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)=}ã(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

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 **e**^{x}

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)

**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).

Simplify the definition of differentiation for sin(x).

The limit of 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 by cos(h)+1 and transform the expression.

When h is positive and small,

< because sin(h)<h.

As h approaches 0, cos(h) approaches 1 and approaches 0.

Thus, becomes 0.

Finally the limit of sin(x) at h=0 is,

lim( )

= lim( ) + lim( )

= cos(x) + 0

= cos(x)

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

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

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

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

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

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

Apply logarithm properties.

Calculate the limit.

We will transform the expression and let t = **e**^{h}-1.

Apply logarithm properties.

Calculate with the definition .

We can differentiate a^{x} using the product rule by first transforming a^{x} into exp^(x*ln(a)).

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.

ClassPad has a tanLine function to have tangent easily and a normal function to have a normal 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, ã(3)/2) is -ã(3)/3, and the tangent line is y = -ã(3)/3x-ã(3)/3.

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

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

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).

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

(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.

This recurrence shows us the solution approaches to 1.618.

The Cobweb diagram gives another way to use Newton's method and the graph.

The differentiation for parametric equations such as (x(t), y(t)) is,

The following example uses {u(t)=7t+2, v(t)=t^3-12t}.

The second derivative is as follows.

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

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

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