

Calculus -

Everything you need to know about calculus is on this page.

Remember this: The whole purpose of calculus is to make very difficult calculations easier.  Yes, sometimes down right easy or at least somewhat easier. Calculus is an amazing tool.

Learn calculus the easy way -
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Below are sample video clips from the above Ultimate Calculus DVD's


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You're all set to master "Calculus"'

Free TI-89 calculus application program.

Scared of calculus? Scared of calculus symbols?  No need to be as they are not meant to scare you.  They are really very simple once you know how to think about them and know what they represent.  For example, often you will see the symbol d or perhaps dx in a formula.  Well, d simply means a small amount of something. So, dx simply means a small amount of whatever x represents.  Don't try to multiply the two (d and x), they are not meant for that, just think of dx as a small amount of x, period.   The symbol dx is called a differential.  Also, you might have seen this symbol, called an integral.   Now that is scary, right?  No, not any more because I know it is simply a tall skinny S.  Now how can a tall skinny S scare anyone?  If you think of as meaning "the sum of" (the word sum starts with an S) well, that isn't scary either.  I know that 4 is the sum of 2 + 2 already.  Let's say you wanted to add up all the little bits of x and determine the sum of all the dx's you haveNow putting these two symbols together,dx simply represents "the sum of"all the "little bits of x" that you have.  This process is often called integration.  Integral calculus involves adding up little bits of things.  A better definition might be, "the part of calculus that deals with integration and its application in the solution of differential equations and in determining areas or volumes etc."  For more information and explanation of the definitions of integral and differential calculus see this page - HERE - and more HERE.

The whole purpose of calculus is to make very difficult calculations easier.  Yes, sometimes down right easy or usually at least somewhat easier.  Most people think calculus is designed to make simple calculations difficult to impossible.  But that is only because they really don't speak or understand calculus.  It is sort of a foreign language.  Learn to understand the language like we did above and calculus gets a lot easier.  One example is calculating a transformer rate of change in output voltage at any one given instant.  A much easier problem to solve if you use calculus.  Who dreamed this calculus stuff up any way?  If you want another clear explanation of calculus read this - HERE.

A function is something whereby you can put in some variable and get a different, dependant variable out. So, if f(x)=2x-3, you can put in some value, say 6, and get f(6)=2(6)-3=9.

Differentiation of a function is the generation of another function for which the "y-value" (value of the dependant variable at a given "x-value," or
independent variable) of the second is equal to the gradient, or slope, of the first.

For example, take the function y=f(x)=x^2. For any given x, there is a y that is equal to x^2. The derivative of this function happens to be f1(x)=2x, meaning that for a given point on the original curve, its slope can be represented by 2x. So, at x=4, f(x)=4^2=16, and its slope at that point, f1(x)=2(4)=8, or 8 units up for every 1 unit over.

The dy/dx means instantaneous change in y divided by instantaneous change in x. An explanation: Slope is measured by change in y divided by change in x. So between two points on a curve, the y-value of the second minus the y -value of the first, all divided by the x-value of the second divided by the x-value of the first, will give you the slope of the straight line between those two points, also called the secant. But we want the slope at a point, which poses some problems. How can there be any change at one point? Well, there can't, really, but what we can do is find the change between two points which are closer to one another than any finite distance. We can determine through algebra that as you make the distance between them smaller and smaller, the change in y over change in x gets closer and closer to some definite ratio, which is the "limit" as the distance between them "approaches zero." Thus, the "dy/dx" is that ratio at an infinitely small distance, thereby effectively being the slope at one point

If it's an upper case sigma then that means the sum of a sequence.

It's got everything to do with integrals. An integral is the sum of the rectangles under the curve, change in x (width) times height, the change in width approaches zero and the number of rectangles approaches infinity. Sums are where integrals come from. It's basically "the sum of all y-values."

For AC electronics, designing circuits is easily done, using complex numbers.

Imagine a voltage source with a angular frequency ω and amplitude A, so as function of time you have V(t) = A*cos(ωt).

Now, replace this with a voltage X(t) = A*exp(ωt). Now, the real voltage can be written as the real part of X(t), being Re(X(t)) = A*cos(ωt).

Using this formalism, you can treat every passive linear component as a complex resistor Z. For lumped devices there are basically three types:

Capacitor with capacity C: Z = 1/jωC
Resistor with resistance R: Z = R
Inductor with inductance L: Z = jωL

Here the number j has the property j² = -1.

