Topic 9: Calculus Option
Limits of Sequences and Functions
lim [𝑓(𝑥) + 𝑔(𝑥)] = lim 𝑓(𝑥) + lim 𝑔(𝑥)
$→&
Algebraic
limit laws:
$→&
lim [𝑓(𝑥) − 𝑔(𝑥)] = lim 𝑓(𝑥) − lim 𝑔(𝑥)
$→&
$→&
$→&
lim [𝑓(𝑥) × 𝑔(𝑥)] = lim 𝑓(𝑥) × lim 𝑔(𝑥)
$→&
$→&
lim 2
$→&
$→&
$→&
lim 𝑓(𝑥)
𝑓(𝑥)
3 = $→&
𝑔(𝑥)
lim 𝑔(𝑥)
$→&
lim [𝑐𝑓(𝑥)] = 𝑐 lim 𝑓(𝑥)
$→&
$→&
L’Hopital’s rule states if the following occurs:
𝒇(𝒙) 𝟎
𝐥𝐢𝐦
=
𝒙→𝒂 𝒈(𝒙)
𝟎
𝑓(𝑥) ±∞
lim
=
$→& 𝑔(𝑥)
±∞
or
Differentiate top and bottom
fraction separately
𝑓(𝑥) 𝐿′𝐻
𝑓′(𝑥)
lim
$→& 𝑔(𝑥) = $→& 𝑔′(𝑥)
lim
The Squeeze Theorem states if:
Use when involving trigonometric functions
𝒈(𝒙) ≤ 𝒇(𝒙) ≤ 𝒉(𝒙)
and
$→&
1
=∞
0>
1
=0
∞
∞
= 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
∞
0
= 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
0
A closed
interval [𝑎, 𝑏]
includes end
points
An open
interval (𝑎, 𝑏)
excludes end
points
A function is continuous at 𝑥 = 𝑎 when:
lim 𝑓(𝑥) = lim 𝑓(𝑥) = limR 𝑓(𝑥)
lim 𝑓(𝑥) = 𝐿
Then:
1
= 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
0
Interval
Notation:
lim 𝑔(𝑥) = lim ℎ(𝑥) = 𝐿
$→&
1
= −∞
07
$→&Q
$→&
$→&
$→&
if lim |𝑎S | = 0, then lim 𝑎S = 0
The Absolute Value Theorem states:
S→T
S→T
𝑓(𝑥) must be continuous at 𝑥V
A function is differentiable at a point 𝒙𝟎 if:
𝑓(𝑥) must not have a “sharp point” at 𝑥V
the tangent to 𝑓(𝑥) at 𝑥V must be vertical
The derivative of a function 𝒇(𝒙) at the point 𝒙𝟎 can be found if the limit exists by:
If not, then the function is not differentiable at that point
𝒇(𝒙𝟎 + 𝒉) − 𝒇(𝒙𝟎 )
𝒉→𝟎
𝒉
𝒇′(𝒙𝟎 ) = 𝐥𝐢𝐦
or
𝑓(𝑥) − 𝑓(𝑥V )
X→V
𝑥 − 𝑥V
𝑓′(𝑥V ) = lim
𝑓(𝑥) is continuous on [𝑎, 𝑏]
Rolle’s Theorem requires three
conditions:
a and b are found from the given
interval
𝑓(𝑥) is differentiable
on (𝑎, 𝑏)
𝑓(𝑥) is continuous on [𝑎, 𝑏]
Mean Value Theorem
requires three conditions:
𝑓(𝑥) is differentiable on ]𝑎, 𝑏[
a and b are found from the given
interval
𝑓(𝑎)= 𝑓(𝑏)
If three conditions are met then there must be a point 𝒄 such
that 𝒇[ (𝒄) = 𝟎
𝑐 ∈]𝑎, 𝑏[
If these conditions are met, then there exists a point 𝑐 such that:
𝑓 [ (𝑐) =
𝑓(𝑏) − 𝑓(𝑎)
𝑏−𝑎
Topic 9: Calculus Option
Improper Integrals
The Fundamental Theorem of Calculus states if 𝒇(𝒙) is continuous
on [𝒂, 𝒃] where:
𝑭(𝒙) =
𝒙
∫𝒂 𝒇(𝒕) 𝒅𝒕
𝒂≤𝒙≤𝒃
then
The p-series states for:
T
]
𝐹′(𝑥) = 𝑓(𝑥)
_
Converges if 𝑝 > 1
1
𝑑𝑥
𝑥^
Diverges if 𝑝 ≤ 1
And for any function where 𝒈(𝒙) such that 𝑭′(𝒙) = 𝒇(𝒙)
g
h
g
g
] 𝑓(𝑥) 𝑑𝑥 = ] 𝑓(𝑥) 𝑑𝑥 + ] 𝑓(𝑥) 𝑑𝑥
𝒃
] 𝒇(𝒙) 𝒅𝒙 = 𝑭(𝒃) − 𝑭(𝒂)
𝒂
&
Where 𝐹(𝑥) is the
antiderivative of 𝑓(𝑥)
&
h
&
] = −]
&
g
T
T
Integrals in the form ∫𝒂 𝒇(𝒙) 𝒅𝒙 are known as improper integrals
Convergent when ∫& 𝑓(𝑥) 𝑑𝑥 = 0 𝑜𝑟 𝑓𝑖𝑛𝑖𝑡𝑒 𝑣𝑎𝑙𝑢𝑒
Improper integrals are either convergent or divergent:
T
Divergent when ∫& 𝑓(𝑥) 𝑑𝑥 = ±∞
When integrating improper integrals, use 𝐥𝐢𝐦 and replace ∞ with 𝒕
𝒕→T
T
o
] 𝑓(𝑥) 𝑑𝑥 = lim ] 𝑓(𝑥) 𝑑𝑥 =
o→T &
&
If 0 ≤ 𝑓(𝑥) ≤ 𝑔(𝑥) for all 𝑥 ≥ 𝑎 then:
The Comparison Test for improper
Integrals states:
T
T
∫& 𝑓(𝑥) 𝑑𝑥 is convergent if ∫& 𝑔(𝑥) 𝑑𝑥 is convergent
