Mathematics Dictionary
Dr. K. G. Shih
Senior High Mathematics
Study Tips and Home Works
How to do the home work ?
1. Answr the questions first in each home work.
2. Then read the related keywords of the given title in SM.
3. Do the questions again as home work.
Symbol Defintion
......... Example : 2*3 means 2 times 3
Home Work by keywords
.... Alphabetic order
S81 |
Equaliteral triangle inscribed unit circle, PA^2+PB^2+PC^2 = 2*3
S82 |
Square ABCD inscribed unit circle, PA^2+PB^2+PC^2+PD^2 = 2*4
S83 |
Hexagon ABCDEF inscribed unit circle, PA^2+PB^2+PC^2+... = 2*6New
S84 |
Bisector of triangle AD = 2*b*c*cos(A/2)/(b+c)
S85 |
Bisectors of triangle make angles x,y,x with sides. a*sin(2*x)+b*sin....= 0
S86 |
Solve x^5 + 1 = 0 by construction
S87 |
Sketch y = x^2 + 1/x
S88 |
Use 3 points to sketch y = a*x^2 + b*x + c
S89 |
Use GC command Inverse to study the inverse of quadratic functions
S90 |
Study y = x^4 + x^3 - 4*x^2 + x + 1
S91 |
Study y = x^5 + 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12
S92 |
Five points method : Sketch y - 2 = 3*sin(2*x + pi/4)
S93 |
Five points method : Sketch y - 2 = 3*cos(2*x + pi/4)
S94 |
Three points and two lines method : Sketch y = tan(x)
S95 |
Three points and two lines method : Sketch y = sec(x)
S96 |
Solve x^4 - i = 0 by construction
S97 |
Solve x^4 + i = 0 by construction
S98 |
Sketch y = x^3 + 1/x
S99 |
Study y = x^3 - 1/x
S100 |
Use graphs to compare the functions of y = x^3 + 1/x and y = x^3 - 1/x
Home Works
S81. Equaliteral triangle inscribed unit circle, PA^2+PB^2+PC^2 = 2*3
Keyword
Cosine law
Question
P is a point on unit circle
Prove that PA^2 + PB^2 + PC^2 = 2*3
Solution
Study subjects :
Solution : TR 27 04
Go to Begin
S82. Square ABCD inscribed unit circle, PA^2+PB^2+PC^2+PD^2 = 2*3
Keyword
Cosine law
Question
P is a point on unit circle
Prove that PA^2 + PB^2 + PC^2 + PD^2 = 2*4
Solution
Study subjects :
Solution : TR 27 05
Go to Begin
S83. Hexagon ABCDEF inscribed unit circle, PA^2+PB^2+PC^2+PD^2+... = 2*6
Keyword
Cosine law
Question
P is a point on unit circle
Prove that PA^2 + PB^2 + PC^2 + PD^2 + PE^2 + PF^2 = 2*6
Solution
Study subjects :
Solution : TR 27 05
Go to Begin
S84. Bisector of triangle AD = 2*b*c*cos(A/2)/(b+c)
Keyward
Area of triangle = b*c*sin(A)/2
Solution
Study subjects :
Solution : TR 27 01
Go to Begin
S85. Bisectors of triangle make angles x,y,z with sides
Keyward
Let AD = bisector of angle A
In triangle ABD : angle x = 180 - B - A/2 and also x = C + A/2
a = 2*R*sin(A)
Solution
Study subjects :
Solution : TR 27 02
Go to Begin
S86. Solve x^5 + 1 = 0 by construction
Reference
Study subjects :
Solution : AL 18 07
Application
Solve x^4 - x^3 + x^2 - x + 1 = 0 by construction
See Al 18 10
Go to Begin
S87. Sketch y = x^2 + 1/x
Solution
Method 1 : Use signs of y at various range and asymptote
Method 2 : Use y' and y" with asymptote
Study subjects :
Solution : AL 27 07
Discussion
Method 2 can determine extreme points and points of inflexion
Draw y = x^2 with the diagram to understand the parabol asymptote
Go to Begin
S88. Use three points to sketch y = x^2 + x - 12
Solution
Three case : b^2 - 4*a*c GT 0 or EQ 0 or LT 0 are given
Study subjects :
Solution : AL 27 09
Question
What is principal axis of y = a*x^2 + bx + c ?
How to find the vertex ?
Estimate the vertex from your plot to see the accuracy of your sketch
Go to Begin
S89. Use GC demo command Inverse to study the inverse of quadratic functions
Start the GC program
Click Inverse command 4 times
Question
How many intersections of y = a*x^2 + b*x + c with its inverse ?
Go to Begin
S90. Study y = x^4 + x^3 - 4*x^2 + x + 1
Study subjects :
Solution : AL 27 10
Questions
Solve y' = 0
Solve y' > 0
Solve y' < 0
Solve y" > 0
Solve y" < 0
Go to Begin
S91. Study y = x^5 + 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12
Study subjects :
Solution : AL 27 11
Questions
1. Solve x^5 + 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12 = 0
2. Solve x^5 + 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12 < 0
3. Solve x^5 + 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12 > 0
Go to Begin
S92. Five points method : Sketch y - 2 = 3*sin(2*x + pi/4)
Study subjects :
Solution : TR 18 03
Go to Begin
S93. Five points method : Sketch y - 2 = 3*cos(2*x + pi/4)
Study subjects :
Solution : TR 18 04
Go to Begin
S94. Three points and two lines method : Sketch y = tan(x)
Study subjects :
Solution : TR 18 05
Go to Begin
S95 : Three points and two lines method : Sketch y = sec(x)
Study subjects :
Solution : TR 18 05
Go to Begin
S96. Solve x^4 - i = 0 by construction
Questions
Study subjects :
Solution : AL 18 11
Go to Begin
S97. Solve x^4 + i = 0 by construction
Questions
Study subjects :
Solution : AL 27 12
Go to Begin
S98. Sketch y = x^3 + 1/x
Solution
Sketch using asymptotes x = 0 and y = x^3 as well as the signs of y
Text are given in AL 27 12
Study subjects :
Solution : AL 27 12
Go to Begin
S99. Study y = x^3 - 1/x
Solution
Estmate zeros of y, y' and y" from diagram
Calculate the zeros of y, y' and y" using formula
Find the domain for the points of inflextions
Enter AL 27 and see diagram
Study subjects :
Solution : AL 27 13
Go to Begin
S100. Use graphs to compare the functions of y = x^3 + 1/x and y = x^3 - 1/x
Solutions
1. Mathematic analytic solutions are in AL 27 12 and AL 27 13
2. Solution based on y' and y" are given in AL 27 14
3. Solution based on graphs are given in AL 27 14
Enter AL 27 and see diagram
Study subjects :
Solution : AL 27 12, AL 27 13 and AL 27 14
Go to Begin
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