Now I'll give an example with three nodes, GND, VIN, VOUT. Between GND and VIN there is a voltage source X(t). Between VIN and VOUT there is a resistor R. Between VOUT and GND is a capacitor C. What is the output voltage as function of input voltage?

This now can be easily solved. We introduce a complex voltage XOUT and XIN.

We have a series connection of two resistors. Using basic circuitry for resistors you find

XOUT = XIN * (ZC / (ZC + ZR)), where ZC is the capacitor's complex resistance and ZR is the resistor's complex resistance.

Now XOUT = XIN *(1/jωC) / (R + (1/jωC)) = XIN / (1 + jωRC)

So, you have XOUT as function of XIN and the angular frequency ω.

The amplification as function of frequency ω can be written as 1/sqrt(1+ω²R²C²). There also is a phase shift, between input and output. That is -arg(1 + jωRC). For small ω (close to DC), the phase shift is close to 0, for high ω, the phase shift is almost 90 degrees.

If you understand complex numbers, then this should be easy to grasp, otherwise it indeed will be very difficult for you to determine transfer functions of capactive and inductive circuits.

The key to understanding these things is

"transfer function"
"complex arithmetic"
"bode plot"
"poles and zeros"
"laplace transform"

Complete calculus course on video - FREE:
Khan Academy

Topics covered in the first two or three semester of college calculus. Everything from limits to derivatives to integrals to vector calculus. Should understand the topics in the pre-calculus play list first (the limit videos are in both play lists)

Pre-Calculus (Non-trigonometry pre-calculus topics)

  1. Introduction to Limits
  2. Limit Examples (part 1)
  3. Limit Examples (part 2)
  4. Limit Examples (part3)
  5. Limit Examples w/ brain malfunction on first prob (part 4)
  6. Squeeze Theorem
  7. Proof: lim (sin x)/x
  8. More Limits
  9. Sequences and Series (part 1)
  10. Sequences and series (part 2)
  11. Permutations
  12. Combinations
  13. Binomial Theorem (part 1)
  14. Binomial Theorem (part 2)
  15. Binomial Theorem (part 3)
  16. Introduction to interest
  17. Interest (part 2)
  18. Introduction to compound interest and e
  19. Compound Interest and e (part 2)
  20. Compound Interest and e (part 3)
  21. Compound Interest and e (part 4)
  22. Exponential Growth
  23. Polar Coordinates 1
  24. Polar Coordinates 2
  25. Polar Coordinates 3
  26. Parametric Equations 1
  27. Parametric Equations 2
  28. Parametric Equations 3
  29. Parametric Equations 4
  30. Introduction to Function Inverses
  31. Function Inverse Example 1
  32. Function Inverses Example 2
  33. Function Inverses Example 3


