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)=±√(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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■Basic Calculation


  

Recurring decimals


Recurring decimals can be converted into fraction.
Recurring decimals 1.405 405 405 405 circulates below the decimal point.
The operation that converts it into the fraction is as follows.

classpad recurring decimal

It pays attention only to the fractional portion that circulates first in this calculation, it converts into the fraction, and the integer part is added at the end.

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Continued Fraction


Continued fractions provide estimation for an irrational number. Continued fractions have also proved useful in the proof of certain properties of numbers such as e and classpad pi. Here are some examples for √(2), √(3), e and classpad pi.

classpad continued fraction

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Reminder theorem -collect-


We can use the collect function to obtain the reminder from the division of two polynomials. This example shows that the reminder is 31x-13 when the original expression is divided by x^2-3x+1.

classpad Reminder theorem correct

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Completing the square -collect-


Collect can be used to find the vertex point of a quadratic equation by transforming it into the completed square form. This example shows that the vertex point for the equation x^2-x-3 is
(1/2, -13/4).

classpad Completing the square collect

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Partial fraction decomposition -expand-


This is an excellent method to make characteristics of a function easy to see. A partial fraction expansion can be produced by selecting the Partial Fraction option within the expand function. In the following example, we can easily see that there is a vertical asymptote at x = -1 and a horizontal asymptote at y = 1 from the expanded expression.

classpad Partial fraction decomposition expand interactive menu classpad Partial fraction decomposition expand example

Other example
classpad Partial fraction decomposition and graph classpad Partial fraction decomposition drag and drop

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Is it true? sin(x)=±√(1-cos(x)^2)


Transform the equation sin(x)^2+cos(x)^2=1, solve it for sin(x), remove the absolute value and graph them. The graph isn't sin(x)! This example teaches us we need to take care that the trigonometric function takes different sign according to the domain. When the absolute value is removed, it is necessary to note it.

classpad trigonometric function sin and cos classpad trigonometric function domain

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Symmetric function to y = mx+n


The inverse of a function is a symmetric function about y = x. As a next step, we will study the symmetric function about any line, y = mx+n, and find the expression written as a parametric equation.

Suppose C(a, b) is a point on the function f(x), and C' (X, Y) is the point of symmetry about
y = mx+n. We will try to find C' (X, Y) by using the fact that the distance from C and C' to y = mx+n is the same.

classpad symmety figure geometry

The calculations needed to find C' are as follows:

classpad symmetry calculation example

Finally, we have the coordinate of C':
classpad symmetry calculation result

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The following example displays the graph y = √(x) and its symmetric graph about y = x+2.

classpad symmetric curve eactivity parametric equations

classpad symmetric curve parametric equations

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Linear Programming


Minimize f(x,y) = 10x+5y,
subject to y ≥ -4x+10, y ≥ -x+6, y ≥ (10-x)/4,
by finding the minimum value and the coordinate.

To solve this problem, we will draw graphs, find each intersection and then substitute into f(x,y) to find which produces a minimum value.

1. Input the inequalities into the Graph Editor and draw them.

classpad Linear Programming equation type classpad Linear Programming editor classpad Linear Programming graph

2. Find an intersection. Once found, use the cursor pad to jump to another.

classpad Linear Programming intersect classpad Linear Programming find intersection classpad Linear Programming find intersection

3. Calculate the value of the objective function at each intersection.

classpad Linear Programming calculate the objective function

The objective function has a minimum value of 36.65 at (1.33, 4.67).

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Another method that can be used to find the minimum value of an objective function is to solve the system of equations and substitute into the objective function.
Drag and drop each equation to a graph window; very useful to clarify the problem.

classpad Linear Programming calculation classpad Linear Programming graph

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Trigonometric addition theorem


The formulas are proved by the figure below. When we have two points A(cos(a), sin(a)) and B(cos(b), sin(b)) on the unit circle, the distance between A and B is the same as that between (cos(a-b), sin(a-b)) and (cos(0), sin(0)).

classpad Trigonometric addition theorem proof

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Trigonometric addition theorem and Euler's formula


Following calculation shows the trigonometric addition theorem proved by Euler's formula.
Euler's formula: classpad Euler's formula
We will apply Euler's formula to classpad complex number. By expanding and simplifying, we can see the addition theorem within the real and complex part.

classpad Trigonometric addition theorem and Euler's formula proof

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Gaining a sense of Euler's formula


We can imagine Euler's formula by examining the Taylor expansion for sin(x) and cos(x).
Expand sin(x) and cos(x) using the Taylor expansion and compare the result with the Taylor expansion for exp^(i*x). It is easy to imagine exp^(i*x)=cos(x)+sin(x)*i !!

classpad a sense of Euler's formula

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Transformation of sums of sines


The sum of two sines is verified by using the trigonometric addition theorem.

classpad Transformation of sums of sines

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sin(x)/x and the limit as x approaches 0


sin(x)/x is an indeterminant form at x = 0 because it becomes 0/0. However it's a function that becomes 1 when we calculate the limit as x approaches 0. To prove this, we can apply L'Hospital's rule quickly.

