Semua fundamental dalam bermatematika dan juga dapat dipraktikan dengan soal soal yang tersedia. Matematika bukanlah hal yang sulit dalam buku ini dan baik digunakan pada semua kalangan.

1. Yayan Sugianto (dirangkum dari berbagai sumber)
2016

2. Pembahasan Materi
Vektor
Transformasi
Exp & Log
Irisan kerucut

3. Vektor
 Vektor biasanya digambarkan sebagai garis yang di salah satu
ujungnya ada mata panah (yang mempunyai panjang dan
arah).
Panjang vektor atau norm
v=(v1,v2,v3), atau
v= v1 i + v2 j + v3 k
i, j, k = vektor satuan

4. Penjumlahan vektor
v = (v1, v2) dan w = (w1, w2)
v + w = (v1+w1, v2+w2)

5. Perkalian Titik
(dot product)
(= scalar)

6. u = w1 + w2
Proyeksi ortogonal
= u − w1

8. Point-line and point-plane
Distance formulas

10. Cross Product

11. 
sin
, AB
B
A
B
B
B
A
A
A
k
j
i
B
A
z
y
x
z
y
x =

=





Cross Product

12. Cross Products and Dot Products

13. Geometry of the Cross Product

14. z
y
x
z
y
x
z
y
x
C
C
C
B
B
B
A
A
A
C
B
A =

 )
(



“Volume of the parallelepipe”

cos
A
height =
Triple Scalar Product

15. Transformasi

20. Rotation Operators

22. Dilations and Contractions

24. Expansion or Compression

25. Shear

26. Geometry of Matrix Operators on R2

27. Exp & Log

28. The Base e
 Exponential functions to the base e, where e is an
irrational number whose value is 2.7182818…, play an
important role in both theoretical and applied
problems.
 It can be shown that
1
lim 1
m
m
e
m
→
 
= +
 
 

29. 5
3
1
Grafik Exp(x) dan Exp(-x)
x
y
– 3 – 1 1 3
f(x) = ex
f(x) = e–x

30. Logarithmic Functions
1
x
y
1
y = ex
y = ln x
y = x

31. Logarithmic Notation
log x = log10 x Common logarithm
ln x = loge x Natural logarithm
( )
log if and only if 0
y
b
y x x b x
= = 

32. Laws of Logarithms
log log log
b b b
mn m n
= +
log log log
b b b
m
m n
n
= −
log log
n
b b
m n m
=
log 1 0
b =
log 1
b b =

33. Examples
 Expand and simplify the expression:
2 2
1
ln x
x x
e
− 2 2 1/2
2 2 1/2
2
2
( 1)
ln
ln ln( 1) ln
1
2ln ln( 1) ln
2
1
2ln ln( 1)
2
x
x
x x
e
x x e
x x x e
x x x
−
=
= + − −
= + − −
= + − −

34. Transformation of functions apply
to log functions just like they apply
to all other functions so let’s try a
couple.
( ) x
x
f 10
log
=
( ) x
x
f 10
log
2 +
=
up 2
( ) ( )
1
log10 +
= x
x
f
left 1
( ) x
x
f 10
log
−
=
Reflect about x axis

35. Remember our natural base “e”?
We can use that base on a log.
7182828
.
2
loge
What exponent do you put on e to
get 2.7182828?
1
7182828
.
2
log =
e
Since the log with this base occurs in nature
frequently, it is called the natural log and is
abbreviated ln.
ln
1
7182828
.
2
ln =
Your calculator knows how to find natural logs. Locate the ln button on
your calculator. Notice that it is the same key that has ex above it. The
calculator lists functions and inverses using the same key but one of them
needing the 2nd (or inv) button.