  1. Introduction to Limits
  2. Limit Examples (part 1)
  3. Limit Examples (part 2)
  4. Limit Examples (part3)
  6. Squeeze Theorem
  7. Proof: lim (sin x)/x
  8. More Limits
  9. Epsilon Delta Limit Definition 1
  10. Epsilon Delta Limit Definition 2
  11. Calculus: Derivatives 1 (new HD version)
  12. Calculus: Derivatives 2 (new HD version)
  13. Calculus: Derivatives 2.5 (new HD version)
  14. Calculus: Derivatives 1
  15. Calculus: Derivatives 2
  16. Calculus: Derivatives 3
  17. Calculus: Derivatives 4: The Chain Rule
  18. Calculus: Derivatives 5
  19. Calculus: Derivatives 6
  20. Derivatives (part 7)
  21. Derivatives (part 8)
  22. Derivatives (part 9)
  23. Proof: d/dx(x^n)
  24. Proof: d/dx(sqrt(x))
  25. Proof: d/dx(ln x) = 1/x
  26. Proof: d/dx(e^x) = e^x
  27. Proofs of Derivatives of Ln(x) and e^x
  28. Extreme Derivative Word Problem (advanced)
  29. Implicit Differentiation
  30. Implicit Differentiation (part 2)
  31. More implicit differentiation
  32. More chain rule and implicit differentiation intuition
  33. Trig Implicit Differentiation Example
  34. Calculus: Derivative of x^(x^x)
  35. Maxima Minima Slope Intuition
  36. Inflection Points and Concavity Intuition
  37. Monotonicity Theorem
  38. Calculus: Maximum and minimum values on an interval
  39. Calculus: Graphing Using Derivatives
  40. Calculus Graphing with Derivatives Example
  41. Graphing with Calculus
  42. Optimization with Calculus 1
  43. Optimization with Calculus 2
  44. Optimization with Calculus 3
  45. Optimization Example 4
  46. Introduction to rate-of-change problems
  47. Equation of a tangent line
  48. Rates-of-change (part 2)
  49. Ladder rate-of-change problem
  50. Mean Value Theorem
  51. The Indefinite Integral or Anti-derivative
  52. Indefinite integrals (part II)
  53. Indefinite Integration (part III)
  54. Indefinite Integration (part IV)
  55. Indefinite Integration (part V)
  56. Integration by Parts (part 6 of Indefinite Integration)
  57. Indefinite Integration (part 7)
  58. Another u-subsitution example
  59. Introduction to definite integrals
  60. Definite integrals (part II)
  61. Definite Integrals (area under a curve) (part III)
  62. Definite Integrals (part 4)
  63. Definite Integrals (part 5)
  64. Definite integral with substitution
  65. Integrals: Trig Substitution 1
  66. Integrals: Trig Substitution 2
  67. Integrals: Trig Substitution 3 (long problem)
  68. Introduction to differential equations
  69. Solid of Revolution (part 1)
  70. Solid of Revolution (part 2)
  71. Solid of Revolution (part 3)
  72. Solid of Revolution (part 4)
  73. Solid of Revolution (part 5)
  74. Solid of Revolution (part 6)
  75. Solid of Revolution (part 7)
  76. Solid of Revolution (part 8)
  77. Sequences and Series (part 1)
  78. Sequences and series (part 2)
  79. Polynomial approximation of functions (part 1)
  80. Polynomial approximation of functions (part 2)
  81. Approximating functions with polynomials (part 3)
  82. Polynomial approximation of functions (part 4)
  83. Polynomial approximations of functions (part 5)
  84. Polynomial approximation of functions (part 6)
  85. Polynomial approximation of functions (part 7)
  86. Taylor Polynomials
  87. Exponential Growth
  88. AP Calculus BC Exams: 2008 1 a
  89. AP Calculus BC Exams: 2008 1 b&c
  90. AP Calculus BC Exams: 2008 1 c&d
  91. AP Calculus BC Exams: 2008 1 d
  92. Calculus BC 2008 2 a
  93. Calculus BC 2008 2 b &c
  94. Calculus BC 2008 2d
  95. Partial Derivatives
  96. Partial Derivatives 2
  97. Gradient 1
  98. Gradient of a scalar field
  99. Divergence 1
  100. Divergence 2
  101. Divergence 3
  102. Curl 1
  103. Curl 2
  104. Curl 3
  105. Double Integral 1
  106. Double Integrals 2
  107. Double Integrals 3
  108. Double Integrals 4
  109. Double Integrals 5
  110. Double Integrals 6
  111. Triple Integrals 1
  112. Triple Integrals 2
  113. Triple Integrals 3
  114. (2^ln x)/x Antiderivative Example
  115. Introduction to the Line Integral
  116. Line Integral Example 1
  117. Line Integral Example 2 (part 1)
  118. Line Integral Example 2 (part 2)
  119. Position Vector Valued Functions
  120. Derivative of a position vector valued function
  121. Differential of a vector valued function
  122. Vector valued function derivative example
  123. Line Integrals and Vector Fields
  124. Using a line integral to find the work done by a vector field example
  125. Parametrization of a Reverse Path
  126. Scalar Field Line Integral Independent of Path Direction
  127. Vector Field Line Integrals Dependent on Path Direction
  128. Path Independence for Line Integrals
  129. Closed Curve Line Integrals of Conservative Vector Fields
  130. Example of Closed Line Integral of Conservative Field
  131. Second Example of Line Integral of Conservative Vector Field
  132. Green's Theorem Proof Part 1
  133. Green's Theorem Proof (part 2)
  134. Green's Theorem Example 1
  135. Green's Theorem Example 2
  136. Introduction to Parametrizing a Surface with Two Parameters
  137. Determining a Position Vector-Valued Function for a Parametrization of Two Parameters
  138. Partial Derivatives of Vector-Valued Functions
  139. Introduction to the Surface Integral
  140. Example of calculating a surface integral part 1
  141. Example of calculating a surface integral part 2
  142. Example of calculating a surface integral part 3
  143. Introduction to L'Hopital's Rule
  144. L'Hopital's Rule Example 1
  145. L'Hopital's Rule Example 2
  146. L'Hopital's Rule Example 3

