Comprehensive Calculus 1 Guide: Theory, Examples, and Exercises with Solutions

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) = 2𝑥 − 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(𝑥) = 𝑥 − 1

𝐷𝑜𝑚 = 𝑦 ∈ ℝ, 𝐼𝑚𝑔 = 𝑦 ∈ ℝ



Exemple 1.2:

If we define f 𝑓 as 𝑓(𝑥) = √𝑥 − 1

𝑥−1 ≥0
𝑥 ≥ 1

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:
|𝑥| < 𝑎

𝑋 ∈ (−𝑎, 𝑎)



Observation 1.2: Even Function: 𝑓(𝑥) = 𝑓(−𝑥); √𝑥2 = |𝑥|.

,Polynomial Functions of Order "N": A function “f” is called polynomial if it can
be written in the following form: 𝑓(𝑥) = 𝑎0𝑥𝑛 + 𝑎1𝑥𝑛 − 1 + ⋯ + 𝑎𝑛 − 1𝑥, onde
𝑎0 ≠ 0, 𝑎1, 𝑎2 , …, n are constants, 𝑥 ∈ ℝ.



Exemple 2.1:

f(x) = 1, 𝑥 ∈ ℝ (order zero)
f(𝑥) = 2x + 1, 𝑥 ∈ ℝ (order 1)
f(x) = x 2 − 2𝑥 + 1, 𝑥 ∈ ℝ (order 2)
𝑓(𝑥) = 𝑥 2 , 𝑥 ∈ ℝ (order 2)
𝑓(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑥 + 1, x ∈ ℝ (order 3)



Exemple 2.2: Consider the function “f” defined as 𝑓(𝑥) = 𝑥 3

The function defined by
𝑓(𝑥) = 𝑥 3 is an odd
function, meaning 𝑓(−𝑥) =
𝑓(−𝑥) for all 𝑥 ∈ ℝ, thus
its graph is symmetric with
respect to the point (0,0).

, Rational Functions:


𝑝(𝑥)
𝑓(𝑥) = , 𝑞(𝑥) ≠ 0 , where p, q, are polynomial functions.
𝑞(𝑥)




Exemple 3.1:
𝑥 2 −1 02 −1 −1
𝑓(𝑥) = , for 𝑥 = 0, 𝑓(0) = = = 1
𝑥−1 0−1 −1

𝐷𝑜𝑚{ 𝑥 ∈ ℝ; 𝑥 ≠ 1 }




Limit:
We say that the limit of a function is equal to a value L, when 𝑥 tends a 𝑥0 , if the values
of 𝑓(𝑥) approach the value of L (as close as we want), as we assign values to 𝑥
sufficiently close to 𝑥0 Thus, we can define the limit as: lim 𝑓(𝑥) = 𝐿, 𝐿 ∈ ℝ
𝑥→𝑥0

When the lateral limits have 2 results, we say that there is a lateral limit (values tending
from the positive side / to the right, and negative side / to the left, of the axis).

Formal Definition of Limit: Given 𝜀 > 0 there exists an ℒ, such that |𝑓(𝑥) − ℒ| > 𝜀



Properties of Limit:


lim [𝑓(𝑥) ± 𝑔(𝑥)] = lim 𝑓(𝑥) ± lim 𝑔(𝑥)
𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
lim [𝑐 ∗ 𝑓(𝑥)] = 𝑐 ∗ (lim 𝑔(𝑥) )
𝑥→𝑎 𝑥→𝑎
lim [𝑓(𝑥) ∗ 𝑔(𝑥)] = lim 𝑓(𝑥) ∗ lim 𝑔(𝑥)
𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
𝑛
lim [𝑓(𝑥)]𝑛 = [lim 𝑓(𝑥)]
𝑥→𝑎 𝑥→𝑎
lim (ln 𝑥 ∗ 𝑓(𝑥)) = ln 𝑥 ∗ (lim 𝑓(𝑥))
𝑥→𝑎 𝑥→𝑎
lim [sin(𝑥) ∗ 𝑓(𝑥)] = sin 𝑥 ∗ [lim 𝑓(𝑥)]
𝑥→𝑎 𝑥→𝑎
lim 𝑓(𝑥)
lim (𝑒) 𝑓(𝑥) = (𝑒)𝑥→𝑎
𝑥→𝑎
𝑛
lim √𝑓(𝑥) = 𝑛√ lim 𝑓(𝑥)
𝑥→𝑎 𝑥→𝑎