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MAT-14903
Tutorial 1
Inverse functions
Exponential functions e , 2 , x x a
bx
Logarithms
Logarithmic scales
Inverse function
Inverse function of (
y
¿
f
x )
is (
x
¿
g
x )
for which ( f
g
( x ) )
¿
and f
( g (
¿
y ) )
x y
Examples:
(
f
x
¿
4
x
)
:
+ ¿ 9
x g
y
¿
4
x
so ¿
y −9
and ¿
(
y )
+ ¿ 9 y −9
4 4
(
f
x )
¿
2
with : x
¿
x 0
y
¿
x2
and ¿ , so
√ y (
g
y )
¿
x √ y
Check for (
f
x )
¿
x
2
and (
g
y )
¿
that f
( g (
¿
y ) ) :
√ y y
f
( g ( y ) )
¿
√ y
2
¿
y
Exponential functions
Exponential functions: functions for which the variable (in this case 𝑥) is in
f
(
x )
¿
a
x
the exponent
Examples: 2x, 3x and ex (𝑥 in exponent)
Parameter 𝑎 is called the base of the exponential function
Do not confuse these with power functions:
Examples: 𝑥4 and 𝑐 𝑥d (𝑥 in base)
Calculation rules for power functions and exponential functions are on the formula
sheet
The graph of an exponential function
f
( t )
¿
c
at
Has (
f
0
¿
c
a0
¿
)
c
Increases for a > 1 (d 0)
¿
(examples for the right)
Decreases for 0 < a < 1 (d 0)
¿
,Logarithms
Logarithmic functions are inverse functions of exponential functions
If ¿
y
then log 2 ( y )
¿
2x x
If y
¿ then log e ( y ) ¿ ( ln ( y ) )
¿
ex x
You can find rules for logarithms on the formula sheet, e.g.:
log ( a b )
¿
lo g a + ¿ ¿ log b
a
log
b
¿ log a
−log b
b
log a =¿ ¿ b
log a
Logarithmic scales
For points 𝑎, 𝑏, 𝑐 and 𝑑 on a logarithmic scale the distance between 𝑎 and 𝑏 is equal
to the distance between 𝑐 and 𝑑 if
a
b
¿
c
d
a
This follows from log
b
¿ log
a
−¿ log b
, MAT-14903
Tutorial 2
Trigonometric functions: sine, cosine and tangent
Definition of the derivative
Rules for finding derivatives
Local extrema: minima and maxima
Repetition: definition sin(x), cos(x) and tan(x)
At the right the unit circle is shown
The point at the intersection of radius with
angle α and the circle is (cos(α), sin(α))
The point at the intersection of
radius with angle α and line is x
¿
1
(1, tan(α))
This shows that
tan ( α )
¿
sin ( α )
cos ( α )
( si n ( α ) )2
2
+ ¿ ( c os ( α ) )
¿
1
Trigonometric functions: properties
Periodicity:
A function is periodic with period T > 0 if for all values of t holds that
f
( t + T )
¿
f
( t )
For the function f
( t
¿
a
)
:
si n ( b t )
The amplitude is a , because the function values vary between −¿a and
a
T
The period is 2¿π
b
The frequency (defined as 1 with period T is b
T 2π
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