Math Note

    Education does not mean quantity of knowledge,
       but the power to assemble and manage it.

-- Classpad founder --




”Basic Calculation
  Recurring decimals
  Continued Fraction
  Reminder theorem -collect-
  Completing the square -collect-
  Partial fraction decomposition -expand-
  Is it true? sin(x)=}ć(1-cos(x)^2)
  Symmetric function to y = mx+n
  Linear Programming
  Trigonometric addition theorem
  Trigonometric addition theorem and Euler's formula
  Gaining a sense of Euler's formula
  Transformation of sums of sines
  sin(x)/x and the limit as x approaches 0
  Natural exponential base e
  Angle classpad pi
  Bisection method
  Transfer function (Bode Diagram, Nyquist Path) -Single logarithm graph-
  Equation of a circle in Polar coordinates
  Conic sections in polar form
  Asymptote
  Finding asymptotes for a hyperbola

”Statistics
  Normal density function and its graph
  Central limit theorem
  Approximation of the Binomial distribution using the Normal distribution
  Regression analysis using matrix operations (Least Squares Regression Line)

”Geometry
  Circular measure of angle (Radian)
  Area of a circle
  Geometric figures and Matrices (shrink, expand, shift, symmetry, rotation)
      Matrix used to express a geometric figure
      Shift
      Shrink or expand
      Rotation
      Symmetry
  Affine transform
  Rotation about the origin
  Rotation of Axes

”Differentiation
  Differentiation of sin(x), cos(x) and tan(x)
  Differentiation of inverse trigonometric functions: sin-1(x), cos-1(x) and tan-1(x)
  Differentiation of ln(x)
  Differentiation of ex
  Tangent line to y = f(x) and f(x, y) = 0
  Differentiation of y = f(x) or f(x, y) = 0 and its graph
  Newton's Method
  Differentiation presented by parametric equations
  Slope in Polar form

”Integration
  Riemann sum
  Double integrals
  Volume of a solid of revolution
  Length of curve and surface area of revolution for y = f(x)
  Area and length of curve given by parametric equations (x(t), y(t))
  Volume of revolution given by parametric equations (x(t), y(t))
  Area of a surface of revolution given by parametric equations (x(t), y(t))
  Length and area of curve given on the polar coordinates
  Area of a surface of revolution on the polar coordinates

”Differential equation
  Solve a differential equation using Laplace transform

”Fourier Series and FFT
  Fourier Series
  FFT (Fast Fourier Transform)
  FFT Example with EA200 (Enjoy My Piano and Guitar Sound)

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”Differentiation


  

Differentiation of sin(x), cos(x) and tan(x)


sin(x)
It is not difficult to differentiate a function such as x^2 based on the definition. However, we need to be more creative to differentiate a trigonometric function such as sin(x).

classpad definition of differentiation
Simplify the definition of differentiation for sin(x).

classpad proof differentiation of sin(x)

The limit of cos(x)sin(h)/h at h=0 is cos(x) because the limit of sin(x)/x at x=0 is 1.
(See the limit of sin(x)/x earlier in this document.)
Multiply the denominator and numerator of sin(x)arccos(h)/h by cos(h)+1 and transform the expression.

transform the expression

When h is positive and small,

transform the expression

equation < equation because sin(h)<h.
As h approaches 0, cos(h) approaches 1 and equation approaches 0.
Thus, equation becomes 0.

Finally the limit of sin(x) at h=0 is,
lim( definition of differentiation )
    = lim( equation ) + lim( equation )
    = cos(x) + 0
    = cos(x)

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cos(x)
A similar proof can be used to prove the differentiation of cos(x).

proof of a differentiation of cos(x)
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tan(x)
We can use the quotient rule to prove the differentiation of tan(x).

proof of a differentiation of tan(x)

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Differentiation of inverse trigonometric functions: sin-1(x), cos-1(x) and tan-1(x)


the differentiation of inverse sin(x)

proof of the differentiation of inverse sin(x)

the differentiation of inverse cos(x)

proof of the differentiation of inverse cos(x)

the differentiation of inverse tan(x)

proof of the differentiation of inverse tan(x)

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Differentiation of ln(x)


It is important to remember the definition of the natural logarithm base e.
definition of logarithm base

The definition of the differentiation of ln(x) is:
definition of the differentiation of ln(x)

We will transform the expression with the conditions that x>0 and h>0.

transform the expression

Suppose h=t*x and that t approaches 0 as h approaches 0.
transform the expression

Apply logarithm properties.
logarithm properties

Calculate the limit.
final transformation

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Differentiation of ex


Differentiation of e^x

We will transform the expression and let t = eh-1.

transform the expression

Apply logarithm properties.
Apply logarithm properties

Calculate with the definition transform the expression .
transform the expression

We can differentiate ax using the product rule by first transforming ax into exp^(x*ln(a)).
final transformation of the expression

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Tangent line to y = f(x) and f(x, y) = 0


The differentiation gives the slope of curve. For example, the slope of y = x^2 is d/dx(x^2 )= 2x at the point (x, x^2).
Suppose the tangent line is y = mx+b. The tangent line at (1, 1) is y = 2x+b because the slope is 2. Substitute (1, 1) to y = 2x+b, we find b = -1.

find Tangent line

ClassPad has a tanLine function to have tangent easily and a normal function to have a normal line.

classpad function Tangent line

When the given function is implicit, ClassPad has an impDiff function to differentiate the implicit function such as Unit circle x^2+y^2 = 1.
When doing impDiff(x^2+y^2 = 1, x, y), you have y' = -x/y. It's a slope of the circle.
The slope at (1/2, ć(3)/2) is -ć(3)/3, and the tangent line is y = -ć(3)/3x-ć(3)/3.

differentiate the implicit function differentiate the implicit function graph

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Differentiation of y = f(x) or f(x, y) = 0 and its graph


The differentiation of y = x^2 is y' = 2x which is a line.

The differentiation of function and graph

The differentiation of Unit circle x^2+y^2 = 1 is y' = -x/y which are two curves, y = x/ć(1-x^2) and y = -x/ć(1-x^2).

The differentiation of Unit circle and its graph

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Newton's Method


Newton's method is a method used to find the x-intercept of the function y = f(x). The recurrence equation used is Xn+1=Xn-f(Xn)/f'(Xn).

Newton's method eactivity

We find that there are four roots for the graph of f(x) = x^4-3x^2+sin(x).

find the root by graphing
(You can approximate the roots by pressing on the graph.)

An initial value for Newton's method is assumed to be 2 because the solution of the rightmost is almost 1.6.

Newton's method by repeating calculation

This recurrence shows us the solution approaches to 1.618.
The Cobweb diagram gives another way to use Newton's method and the graph.

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Differentiation presented by parametric equations


The differentiation for parametric equations such as (x(t), y(t)) is,
The differentiation for parametric equations
The following example uses {u(t)=7t+2, v(t)=t^3-12t}.

classpad example The differentiation for parametric equations

The second derivative is as follows.
the second derivative for parametric equations

The following example is the first and second derivative for {x(t)=t^2, y(t)=t^4+1}.

classpad example the first and second derivative for parametric equations

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Slope in Polar form


The slope in Polar form r = f(ƒĘ) is as follows.
slope in Polar form

The next example shows how to calculate the slope at ƒĘ=classpad pi/6 for r = 1+cos(ƒĘ).

example of the slope in Polar form

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