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Lecture Notes– Differential-integral calculus:
Function
A function “𝑓” is a rule that relates a variable (real number) “𝑥” to a real value “𝑦”. The
values that form a set of inputs for the function “𝑥” form the image set of the function.
For example, let a function “𝑓” be defined as 𝑓(𝑥) = 2 𝑥 − 1 , se 𝑥 = 0 , então, 𝑦 =
𝑓( 0 ) = 2 ∗ 0 ∗ (− 1 ) = 0 ∗ 1 = − 1. The image of the function “𝑓” is also the set of real
numbers.
Exemple 1. 1 :
If we consider 𝑓
Exemple 1. 2 :
If we define f 𝑓 as 𝑓
Therefore, 𝐷𝑜𝑚 = 𝑥 ∈ ℝ. 𝑥 ≥ 1.
Observation 1. 1 : The modular function is intended to measure the distance between the
value x and the origin. For any interval:
Observation 1. 2 : Even Function: 𝑓
Polynomial Functions of Order "N": A function “f” is called polynomial if it can
be written in the following form: 𝑓(𝑥) = 𝑎 0 𝑥𝑛 + 𝑎 1 𝑥𝑛 − 1 + ⋯ + 𝑎𝑛 − 1 𝑥, onde
𝑎 0 ≠ 0 , 𝑎 1 , 𝑎 2 , …, n are constants, 𝑥 ∈ ℝ.
Exemple 2.1:
f(x) = 1 , 𝑥 ∈ ℝ (order zero)
f(𝑥) = 2 x + 1 , 𝑥 ∈ ℝ (order 1)
f
x
= x
2
− 2 𝑥 + 1 , 𝑥 ∈ ℝ (order 2)
2
, 𝑥 ∈ ℝ (order 2)
3
2
Exemple 2.2: Consider the function “f” defined as 𝑓
3
The function defined by
3
is an odd
function, meaning 𝑓(−𝑥) =
for all 𝑥 ∈ ℝ, thus
its graph is symmetric with
respect to the point ( 0 , 0 ).
Asymptotes:
Graphically, it is when a
function approaches a
value, similar to a straight
line.
Continuity:
Let 𝑓: 𝑥 ≤ ℝ → ℝ , 𝑎 ∈ 𝑥
We say that the function 𝑓 is continuous at 𝑎 if: lim
𝑥→𝑎
Intermediate Value Theorem (IVT):
If 𝑓 is a continuous function on a closed interval [𝑎, 𝑏], then 𝑓 assumes all values
between 𝑓(𝑎) and 𝑓(𝑏).
Exemple 4.1: Show that the function 𝑓
3
− 4 𝑥 + 3 has at least one real root
in the interval [− 4 , 0 ].
The function 𝑓
3
− 4 𝑥 + 3 is continuous in
[
]
3
3
By IVT, f assumes all values between 𝑓
and 𝑓
, therefore there exists a value
𝑐 ∈ [− 4 , 0 ], such that 𝑓( 0 ) = 0.
Cosine Function:
𝑓(𝑡) = 𝑐𝑜𝑠𝑡, 𝑡 ∈ ℝ, 𝑐𝑜𝑠𝑡 ∈ [− 1 , 1 ]
Tangent Function:
= tan 𝑡, tan 𝑡 ≠ ±
𝜋
2
3 𝜋
2
5 𝜋
2
7 𝜋
2
(valores onde denominadores são igual
a zero).
Squeeze Theorem:
lim
𝑥→𝑎
lim
𝑥→𝑎
Therefore:
lim
𝑥→𝑎
Observation 3.1:
Fundamental Trigonometric Limit: lim
𝑥→ 0
sin 𝑥
𝑥
Função Exponencial: It is a function of the type 𝑓(𝑥) = 𝑎
𝑥
, 𝑥 ∈ ℝ, generally used
for calculations in financial mathematics. It can be observed in:
Simple Interest: 𝑀 = 𝐶 + 𝐶 ∗ 𝑖, where C is the capital, and i is the interest rate.
Compound Interest: 𝐶 (𝑖 + 𝑖)
𝑡
, where t is my time.
