+x +y +x +y +x +y slope of tangent = dy/dx. y a x y a x y a x
TRANSCRIPT
+X
+Y
+X
+Y
+X
+Y
1 1, ( )P x f x
1 1, ( )Q x x f x x
1 1, ( )P x f x
1 1, ( )P x f x
1 1, ( )Q x x f x x
slope of tangent = dy/dx
1 1
2 1
( ) ( )slope chord PQ
f x x f x
x x
0x
x-1 0 1 2 3
y-10
0
10
20function y = 2x3 - 5x2 - 8x + 12
x-1 0 1 2 3
dy/d
x
-20
0
20derivative dy/dx = 6x2 - 10x - 8
0dy
dx
0dy
dx
0dy
dx
3 22 5 8 12y x x x
2/ 6 10 8 0dy dx x x
y
a
x
2 2a x
sinx
ya
2 2
cosa x
ya
y
a
x
2 2a x
cosx
ya
2 2
sina x
ya
y
a
x
2 2a x
tanx
ya
2 2
22
2 2
cos
cos
ay
a x
ay
a x
xN+1 = xbx1 = xa
xi = xa + (i-1)x X
y
5
3
4
3/ 4y x
Y
X
u
x
1
21 x
2
tan( )1
sin( )1
xu
xu
x
opposite side a
adjacent side b
hypotenuse c
2 2 2c a b
rad2
1
12
o45 rad4
21
3o
o
30 rad6
60 rad3
+X
+Y
a
(0, 0)
(xC, yC)
o1 revolution 360 2 rad
a
a
a
1 rad
a
a s
angle [degrees]-360 -315 -270 -225 -180 -135 -90 -45 0 45 90 135 180 225 270 315 360
y
-1
-0.5
0
0.5
1
1st
quadrant2nd
quadrant3rd
quadrant4th
quadrant
0 1y 0 1y 1 0y 0 1y
1/ 2 0.7071
1/ 2 0.7071
30o 150o-210o-330o
-150o -30o 210o 330o
3 / 2 0.8660
3 / 2 0.8660
60o 120o-240o-300o
300o240o-60o-120o
45o
-45o-135o
135o-225o
225o 315o
-315o
sin sin( )
ODD FUNCTION
-2 - 2/2-/2angle
[degrees / radians] -3/2 3/2
angle [degrees]-360 -315 -270 -225 -180 -135 -90 -45 0 45 90 135 180 225 270 315 360
y
-1
-0.5
0
0.5
1
1stquadrant
2ndquadrant
3ndquadrant
4thquadrant
angle [degrees / radians]
-2 - 2/2-/2
3 / 2 0.8660
1/ 2 0.7071
1/ 2 0.7071
3 / 2 0.8660
0 1y 0 1y 1 0y 1 0y
-3/2 3/2
-360 -315 -270 -225 -180 -135 -90 -45 0 45 90 135 180 225 270 315 360
y
-20
-15
-10
-5
0
5
10
15
20
angle [degrees / radians] -2 - 2/2-/2-3/2 3/2
A
c
ba
C
B
base b
height h
angle [degrees / radians]
-2 - 2/2-/2-3/2 3/2
sin( ) cos( / 2)y cos( ) sin( / 2)y
-360 -315 -270 -225 -180 -135 -90 -45 0 45 90 135 180 225 270 315 360
y
-1
-0.5
0
0.5
1
O(0, 0)
+Y
+X
, ,P P P PP x y P r
Px
Py
Pr
P
2 2 2
cos sin
tan
P P P P P P
PP P P P
P
x r y r
yr x y
x
O(0, 0)
+Y
+X1x
2y
1y
2x
b
y m x b
intercept 0x y b
2 1
2 1
slope or gradienty y
mx x
O(0, 0)
+Y
+X
b
y m x b
y intercept
0x y b slope or gradient
rise ym
run x
x intercept
0 x
by b
m
x
y
x-15 -10 -5 0 5 10 15
y
-10
-5
0
5
10
15
20
25
30
3 5y x
2 10y x
120
2y x
3,4P
4,18Q
10,25R
O(0, 0)
+Y
+X
y m x y x
x and y are directly proportional to each other (doubling the value of x doubles the value of y)
0y m x b b
x and y are not directly proportional to each other (doubling the value of x does not double the value of y)
+Y
O(0, 0) +X
x and y are inversely proportional to each other only if
constantk
y kx
doubling the value of x then the value of y is halved
+X
+Y
b
a +X
+Y
b
a +X
+Y
a
+X
+Y
ba(x1, y1)
a > b a = ba < b
+X
+Y
b
a
a > b
cos , sina b
time t [a.u.]0 10 20 30 40 50 60 70 80 90 100
disp
lace
men
t y
[a.u
.]
0
2
4
6
8
10
12
0 max
2sin
26 4sin / 4
25
y y y tT
y t
0y
maxy
maxy
T
T
T
T
0 max6 4
25 / 4 rad
y y
T
( 0) 6 4sin / 4 6 4 0.7071 8.8284y t
y = 8.8284