Top

## ■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.

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.

Top

### 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 . Here are some examples for √(2), √(3), e and .

Top

### 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.

Top

### 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).

Top

### 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.

Other example

Top

### 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.

Top

### 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.

The calculations needed to find C' are as follows:

Finally, we have the coordinate of C':

Top

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

Top

### 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.

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

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

Top

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.

Top

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)).

Top

### Trigonometric addition theorem and Euler's formula

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

Top

### 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 !!

Top

### Transformation of sums of sines

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

Top

### 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.

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

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.

Top

### 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 . 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.

Top

### Angle

The number is the area of unit circle.
We can also calculate using .

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

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.

Top

### 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).

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

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

Top

### 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 and the phase is . Single logarithm coordinates are used for drawing the graph.

Top

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

Top

### Equation of a circle in Polar coordinates

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

R^2 is calculated by Pythagoras' theorem.

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

Top

### Conic sections in polar form

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.

Top

### 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 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.

Top

### Finding asymptotes for a hyperbola

Find the asymptotes for 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.

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 .

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

Top

Next page