Exemple 4 .1: Let a function 𝑓
2
Tangent Line:
The tangent line to a curve at a point is the line that best approximates the curve at
that point, representing its slope at that instant. For each point on the curve, there
is a unique tangent line with a determined slope. The slope of the secant line “s”
that passes through points 𝑃 (𝑥 0
0
), is given by:
𝑠
= tan 𝛼 =
0
0
The slope of the tangent line to the curve 𝑦 = 𝑓(𝑥) no ponto (𝑥
0
0
)) is given by:
𝑡
= lim
𝑥→𝑥
0
0
0
The equation of the tangent line to the curve 𝑦 = 𝑓(𝑥) at the point (𝑥
0
0
)) é:
0
0
With this concept of tangent line, we can define another concept, derivatives. A
derivative is the slope of the tangent line to the curve 𝑦 = 𝑓(𝑥) at the point
0
0
) é 𝑎
𝑝
= lim
𝑥→𝑥
0
𝑓(𝑥) − 𝑓(𝑥
0
))
𝑥−𝑥 0
, 𝑖𝑓 𝑡ℎ𝑒 𝑙𝑖𝑚𝑖𝑡 𝑒𝑥𝑖𝑠𝑡𝑠. The derivative measures
the rate of change, and is calculated mechanically, using the following "formula":
𝑛
, my derivative function is: 𝑛 ∗ 𝑥
𝑛− 1
. Of course, this is the summarized form to
calculate the derivative, the derivative comes from the following expression:
′
(𝑥) = lim
𝛥𝑥→ 0
𝑓(𝑥+𝛥𝑥)−𝑓(𝑥)
𝛥𝑥
Exemple 5.1:
3
2
′
2
Rules/Properties of differentiation:
Multiplication by scalar: [𝑘 ∗ 𝑓(𝑥)]
′
′
Quotient Rule: (
𝑓
( 𝑥
)
𝑔
( 𝑥
)
′
𝑓
( 𝑥
)
′
∗𝑔
( 𝑥
) −𝑓
( 𝑥
) ∗𝑔
( 𝑥
)
′
𝑔
( 𝑥
)
2
Product Rule: (𝑓
′
′
′
Observation 4.1:
′
′
Observation 4.2: In the case of the tangent, we can consider it as the division of
two functions, since, tan 𝑥 =
𝑠𝑖𝑛(𝑥)
𝑐𝑜𝑠(𝑥)
, logo, 𝑓(𝑥) = 𝑠𝑖𝑛(𝑥) e 𝑔(𝑥) = 𝑐𝑜𝑠(𝑥). We can
solve using the quotient rule, thus, we get that tan 𝑥 = 𝑠𝑒𝑐
2
Observation 4.3: The derivative can also be written with the notation
𝑑𝑦
𝑑𝑥
Maxima and Minima:
Theorem 1.1: A continuous function f on a closed interval [a,b] has an absolute
maximum or minimum.
Theorem 1.2: The derivative of the function being zero or not existing is called a
critical point.
Definition 1.1: A point 𝑥 0
∈ (𝑎, 𝑏) where 𝑓(𝑥
0
′
= 0 ou 𝑓(𝑥
0
′
does not exist is
called a critical point of the function.
Finding absolute extrema of continuous functions on a closed interval [a,b].
a) Calculate 𝑓(𝑎) e 𝑓(𝑏) (the values can be at the function's endpoints).
b) Calculate the critical points of 𝑓 in (𝑎, 𝑏) and calculate the values of 𝑓 at
these critical points
c) Compare the values of 𝑓 found in the previous steps.
Exemple 7.1: Find the absolute maximum and minimum of 𝑓
3
− 3 𝑥 + 1 em
[
]
3
3
′
2
′
2
The Critical points are 𝑥 = − 1 e 𝑥 = 1
does not belong to the interval
[
]
, therefore, the only critical point for 𝑓
is 𝑥 = 1.
3
Therefore, we have that:
( 3 , 19 ) is the absolute maximum point of 𝑓 em [ 0 , 3 ].
is the absolute minimum point of f 𝑓 in
[
]
Observation 5.1: A derivative with a positive sign means that the function is
increasing; A derivative with a negative sign means the function is decreasing.
When it is zero, it means that it has reached one of its critical points.
Observation 5.2: If necessary, a graph can be estimated by analyzing the derivative,
examining the midpoint between the points.
That is, 𝑥 = 2 ou 𝑥 = − 2. Observing the intervals, the function is increasing for 𝑓 is
increasing for 𝑥 < − 2 e 𝑥 > 2 , and decreasing for − 2 < 𝑥 < 2.
Rates and Antiderivatives:
Exemple 8.1The displacement (in meters) of a particle moving along a straight line
is given by the equation 𝑠(𝑡) = 𝑡
2
− 8 𝑡 + 18 , where t is measured in seconds. Find
the instantaneous velocity for 𝑡 = 4 𝑠.
2
′
Observation 6.1: To find the position instead of the velocity, the integral is used,
which acts as the "antiderivative".
Exemple 8.2: A particle moves on a straight line with acceleration given by 𝑎
6 𝑡 + 4. Its initial velocity is 𝑣
= − 6 𝑚/𝑠 and its initial displacement is 𝑠
9 𝑐𝑚. Find its position function.
2
2
2
3
2
2
2
3
2
Definição 2.1: A Function 𝑓 is an antiderivative of 𝑓 if 𝑓(𝑥)
′
= 𝑓(𝑥). We use the
following notation: ∫ 𝑓
′
Exemple 9.1: ∫
2
𝑥
3
3
- 𝑐 , where c is my constant of integration.
Rules of integration for some cases:
a) ∫
𝑎
𝑥
𝑎+ 1
𝑎+ 1
b) ∫
1
𝑥
𝑑𝑥 = ln |𝑥| 𝑑𝑥 + 𝑐
c) ∫
𝑥
𝑥
d) ∫
𝑘 𝑑𝑥 = 𝑘𝑥 + 𝑐 , where 𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
e) ∫
f) ∫
Trigonometric Substitution:
2
2
2
2
2
2
Inversa de 𝑦 = 𝑎𝑟𝑐𝑠𝑖𝑛(𝑥)
2
2
2
2
2
2
2
2
2
2
Partial Fractions/Decomposition of Fractions:
2
2
= ln 𝑥
2
2
(ln(𝑥) − 1 ) 𝑑𝑥 +
(ln(𝑥) + 1 ) 𝑑𝑥
