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Summary Mathematics 144 Summaries

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  • Course
  • Mathematics 144 (MAT144)
  • Institution
  • Stellenbosch University (SUN)
  • Book
  • Calculus

A neatly digitally summarised document covering all the work done in semester 2: weeks 1 -12, including linear algebra and calculus. (Ends with 10.5 Conic Sections) - all work needed for the A2 and A1 exams.

Last document update: 1 year ago

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  • No
  • All the content covered in semester 2, from weeks 1 to 12.
  • September 12, 2022
  • October 23, 2022
  • 80
  • 2022/2023
  • Summary

Subjects

  • maths 144
  • mathematics 144
  • mathematics 144 summaries
  • maths 144 summaries
  • calculus notes
  • calculus
  • mathematics 144 notes
  • semester 2
  • first year maths
  • maths

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  • MAT1512 - Exam Questions PACK (2010-2022)
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  • Stellenbosch University (SUN)
  • Mathematics 144 (MAT144)
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Week 1 : Revision



4.5 . Substitution Rule


/ fcgcx ) ) gllxldx =
/flu) du
if 4=41×1 is a differentiable function




set u to
your inner function
•




•
when e is involved set u = the power of e
,


•
differentiate U


◦

manipulate so that function cancels out

" "
•
remember + C




6. 1 . Inverse functions


only one lone output for each input )
•
one -
to -




•
horizontal line -
test




f- ( x ) reflection
'
f- ( y ) = x
y
= in ↳= x




if is with domain A and B then f-
'
has domain B and
f l l
range range A
• -




,




cancellation
equations
"
f- ( fix ) ) = x V ✗ C- A ( Domain of inner function )


f- ( f-
'
( x)) = X U KE B




Find the inverse :


① Let
y
= f- ( x )


② Find domain and range of fcx )

③ Solve for have find the
equation x i. t.co .

y ( sometimes
you to
square )
④ Swap x and
y to find
'
f- ( x )

, Derivative of an inverse function

* If f is a 1-1 continuous function, then f-
'
is also continuous


slope of inverse
f at a =
IF
f is an odd function :


9
'

(
f 1)
-


I
(a) = = I

f- (b)
'
f ' ( fila ) ) fcx ) - DX = 0


I



f-
'

f- (b) = a (a) = b




6. G. Inverse trig functions

•

trig functions are not 1- I


•
we must restrict their domains to make them 1- I




Arcsine


sin
_ '

✗ =
y siny=x and
-



≤
y≤ ¥
sin
_ '
Csinx ) = ≥ for
-

É ≤ ✗ ≤ ¥
'
for ≤ 1
sin ( sin
_

-1 ≤ ✗
x ) = x


Input domain :
-
I ≤ ✗ ≤ i


sink =
y




±z
I

'



off
I
( sin
-



x ) =


,
1- ✗ 2 -




-
I




Arccos

'
IT for
-


◦ ≤ x ≤ cos -1 ≤
cosx =
y ,
y = x
y ≤ 1


'
( COSI ) for IT
-



COS = x 0 ≤ ✗ ≤



COS ( COS
'
) for
_


x = x
-
I ≤ × ≤ I



( cos
- '
x ) = -
1 -
I < x e I
;

I 2
-
✗

,Arctan


tan
- '
✗ =
y any
+ =x and
-
¥ <
y < E
( tan
_
'
X) =
I

, 1-1×2




method
triangle
I
' '
Prove sin cos =
_ -



e. ✗ x
g. + 2



¥ Iz
-


' ≤ a ≤
Let
_


a =
sin x
;

✗ Sina
b
=




,
×
b=
'
Let b
-


cos X O ≤ ≤ it


✗ =
costs
a


1- ✗ 2




at b + ¥ =
IT



i. at b =
¥
"
b = b
"
/ nb




Integration
≥
a

:-/
a
"
DX =
Inca )




( É)
/
I

xz+az
=
ta - arctan
/{ du
= 81h ( IU ) )


/¥ dx = In (1×1)

, week 2




6.7 .
Hyperbolic functions

I

" "
sinhx = e -
e- cosechx =
sinhx
2



I

' "
coshx = e' + e- sechx =
cosh >c
2



COSHX
tanhx = sinhx [ ◦ thx =
sinhx
coshx




Hyperbolic identities

cosh >
sin C -
x > = -
sinhx cos he -
x) = coshx sinhcxty ) = sinh >
ccoshy
+
csinhy
coshzx sinhzx cosh >
cushy
= I 1- tanhzx = sech 2x coshlxty ) = +
sinhxsinhy
-




sinhx
"
coshx + =
e Sinha>c) = Zsinhsccosh >c
of
Properties infinity
as ± K = as

Derivatives of Hyperbolic functions
+ A = A


ddxlcosechx)
d
d-✗ ( sinh >c) =
coshx =
-

cosechxcothx
d d
d-
× ( cosh >c) = sin hide fsechx ) =
-

sechxtanhx A. ( IK ) = ± as if k≠O

d
dI ( tanh >c) SECHZX A A
( coth >c) cosechzx
=
= -

= -




= 0 if k≠o
6.8 .
Indeterminate forms and L' Hopital 's Rule

¥ = ± as if k≠O

if k≠o
8- = indeterminate form
ago =
indeterminate form
= as




% =
indeterminate form I = indeterminate form } y=1n . . .
¥ = 0



9- = A



as
form
=

indeterminate

}
as as :
write
quotient
-


as a


A. 0 = indeterminate form

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