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integrals and derivatives of functions. It is based on the summation of the infinitesimal differences. Calculus is the study of continuous change of a function or a rate of change of a function. It has two major branches and those two fields are related to each other by the fundamental theorem of calculus. The two different branches are:
๏ท Differential calculus ๏ท Integral Calculus
Differential calculus deals with the rate of change of one quantity with respect to another or it as a study of rates of change of quantities. For example, velocity is the rate of change of distance with respect to time in a particular direction.
Function: A function is defined as a relation from a set of inputs to the set of outputs in
which each input is exactly associated with one output. The function is represented by โf(x)โ.
Dependent Variable: The dependent variable is a variable whose value always
depends and determined by using the other variable called an independent variable. The dependent variable is also called the outcome variable. The result is being evaluated from the mathematical expression using an independent variable is called a dependent variable.
Independent Variable: Independent variables are the inputs to the functions that
define the quantity which is being manipulated in an experiment. Let us consider an example y= 3x. Here, x is known as the independent variable and y is known as the dependent variable as the value of y is completely dependent on the value of x.
Domain and Range: The domain of a function is simply defined as the input values of
a function and range is defined as the output value of a function. Take an example, if f(x) = 3x be a function, the domain values or the input values are {1, 2, 3} then the range of a function is given as
Module 1: Differential Calculus
In Mathematics, calculus is a branch that deals with finding the different properties of
f(1) = 3(1) = 3
f(2) = 3(2) = 6
f(3) = 3(3) = 9
Therefore, the range of the function will be {3, 6, 9}.
Interval: An interval is defined as the range of numbers that are present between the
two given numbers. Intervals can be classified into two types namely:
Open Interval โ The open interval is defined as the set of all real numbers ๐ฅ such that ๐ < ๐ฅ < ๐. It is represented as (๐, ๐)
Closed Interval โ The closed interval is defined as the set of all real numbers ๐ฅ such that ๐ โค ๐ฅ and ๐ง โค ๐, or more concisely ๐ โค ๐ฅ โค ๐, and it is represented by [๐, ๐]
Limit
Let ๐ฅ be a variable and โ๐โ be a constant. Since โ๐ฅโ is a variable we can change its value at pleasure. It can be changed so that its value comes nearer and nearer to a. Then we say that ๐ฅ approaches โ๐โ and it is denoted by ๐ฅ โถ ๐.
Definition: Let ๐(๐ฅ) be a function defined on an interval that contains ๐ฅ = ๐, except
possible at ๐ฅ = ๐. Then we say that, (^) ๐ฅโ๐lim ๐(๐ฅ) = ๐ฟ.
If for every number โ> 0 there is some number ๐ฟ > 0 such that |๐(๐ฅ) โ ๐ฟ| <โ whenever 0 < |๐ฅ โ ๐| < ๐ฟ.
We say (^) ๐ฅโ๐limโ ๐(๐ฅ)^ is expected value of ๐ at ๐ฅ = ๐ given the values of ๐ near ๐ฅ to the
left to ๐. The values is called left hand limit.
โด ๐ฟ๐ป๐ฟ = lim ๐ฅโ๐โ ๐(๐ฅ)
(^) ๐ฅโ๐lim[๐(๐ฅ) + ๐(๐ฅ)] = lim ๐ฅโ๐ ๐(๐ฅ) + lim ๐ฅโ๐ ๐(๐ฅ)
(^) ๐ฅโ๐lim[๐(๐ฅ) โ ๐(๐ฅ)] = lim ๐ฅโ๐ ๐(๐ฅ) โ lim ๐ฅโ๐ ๐(๐ฅ)
(^) ๐ฅโ๐lim[๐(๐ฅ) โ ๐(๐ฅ)] = lim ๐ฅโ๐ ๐(๐ฅ) โ lim ๐ฅโ๐ ๐(๐ฅ)
(^) ๐ฅโ๐lim [๐(๐ฅ)๐(๐ฅ)] = ๐ฅโ๐limlim^ ๐(๐ฅ) ๐ฅโ๐ ๐(๐ฅ)
(^) ๐ฅโ๐lim ๐๐(๐ฅ) = ๐ lim ๐ฅโ๐ ๐(๐ฅ)
Continuity
Continuity of a function at a point: A real function ๐(๐ฅ) is said to be continuous at
a point โ๐โ of its domain if lim ๐ฅโ๐ ๐(๐ฅ) exists and equals to ๐(๐).
OR
A real function โ๐โ is said to be continuous at ๐ฅ = ๐ iff
(i) ๐(๐) exists (ii) lim ๐ฅโ๐ ๐(๐ฅ) exists and (iii) lim ๐ฅโ๐ ๐(๐ฅ) = ๐(๐)
Note: A function โ๐โ is said to be left continuous or continuous on the left of ๐ฅ = ๐ iff
(i) ๐(๐) exists (ii) (^) ๐ฅโ๐limโ ๐(๐ฅ) exists and (iii) (^) ๐ฅโ๐limโ ๐(๐ฅ) = ๐(๐)
A function โ๐โ is said to be right continuous or continuous on the right of ๐ฅ = ๐ iff (i) ๐(๐) exists (ii) (^) ๐ฅโ๐lim+ ๐(๐ฅ) exists and (iii) (^) ๐ฅโ๐lim+ ๐(๐ฅ) = ๐(๐)
Continuity of a function in an internal: If a function โ๐โ is continuous at every point
of an open interval (๐, ๐) then it is said to be continuous on (๐, ๐) and ๐ is said to be continuous on the closed interval [๐, ๐] iff it is continuous in (๐, ๐), right continuous function at left end point ๐ and left continuous at right end point ๐.
Continuous function: A function โ๐โ is said to be a continuous function iff it is
continuous at every point of the domain.
Example: Examine the continuity of the function ๐(๐) = ๐๐๐^ โ ๐ at ๐ = ๐.
Given function ๐(๐ฅ) = 2๐ฅ^2 ๏ญ 1
โด ๐(3) = 2(3^2 ) ๏ญ 1 = 17
lim ๐ฅโ3 ๐(๐ฅ) = lim ๐ฅโ3(2๐ฅ^2 โ 1) = 2(3^2 ) โ 1 = 17 = ๐(3)
โด ๐(๐ฅ) is continuous at ๐ฅ = 3.
Remark
Let ๐(๐ฅ) and ๐(๐ฅ) be two continuous real functions at ๐ฅ = ๐ โ ๐
. Then
- ๐(๐ฅ) + ๐(๐ฅ)^ is continuous at ๐ฅ = ๐
- ๐(๐ฅ) โ ๐(๐ฅ)^ is continuous at ๐ฅ = ๐
- ๐(๐ฅ) โ ๐(๐ฅ) is continuous at ๐ฅ = ๐
- ๐(๐ฅ)๐(๐ฅ) is continuous at ๐ฅ = ๐ (provided ๐(๐) โ 0)
Also, the composition of two continuous functions is a continuous function.
Differentiability
A real function ๐(๐ฅ) is said to be differentiable at ๐ฅ = ๐ if ๐โฒ(๐) = lim ๐ฅโ๐๐(๐ฅ)โ๐(๐)๐ฅโ๐ exists.
Note: A function ๐(๐ฅ) is differentiable at ๐ฅ = ๐ then ๐(๐ฅ) is continuous at ๐ฅ = ๐. The converse of this result is not true i.e., a continuous function need not be differentiable. For example the function ๐(๐ฅ) = |๐ฅ| is continuous but not differentiable.
