Mathematics Dictionary

Calcuator Programs | Program 01 00 : Probability.
Functions
- 01 01 P(n,r)
- 01 02 C(n,r)
- 01 03 Binomial : B(x) = C(n,r)*(p^r)*(q^(n-r))
- 01 04 Hypergeometric : H(x) = C(n,x)*(N-n,n-x)/C(N,n)
  - Sample space is N,
  - Take samples n, expect samples x
  - Input : N,n,x
- 01 05 Hypergeometric : H(x) = C(n,x)*(N-n,s-x)/C(N,s)
  - Sample space is N, defective samples is n
  - Take samples s, expect defective sample x
  - Input : N,n,s,x
- 01 06 Normal : N(z) = standard table
  - Input : z
  - Example : N(z LE 1) = 0.8413
- 01 06 Normal : N(z=(x-u)/d)
  - Input : u,d,x (u = mean, d = standard distribution)
  - Example : See AL 17 11
Examples in AL 17 08, AL 17 09, AL 17 10, AL 17 11

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Q02. Coordinate geometry

Programs

Calcuator Programs | Program 02 00 : Coordinate geometry.
Questions
- 1. Give 3 points, find a,b,c,A,B,C of triangle
- 2. Give 3 points, find area of triangle

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Q03.

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Q04. Exponent and logarithm

Sketh Programs | Find graphic solutions.
Functions
- 04 15 y = sinh(x)
- 04 16 y = cosh(x)
- 04 17 y = tanh(x)
- 04 18 y = csch(x)
- 04 19 y = sech(x)
- 04 20 y = coth(x)
- 04 21 y = exp(x)
- 04 22 y = exp(-x)
- 04 23 y = ln(x) or log(x) base e
- 04 33 y = exp(x) and y = ln(x)
- 04 34 y = exp(x) and y = exp(-x)
- 04 44 Equation : e^x + e^(2*x) + e^y + e^(2*y) = 12
Example : Find the asymptotes of e^x + e^(2*x) + e^y + e^(2*y) = 12
- Start sketch programs
- Click Menu command
- Click Section 4 of exponent and logarithm in upper box
- Click program 44 in lower box
- No data is required. Find asymptote from diagram.
- Reference : See Exponent in Algebra
Demo examples
- All programs are demo
Study example using 04 33
- 1. Find intersections of y=exp(x) with its inverse
- 2. Find intersections of y=ln(x) with its inverse

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Q05. Functions with absolute operation

Sketh Programs | Find graphic solutions.
Functions
- 05 25 y = Abs(a*x + b)
- 05 26 y = a*Abs(x) + b
- 05 27 y = a*x^2 + b*Abs(x) + c
- 05 28 y = Abs(a*x^2 + b*x + c)
- 05 29 y = Abs(a*x^2 + b*Abs(x) + c)
- 05 46 Solve Abs(a*x^2 + b*Abs(x) + c) = d
Example : How many real roots in Abs(x^2 - 6*Abs(x) + 8) = 0.5
- Start sketch programs
- Click Menu command
- Click Section 5 in upper box
- Click program 46 in lower box
- Give data a, b, c, d : 1, -6, 8, 0.5
- Reference : See Inverse in Algebra
Demo examples
- Click Menu and then click Demo
- Select section 5
- Select program

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Q06. Functions and its invsers

Sketh Programs | Find graphic solutions.
Functions
- 06 32 y = a*x + b
- 06 33 y = a*x^2 + b*x + c
- 06 35 y = exp(x)
- 06 35 y = ln(x)
Example : How many intersections of y = a*x^2 + b*x + c with its inverse
- GC | Graphic Calculator : y = F(x).
- Example 1 : None. See GC 06 01
- Example 2 : One.. See GC 06 02
- Example 3 : Two.. See GC 06 03
- Example 2 : four. See GC 06 04
Example : Find intersections of y = x^2 - 6*x + 8 with its inverse from graph

Sketch program in GC 06 10

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Q07. Intersections of quadratic functions with other functions

Functions
- 07 36 with y = 1/x
- 07 37 with y = p*x + q

Example : How many intersections of y = a*x^2 + b*x + c with y = 1/x

GC | Graphic Calculator : y = F(x).
Example 1 : Two.... See GC 06 07
Example 2 : One.... See GC 06 08
Example 3 : Three.. See GC 06 09

Example : Find intersections of y = x^2 - 6*x + 8 with y = 1/x from graph

Sketch program in GC 06 11

Example : Find intersections of y = x^2 - 6*x + 8 with y = x - 2 from graph

Sketch program in GC 06 12

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Q08. Solve x^7+ 2*x^6- 5*x^5- 13*x^4- 13*x^3- 5*x^2+ 2*x+ 1 = 0

Equation | Mehtods

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Q09.

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Q10. Examples

Study notes of y = a*x^3 + b*x^2 + c*x + d

1. Cubic functions : Highest power is 3.
2. Slope of cubic function is y' = 3*a*x^2 + 2*b*x + c (a parabola).
- Curve is increasing if y' is positive.
- Curve is decreasing if y' is negative.
- Curve has critical point if y' = 0
3. Vertex
- It has none vetex.
- It has two vetices.
4. Concavity
- Concave upward upward if 2nd derivative is positive.
- Concave upward downward if 2nd derivative is negative.
- Point of inflexion if 2nd derivative is zero.
5. Zero values of y
- It has 3 zeros.
- It has 2 zeros (One duplicate real roots)
- It has 1 zero. (Two complex roots)
6. Cubic formula : It is used in MD2002 Lesson 17.
7. Example
- What is the expression of the demo function ?
  - Polynomila form
  - Factor form
- What is the y-intercept ?
- what are the zero values of y ?
- What are the coordinates of the critical points ?
- What is the domian for y on graph when y is less than zero ?
7. Exercises
- 1. Sketch y = x^3
- 2. Sketch y = x^3 -2*x^2 - x + 2