36. Irisan Kerucut

39. General Equation of
a Conic Section
2 2
0
where A, B, and C are not all zero.
Ax Bxy Cy Dx Ey F
+ + + + + =

40. Which Conic is it?
 Parabola: A = 0 OR C = 0
 Circle: A = C
 Ellipse: A ≠ C, but both have the same sign
 Hyperbola: A and C have Different signs

41. Example: State the type of conic and write it
in the standard form of that conic
 Conic: A and C same sign, but A ≠ C
 ELLIPSE
 Standard Form:
2 2
2 8 12 4 0
x y x y
+ + − − =
2 2
2 2
( ) ( )
1
x h y k
a b
− −
+ =
2 2
( 8 __) 2( 6 __) 4 __ __
x x y y
+ + + − + = + +
2 2
( 8 16) 2( 6 9) 4 16 18
x x y y
+ + + − + = + +
2 2
( 4) 2( 3) 38
x y
+ + − =
2 2
( 4) ( 3)
1
38 19
x y
+ −
+ =

42. Example: State the type of conic and write it in
the standard form of that conic
 Conic: A = 0
 PARABOLA
 Standard Form:
2
6 2 24 10 0
y x y
+ − + =
2
( )
x a y k h
= − +
2
2 6 24 10
x y y
= − + +
2
3( 4 __) 5 ( 3)(__)
x y y − −
= − − + +
2
3( 4 4) 1
5 2
x y y
− + + +
= −
2
3( 2) 17
x y
= − − +
(y – k)2 = 4p(x – h)

43. Example: State the type of conic and write it
in the standard form of that conic
 Conic: A =C
 CIRCLE
 Standard Form:
2 2
2 2 8 12 4 0
x y x y
+ + − − =
2 2 2
( ) ( )
x h y k r
− + − =
2 2
2( 4 __) 2( 6 __) 4 __ __
x x y y
+ + + − + = + +
2 2
2( 4 4) 2( 6 9) 4 8 18
x x y y
+ + + − + = + +
2 2
2( 2) 2( 3) 30
x y
+ + − =
2 2
( 2) ( 3) 15
x y
+ + − =
**Divide all by 2!**

44. Circles
The set of all points that are the same distance
from the center.
Standard Equation:
2
2
2
)
(
)
( r
k
y
h
x =
−
+
−
CENTER: (h, k)
RADIUS: r (square root)
(h , k)
r

45. Example
81
)
8
(
)
2
( 2
2
=
+
+
− y
x
h
k r²
Center:
Radius: r
)
,
(
9
k (
)
, →
h 8
2 −

46. Ellipses
Basically, an ellipse is a squished circle
Standard Equation: (x - h)
a2
2
+
(y - k)
b2
2
=1
c2
= a2
- b2
Center: (h,k)
a: major radius, length from center to edge of circle
b: minor radius, length from center to top/bottom of circle
* You must square root the denominator
(h , k)
a
b

47. Example
1
4
)
5
(
25
)
4
(
2
2
=
−
+
+ y
x
a²
b2
This must
equal 1
Center: (-4 , 5)
a: 5
b: 2

48. Parabolas
)
(
4
)
( 2
k
y
p
h
x −
=
−
Standard Equations:
)
(
4
)
( 2
h
x
p
k
y −
=
−
p>0 Opens UP Opens RIGHT
p<0 Opens DOWN Opens LEFT
vertex
vertex

49. Example
)
5
(
)
2
(
12
1 2
−
=
+
− y
x
What is the vertex? How does it open?
(-2 , 5)
opens
down
Example 5
2
)
2
(
125
5 −
= y
x
What is the vertex? How does it open?
(0 , 2) opens
right

50. Hyperbolas
Looks like: two parabolas, back to back.
Standard Equations:
1
)
(
)
(
2
2
2
2
=
−
−
−
b
k
y
a
h
x
1
)
(
)
(
2
2
2
2
=
−
−
−
b
h
x
a
k
y
Opens UP and DOWN
Opens LEFT and RIGHT
Center: (h , k)
(h , k)
(h , k)

51. Hyperbolas – Transverse Axis

52. Example
1
4
)
5
(
25
)
4
(
2
2
=
−
−
+ y
x
Center: (-4 , 5)
Opens: Left and right