Differential Equations

  1. Introduction to differential equations
  2. Separable Differential Equations
  3. Separable differential equations 2
  4. Exact Equations Intuition 1 (proofy)
  5. Exact Equations Intuition 2 (proofy)
  6. Exact Equations Example 1
  7. Exact Equations Example 2
  8. Exact Equations Example 3
  9. Integrating factors 1
  10. Integrating factors 2
  11. First order homegenous equations
  12. First order homogenous equations 2
  13. 2nd Order Linear Homogeneous Differential Equations 1
  14. 2nd Order Linear Homogeneous Differential Equations 2
  15. 2nd Order Linear Homogeneous Differential Equations 3
  16. 2nd Order Linear Homogeneous Differential Equations 4
  17. Complex roots of the characteristic equations 1
  18. Complex roots of the characteristic equations 2
  19. Complex roots of the characteristic equations 3
  20. Repeated roots of the characteristic equation
  21. Repeated roots of the characterisitic equations part 2
  22. Undetermined Coefficients 1
  23. Undetermined Coefficients 2
  24. Undetermined Coefficients 3
  25. Undetermined Coefficients 4
  26. Laplace Transform 1
  27. Laplace Transform 2
  28. Laplace Transform 3 (L{sin(at)})
  29. Laplace Transform 4
  30. Laplace Transform 5
  31. Laplace Transform 6
  32. Laplace Transform to solve an equation
  33. Laplace Transform solves an equation 2
  34. More Laplace Transform tools
  35. Using the Laplace Transform to solve a nonhomogenous eq
  36. Laplace Transform of : L{t}
  37. Laplace Transform of t^n: L{t^n}
  38. Laplace Transform of the Unit Step Function
  39. Inverse Laplace Examples
  40. Laplace/Step Function Differential Equation
  41. Dirac Delta Function
  42. Laplace Transform of the Dirac Delta Function
  43. Introduction to the Convolution
  44. The Convolution and the Laplace Transform
  45. Using the Convolution Theorem to Solve an Initial Value Prob

Useful calculus links 

Rules for limits
Derivative of a constant
Common derivatives
Derivatives of power functions of e
Trigonometric derivatives
Rules for derivatives
The antiderivative (Indefinite integral)
Common antiderivatives
Antiderivatives of power functions of e
Rules for antiderivatives
Definite integrals and the fundamental theorem of calculus
Differential equations
Calculus Reference
Calculus Resources - Comprehensive!!
Calculus Tutorial
Calculus Explained with Pictures
Calculus Aids - Reference - Solutions - Formulas
Calculus Without Tears
GREAT CALCULUS Java Applets for Learning

Calculus (Weisstein and Wolfram Research)
Calculus Problems (Alvirne High School)
Calculus Problems (Mathman)
Calculus Resources (Alvirne High School)
Pre-Calculus - High School Level (Homeworkhelp.com)
Calculus Resources (Fife and Husch)
Calculus Resources (ISLMC) 11-99
Calculus Resources - MV (Math Forum)
Calculus Resources -SV (Math Forum)
Graphics for Calculus (Arnold)
Math By Topic or Strand - Grades 10 - 12 (Math Central)
Math Resources by Topic (AOL - Academic Assistance Center)
Pre-Calculus Resources (Fife and Husch)
Pre-Calculus Resources (Math Forum)
Calculus (S.O.S. MATHematics)
Calculus - History and Definitions (Columbia Encyclopedia)
Calculus Basic Formulas (Thinkquest Team 20991)
Preparation for Pre-Calculus and Calculus (Thinkquest Team 20991)
Calculus Edu-Links
Calculus from the MathForum
Calculus Reference Sheet
Dave's Table of Integrals
The Integrator
Karl Hahn's Calculus Tutor
Kevin Brown's Calculus and DiffEq Notes
Learning Calculus
Multivariable Calculus Applets
Quandaries and Queries
Tutor 2000
Visual Calculus

MIT Open Courseware



Calculus 1 & 2



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