Study notes of y = a*x^4 + b*x^3 + c*x^2 + d*x + e

1. Slope of cubic function is y' = 4*a*x^3 + 3*b*x2 + c*x + d.
- Curve is increasing if y' is positive.
- Curve is decreasing if y' is negative.
- Curve has critical point if y' = 0
3. Vertex
- It has one or three vetices.
4. Concavity
- Concave upward upward if 2nd derivative is positive.
- Concave upward downward if 2nd derivative is negative.
- Point of inflexion if 2nd derivative is zero.
5. Zeros of y
- It has no zero of y. (Four complex roots).
- It has one zero of y. (One duplicate real and two complex).
- It has three zeros of y. (One duplicate real and two different real)
- It has four zeros of y. (Four different real roots)
6. Quartic formula :
- It used to find four complex roots
- It is used in MD2002 Lesson 17.

Study the graph of demo question in Program 04

What is the expression of the demo function ?
- Polynomila form
- Factor form
What is the y-intercept ?
what are the zero values of y ?
What are the coordinates of the critical points ?
What is the domian for y on graph when y is less than zero ?

Exercises

1. Sketch y = x^4 + 4*x^3 + 6*x^2 + 4*x + 1
2. Sketch y = x^4 - 4*x^3 - x^2 + 16*x - 12

Study notes of y = a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x + f

1. Slope of quaint function is y' = 5*a*x^4 + 4*b*x3 + 3c*x^2 + 2*d*x + e.
- Curve is increasing if y' is positive.
- Curve is decreasing if y' is negative.
- Curve has critical point if y' = 0
3. Vertex
- It has none, two or four vetices.
4. Concavity
- Concave upward upward if 2nd derivative is positive.
- Concave upward downward if 2nd derivative is negative.
- Point of inflexion if 2nd derivative is zero.
5. Zeros of y
- It has one zero of y. (One real root and Four complex roots).
- It has two zeros of y. (One real, One duplicate real and two complex).
- It has three zeros of y. (One duplicate real and one different real)
- It has four zeros of y. (Three different real and one duplicate)
- It has five zeros of y. (Five different real)
6. Quaint formula : It may not be availble in mathematic field.
7. Study the graph of demo question in Program 01 05
- What is the y-intercept ?
- what are the zero values of y ?
- What are the coordinates of the critical points ?
- What is the domian for y on graph when y is less than zero ?
8. Exercises
- 1. Change y = x^5 - 8*x^4 + 15*x^3 + 20*x^2 - 76*x + 48 to factor production
- 2. Above function have roots p,q,r,s,t. Find p*q*r*s*t (Production)
- 2. Above function have roots p,q,r,s,t. Find p+q+r+s+t (Sum)

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Q00. How to use sketch program

Sketh Programs | Find graphic solutions.
1. Start sketch program
- Clcik sketch program
- Select run at current location
- Select yes to run
2. Sketch y = x^2 - 6*x + 8 which is in 01 02
- Click Menu command
- Click polynomial function in section 1 in upper box
- Click Program 02 in lowe box
- Give coefficients a, b, c : That is 1, -6, 8
3. How change scale and replot ?
- After we get the graph, we select new xmax and ymax in left box
- Click replot
4. What is demo command ?
- Click Menu then click Demo
- Select a section number
- Slect a program
- It will give a plot using default values.
- For example : What is the function of 01 04 ?
5. What is the meaning of 01 04 ?
- First number is the section number in upper box
- Second number is the function number in lower box

Factor form : y = (x-a)*(x-b)*(x-c) in Program 38

1. Change to polynomial using program 38.
2. Method 2 : Use multiplication of polynomials.
3. Find y-intercept.
4. Quick sketch : Use 3 zeros and y-intercept to plot the curve.

Factor form : y = (x-a)*(x-b)*(x-c)*(x-d) in Program 39

1. Change to polynomial using program 39.
2. Method 2 : Use multiplication of polynomials.
3. Find y-intercept.
4. Quick sketch : Use 4 zeros and y-intercept to plot the curve.

Factor form : y = (x-a)*(x-b)*(x-c)*(x-d)*(x-e) in Program 40

1. Change to polynomial using program 40.
2. Method 2 : Use multiplication of polynomials.
3. Find y-intercept.
4. Quick sketch : Use 5 zeros and y-intercept to plot the curve.

Solve (x-a)*(x-b)*(x-c)*(x-d) = e

Study Program | Graphic Solutions of Polynomial Functions.
Graphic method 1 : Sketch y = (x-a)*(x-b)*(x-c)*(x-d) - e
- Zeros of y are the solution (Program 06)
- What is the question ? What are the solutions
Graphic method 2 : Sketch y = (x-a)*(x-b)*(x-c)*(x-d) and y = e
- Intersections of line and curve the solution (Program ??)
- What is the demo question ? What are the solutions ?
Examples
- Program 09 : y = (x-5)*(x-7)*(x+4)*(x+6) - 504
- Program 10 : y = (x+9)*(x-3)*(x-7)*(x+5) - 385
- Program 12 : y = (28x-7)*(x-3)*(x+3)*(2*x+5) -91
- Program 13 : y = 12*x^4 - 56*x^3 +59*x^2 - 56*x + 12
- Program 14 : y = x^4 + x^3 - 4*x^2 + x + 1

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