Fundamental Theorem  -  The other link between differentiation and integration

                         Example 1  -  Integrate a function using the 2nd Fundamental Theorem

                         Example 2  -  Find the derivative of a function with the 2nd Fundamental Theorem

                         Example 3  -  Find the derivative of a function

                         Example 4  -  Find the derivative of a function

                   Average Value on an Interval  -  How to find the average value of a function over a particular interval

                         Example 1  -  Sketch the graph of a function and find its average value

                   MVT for Integrals  -  The Mean Value Theorem as it applies to integrals

                         Example 1  -  Find the mean value of an integral

                   Integration by Substitution  -  The most fundamental and widely used technique of integration

                         Guidelines (jump)  -  A summary of the technique

                         Change of Variables  -  Also known as u-substitution

                                CV for Definite Integrals  -  u-substitution as it applies to the Definite Integral

                                      Example 1  -  Evaluate a definite integral using u-substitution

                                      Example 2  -  Evaluate a definite integral using u-substitution

                         Even and Odd Functions  -  How to integrate even and odd functions

                         General Power Rule  -  The General Power Rule of Differentiation in reverse

                         Integration of Composite Function  -  What u-substitution is really all about

                   Numerical Integration  -  Approximation methods of integration

                         Error  -  The difference between the approximation and the actual value

                         Integrals of Quadratics  -  How to quickly integrate a 2nd degree polynomial

                         Simpson's Rule  -  Approximating an integral using 2nd degree polynomials

                         Trapezoidal Rule  -  Approximating an integral by adding up trapezoids instead of rectangles                 

      Techniques  -  Various techniques for evaluating integrals

             Improper Integrals  -  Integrals involving infinite limits of integration

                   Infinite Discontinuity  -  Place where the integrand equals  for some value c in the interval [a, b]

                   Special Type  -  A special type of improper integral based on geometric series

Indeterminate Forms  -  Numbers involving 0 and  that make it impossible to determine what the number actually is

                   L'Hôpital's Rule  -  Method to transform Indeterminate Forms into determinate forms

             Integration by Parts  -  How to integrate two functions multiplied together

                   Guidelines (jump)  -  A quick summary of IBP

                   Common Integrals  -  A short list of integrals that can be solved by IBP relatively easily

             Partial Fractions  -  A technique for integrating rational functions

                   Guidelines (jump)  -  A quick summary of how to apply Partial Fractions

Trig Substitution  -  Integration technique involving transforming integrands into products of powers of trig functions

                   Special Formulas  -  Commonly occurring integration formulas

             Trigonometric Integrals  -  Integration technique involving products of powers of trig functions

                   Wallis's Formulas (jump)  -  Special formula for evaluating cos(x) raised to some power

                   Integrals w/ Sec/Tan  -  Integration of powers of secant and tangent functions

                   Integrals w/ Sin/Cos  -  Integration of powers of sine and cosine functions

      Applications  -  Real world type applications of the Definite Integral

             Arc Length  -  How arc length of a function is defined

                   Examples (jump)

                         Example 1  -  Finding the length of an arc over a given interval

                   Area of Surface of Revolution  -  How to find the surface area of a function revolved about an axis

                   Surface of Revolution  -  What happens when a function is revolved about an axis

             Area Between 2 Curves  -  How to find the area enclosed by two given functions

                   Example 1  -  Find the area of a shaded region

                   Example 2  -  Find the area of a shaded region shaped like a tie-fighter

             Fluid Pressure and Force  -  The force per unit area exerted by a fluid such as water or air                              

                   Fluid Force  -  The force exerted on a surface by a fluid such as water or air

             Moments and Centers of Mass

                   2-D System Moments  -  Expanding moments of a system from a line into a plane

Linear System Moments  -  Point around which various masses cancel each other out to maintain a certain equlibrium

             Planar Lamina  -  A continuous flat sheet of uniform density

             Theorem of Pappus -  How to find the volume of an object based on its centroid revolved around an axis

      Volume of Revolution  -  What happens when a curve is revolved around an axis

             Shell Method  -  One of two methods used to find the volume of a solid of revolution

                   Example 1  -  Find the volume of a solid of revolution

                   Example 2  -  Find the volume of a solid of revolution

Example 3  -  Find the volume of a solid of revolution, except use both the Shell Method and the Washer Method

                         Solution:  Shell Method  -  One of 2 ways to solve Example 3

                         Solution:  Washer Method  -  One of 2 ways to solve Example 3

             Washer Method  -  The other of two methods used to find the volume of a solid of revolution

Disc Method  -  A special case of the Washer Method, where the inner radius of the washer equals zero

                         Example 1  -  Find the volume of a solid of revolution

                         Example 2  -  Find the volume of a solid of revolution

Solids w/Known Cross Sections  -  Quick summary of formulas for solids whose cross-sections are known

                   Washer Method  -  The full explanation of the Washer Method

                         Example 1  -  Find the volume of a solid of revolution

                         Example 2  -  Find the volume of a solid of revolution

Example 3  -  Find the volume of a solid of revolution, but use both the Shell Method and the Washer Method

                                Solution:  Shell Method  -  One of 2 ways to solve Example 3

                                Solution:  Washer Method  -  One of 2 ways to solve Example 3

Work  -  One of the most common applications of calculus, involving a force moving an object over some distance

             Work Done by a Variable Force  -  The work done by a force that is changing at some given rate

                   Example (jump)                                             

                         Example 1  -  Find the work done by a cable lifting a car up a mine shaft

                         Example 2  -  Find the work required to empty a tank full of water

                         Example 3  -  Find the work required to empty a tank full of gasoline

                   Coulomb's Law  -  Work done by moving charged particles around

                   Hooke's Law -  Work performed by springs

                         Example 1  -  Find the work required to stretch a spring

Law of Universal Gravitation  -  Work performed by any two masses (such as the Earth and the Moon)

                         Example 1  -  Find the work required to put a satellite in geostationary orbit   


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