Graphics Reference
In-Depth Information
Theorem A : x a
may be extended in a unique way to a
qn 19
continuous bijection R R , which is
increasing when 1
a and decreasing when
0 a 1.
T
heorem a
a
a
and a
( a
)
,
qn 20 when 0 a and x , y R.
Definition If 0 a and a y , then log
y x .
Theorem For x , y 0, log
x log
y log
xy .
qn 22
Theorem D : x
( a
1)/ x may be extended in a unique
qns 14, 15,
way to a continuous function on R.
16, 23
Theorem D ( x )
dy / y as x 0.
qns 14, 15,
25
T
heorem If 0
a and A ( x )
a
then
qns 24, 25 A ( x ) A ( x )ยท
dy / y .
Definition
dx / x 1, E ( x ) e exp( x ).
Theorems E ( x ) E ( x ), E (
dx / x ) a ,
dx / x log
a .
qns 28, 29, 30
Definition log x log
x
Theorem When a 0, a e ; log a x log a .
qn 31
Theorem
qn 32
a
n
e .
lim
1
Circular or trigonometric functions
The origins of the functions sine, cosine, tangent, cotangent, secant
and cosecant are geometric. A particle, P , moves around the
circumference of a circle with centre O and radius 1. The angle through
which theradius turns is the length of arc which is traced out by P on
the circumference. If the particle moves anti-clockwise from the point A
to thepoint B , and thearc length from A to B is x , then the
perpendicular from B to OA has length 'sine of x '. Thecosineof x is the
sine of the complementary angle. The tangent of x is thelength along
thetangent at B from B to the point where it meets OA produced. The
cotangent of x is the tangent of the complementary angle.
If we only have the established properties of the real numbers, how
do we define these functions? If logic was all that mattered, we could
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