T
T
∫& 𝑔(𝑥) 𝑑𝑥 is divergent if ∫& 𝑓(𝑥) 𝑑𝑥 is divergent
or a decreasing function 𝑓(𝑥) for all 𝑥 > 𝑎,
then there is an upper and lower sum such
that:
T
T
T
q 𝑓(𝑘) < ] 𝑓(𝑥) 𝑑𝑥 < q 𝑓(𝑘)
&
st&>_
st&
The Rienmann Sum states:
For an increasing function 𝑔(𝑥) for all 𝑥 >
𝑎, then there is an upper and lower sum
such that:
T
T
T
q 𝑔(𝑘) < ] 𝑔(𝑥) 𝑑𝑥 < q 𝑔(𝑘)
st&
&
st&>_
Topic 9: Calculus Option
Series Part 1
T
A series consists of 𝒂𝟏 + 𝒂𝟐 + 𝒂𝟑 …
It is denoted by:
T
q 𝑎S = 𝑆
q 𝑎s = 𝑆S
St_
st_
For a total sum
For a partial sum
If lim 𝑎S ≠ 0 or does not exist then ∑ 𝑎S diverges
S→T
The Divergence Test states:
If lim 𝑎S = 0 then ∑ 𝑎S may converge or may diverge
S→T
Geometric Infinite Series:
Converges if |𝑟| < 1
Diverges if |𝑟| ≥ 1
T
q 𝑎𝑟 S7_
Sum of infinite geometric series:
St_
Key series
types:
Telescoping Series:
Partial Fractions
P-Series:
=
Find 𝑆S equation
𝑎
1−𝑟
Take lim 𝑆S
Simplify 𝑆S
_
S→T
Converges if 𝑝 > 1
S}
_
(Harmonic Series: S)
Diverges if 𝑝 ≤ 1
If 𝑎S ≤ 𝑏S for all of 𝑛 and ∑ 𝑏S converges, then ∑ 𝑎S also converges
The Comparison Test states given for two
series of positive terms:
If 𝑎S ≥ 𝑏S for all of 𝑛 and ∑ 𝑏S diverges, then ∑ 𝑎S also diverges
1. Check if positive (0 ≤ 𝑎S)
2. Find 𝑏S (𝑏S generally is 𝑎S take away a useless term)
Steps:
3. Determine whether 𝑏S is smaller or larger than 𝑎S
3. Find if 𝑏S is convergent or divergent
If ∑
&~
g~
exists and 𝑎S and 𝑏S are
positive to use limit comparison test:
Then both series either converges of
diverges
The Limit Comparison Test States:
1. Find 𝑏S (𝑏S generally is 𝑎S take away a useless term)
Steps:
&
2. Find if 𝑎S and 𝑏S are positive and if ∑ g~ exists
~
T
The Integral Test states for 𝒇(𝒏) = 𝒂𝒏 , if
then
q 𝑎S
St_
Positive
Decreasing
Continuous
1. Check criteria: If it’s positive decreasing
and continuous
T
]
St_
𝑓(𝑥)𝑑𝑥
Will both converge
or diverge
2. Then integrate 𝑎S
Topic 9: Calculus Option
Series Part 2
The Alternating Series Test states for:
T
if lim |𝑎S | = 0 and |𝑎S>_ | < |𝑎S |, then the series will converge
S→T
q(−𝟏)𝒏 𝒂𝒏
𝒏t𝟏
If: ∑ 𝒂𝒏 converges, then 𝐥𝐢𝐦 𝑺𝒏 = 𝑺
𝒏→T
However, if 𝒏 does not approach ∞ then
the Error of a Sum can be found:
The Absolute Converge states: For ∑ 𝑎S, if ∑|𝑎S | =
|𝑎_ | + |𝑎• | + ⋯ is convergent, then ∑ 𝑎S is absolutely
convergent
Use if series is not always
positive, or not
alternating, or when
dealing with trig
1. Write ∑ 𝑎S as ∑|𝑎S |
For alternative series only:
Note:
|𝑹𝒏 | = |𝑺 − 𝑺𝒏 | ≤ 𝒂𝒏>𝟏
Absolute convergence, is stronger than convergence
|𝐸𝑟𝑟𝑜𝑟| ≤ |𝒂𝒏>𝟏 |
If a series is absolute convergent, it must be convergent
𝒂𝒏>𝟏 ≤ 𝒆𝒓𝒓𝒐𝒓 𝒔𝒑𝒆𝒄𝒊𝒇𝒊𝒆𝒅
Conditional convergence is when series converges, but is not absolutely convergent
&~RŒ
1. lim ‹
S→T
The ratio test stages if:
Don’t need to use divergent test for ratio test
&~
‹ < 1, ∑ 𝑎S is absolutely convergent
&~RŒ
2. lim ‹
S→T
&~
&~RŒ
3. lim ‹
S→T
&~
‹ > 1, ∑ 𝑎S is divergent
‹ = 1, ∑ 𝑎S is inconclusive