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All lectures of the course Mathematics 2

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This document contains all lectures and their formulas and information of the course Mathematics 2. It has formulas, pictures and all tutorial lectures of said course.

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  • April 26, 2022
  • 25
  • 2021/2022
  • Class notes
  • Casper zelissen
  • All classes

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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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