classpad sin(x)/x limit Transformation of sums of sinesclasspad sin(x)/x graph

The next step is to prove that the lim(sin(x)/x)=1 as x approaches 0.

classpad sin(x)/x proof limit

Consider a unit circle with the measure of arcEB = h. Let h equal one radian in length. Thus, we have angleEAB = arcEB = h. We also know that EC=sin(h), DB=tan(h) and sin(h)<h<tan(h).
This gives us: sin(h)/h<1 by dividing sin(h)<h by h.
We also have cos(h)<sin(h)/h by dividing h<tan(h) by h and multiplying cos(h).
Therefore, we have cos(h)<sin(h)/h<1 instead of sin(h)<h<tan(h).
The value of cos(h) becomes 1 at h = 0. This helps us better understand that the limit of sin(h)/h becomes 1 as h approaches 0.
This limit is important when we prove differentiation of trigonometric functions.

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Natural exponential base e


The base of the natural logarithm is a transcendental number e and is called Napier's constant. It is defined as classpad Natural exponential base e . The derivative of exp^(x) at x = 0 is 1.
The definition of e is essential for differentiating ln(x) or exp^(x). It seems that the number of Napier was defined to make the logarithmic function and exponential easy to handle.

classpad definition Natural exponential base e and graph classpad Natural exponential base e sum and limit infinity classpad limit calculation

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Angle classpad pi


The number classpad pi is the area of unit circle.
We can also calculate classpad pi using classpad pi sum infinity.

classpad pi integation circumference classpad pi integration infinity classpad pi sum

When we have Taylor expansion of sin-1(x) and substitute x=0.5, we can estimate the value of classpad pi because sin-1(classpad pi/6)=0.5.

classpad taylor expansion estimation

Taylor expansion to x^9 gives the approx value to 5 digits. When we wish to have a value of 10 digits, Taylor expansion to x^27 is required.

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Bisection method


Solve f(x)=x-cos(x) using the bisection method in the interval of 0<x<1.

1) Draw a graph and make sure that the solution stays within 0<x<1.
2) Calculate f(a), f(b) and f((a+b)/2).

classpad example bisection method

Compare the sign of f((a+b)/2) with f(a) and f(b), and replace a or b with (a+b)/2.

classpad example bisection method repeat

After repeating this manipulation, you will find that the distance between a and b becomes smaller and smaller.

classpad example bisection method repeat

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Transfer function (Bode Diagram, Nyquist Path) -Single logarithm graph-


The following example uses a Bode diagram approach to analyze the transfer function and the graph of the Nyquist path using a parametric equation. When the transfer function G(s) is 1/(1+s), the amplitude is classpad tansfer function amplitude and the phase is classpad tansfer function phase . Single logarithm coordinates are used for drawing the graph.

classpad tansfer function eactivity classpad tansfer function graph

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The Nyquist path for the transfer function is graphed with a parametric equation.

classpad The Nyquist path for the transfer function eactivity classpad The Nyquist path for the transfer function graph

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Equation of a circle in Polar coordinates


The general equation of a circle with a center at (r0, α) and radius R is:

classpad The general equation of a circle

clsspad The general equation of a circle figure

R^2 is calculated by Pythagoras' theorem.

classpad The general equation of a circle Pythagoras' theorem.

When the center (r0, α) is (2, 0) and radius R is 2, the equation is classpad polar equation of circle.

classpad calculaton polar equation circle classpad polar equation circle gaph

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Conic sections in polar form


classpad polar form curves

We can find the eccentricity for conic sections such as the Parabola, Ellipse and Hyperbola. ε is defined as the ratio of the distance between P, a point on the curve, and a focus point to the distance between P and a directrix.

classpad conics equation polar classpad conics equation eccentricity

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Asymptote


There are three kinds of asymptotes ( y = a (Horizontal asymptote), x = b (Vertical asymptote), and y = mx+b (Oblique asymptote) ).
For instance, the denominator of classpad rational function with denominator becomes 0 by x = 3 and then becomes a vertical asymptote. If you calculate the limit at x = 3, it becomes infinity from right (+), and it becomes minus infinity from left (-). It is understood that the curve is discontinous. If you expand the expression into partial fractions, you find y = x+1 is an Oblique asymptote.

classpad asymptote limit calculation classpad asymptote graph

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Finding asymptotes for a hyperbola


Find the asymptotes for classpad ellipse equation by calculation.

1. Differentiate the equation with respect to x.
2. Solve the original equation for y and substitute into y'.
3. Find the limit at x = infinity and x = negative infinity.

classpad calculation asymptote ellipse

Suppose the asymptotes are y = ± b/a*x + y0, and rearrange to y0 = y - ± b/a*x.
To find y0, we take the limitation of y - ± b/a*x as x approaches infinity.
First we need y in terms of x. If a>0 and b>0, we have classpad asymptote result .

classpad asymptote limit

From our calculation, y0 is 0. The asymptotes are y = b/a*x and y = -b/a*x.

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