Introduction to Calculus: Calculus Definition, Calculus Types, Calculus Examples and Calculus Formulas, Introduction to Calculus PDF, Calculus Notes PDF.
Calculus is one of the main branches of mathematics. The other branches are algebra, geometry, trigonometry, etc. mathematics is a wide concept in sciences used to solve numerical problems and daily life calculations.
In this post, we will discuss the main branches of calculus along with definitions, types, and examples.
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What is calculus?
Calculus is a branch of mathematics that deals with the properties and evaluation of integration and differentiation of functions, by methods originally based on the summation of infinitesimal differences.
Types of calculus
There are various types of calculus such as:
 Limit calculus
 Integral calculus
 Differential calculus
These types are frequently used as a system of calculation or reasoning. Let us briefly discuss the types of calculus with examples.

Limit calculus
Limit is a branch of calculus that is widely used to define the other branches of calculus such as integration and differential. It is defined as a value that tells a function approach to some value by placing the particular value in the function.
Limit formula
The formula of the limit calculus is:
limh→n p(h) = M

 “h” = a variable of the function.
 P(h) = given function
 “n” = the limit value that a function approaches.
 M = final result after placing the particular value.
A limit calculator with steps can be used to get the stepbystep solution to limit calculus problems according to the above formula.
Types of limits
 Lefthand limit
 Righthand limit
 Twosided limit
Example of limit
Example
Calculate the limit function p(h) = 2h2 – 5h3 + 3h +15, when h approaches 3.
Solution
Step 1: First of all, take the given function and apply the notation of the limit on it.
p(h) = 2h2 – 5h3 + 3h +15
limit value = n = 3
limh→3 p(h) = limh→3 [2h2 – 5h3 + 3h +15]
Step 2: Now apply the notation of limit calculus to each function separately with the help of the sum and difference rules of limits.
limh→3 [2h2 – 5h3 + 3h +15] = limh→3 [2h2] – limh→3 [5h3] + limh→3 [3h] + limh→3 [15]
Step 3: Now take the constant coefficient outside the limit notation.
= 2limh→3 [h2] – 5limh→3 [h3] + 3limh→3 [h] + limh→3 [15]
Step 4: Now apply the power and constant rules of limits.
= 2 [32] – 5 [33] + 3 [3] + [15]
= 2 [9] – 5 [270] + 3 [3] + [15]
= 18 – 135 + 9 + 15
= 117 + 9 + 15
= 93

Integral calculus
The integral calculus is frequently used to find the area under the curve. It is defined as a new function whose original function is a differential or a numerical value by taking the boundary values.
Integral formula
The formulas for the integral are:
 ab p(h) dh = P(b) – P(a) = N
 ʃ p(h) dh = P(h) + C
where,
 ʃ = integral notation
 a & b = boundary values
 p(h) is the integrand function.
 “h” = the integrating variable.
 P(v) – P(u) = applying limit values by the fundamental theorem of calculus.
 N = final result
 C = constant of integration
Types of integral calculus
 Definite integral
 Indefinite integral
Example of integral
Example
Find the integral of p(h) = 3h3 + 4h – 2h4 + 1 with respect to h.
Solution
Step 1: First of all, apply the integral notation to the given function.
p(h) = 3h3 + 4h – 2h4 + 1
ʃ p(h) dh = ʃ [3h3 + 4h – 2h4 + 1] dh
Step 2: Use the sum and difference rule of integral calculus and apply the integral calculus notation to each function.
ʃ [3h3 + 4h – 2h4 + 1] dh = ʃ [3h3] dh + ʃ [4h] dh – ʃ [2h4] dh + ʃ [1] dh
Step 3: Use the constant function rule of integral and take the constant coefficients outside the notation.
= 3ʃ [h3] dh + 4ʃ [h] dh – 2ʃ [h4] dh + ʃ [1] dh
Step 4: Integrate the above expression.
= 3 [h3+1 / 3 + 1] + 4 [h1+1 / 1 + 1] – 2 [h4+1 / 4 + 1] + [h] + C
= 3 [h4 / 4] + 4 [h2 / 2] – 2 [h5 / 5] + [h] + C
= 3/4 [h4] + 4/2 [h2] – 2/5 [h5] + [h] + C
= 3/4 [h4] + 2 [h2] – 2/5 [h5] + [h] + C
= 3h4/4 + 2h2 – 2h5/5 + h + C
To avoid these calculations to find the integral of the function use an antiderivative calculator.

Differential Calculus
The differential calculus is frequently used to find the slope of the tangent line. It is defined as the instantaneous rate of change of a function with respect to the independent variable of the function. It can also be defined by using the limits.
It is denoted by d/dx.
Derivative formula
f’(x) = Limh→0 [f(x + h) – f(x)]/h
Types of differential calculus
 Explicit derivatives
 Implicit differentiation
 Partial derivative
Example of differential calculus
Example
Calculate 5h2 + 2h3 – 8h4 + 3h + 9 with respect to “h”.
Solution
Step I: First of all, write the given differential function and apply the notation of differentiation to it.
p(h) = 5h2 + 2h3 – 8h4 + 3h + 9
d/dh [p(h)] = d/dh [5h2 + 2h3 – 8h4 + 3h + 9]
Step II: Now apply the notation of differentiation to each function separately by using the sum and difference rules of differentiation.
d/dh [p(h)] = d/dh [5h2 + 2h3 – 8h4 + 3h + 9] = d/dh [5h2] + d/dh [2h3] – d/dh [8h4] + d/dh [3h] + d/dh [9]
Step III: Now apply the constant function rule of differential calculus.
= 5 d/dh [h2] + 2 d/dh [h3] – 8 d/dh [h4] + 3 d/dh [h] + d/dh [9]
Step IV: Now find the derivative of the above expression with respect to “u”.
= 5 [2 * h21] + 2 [3 * h31] – 8 [4 * h41] + 3 [h11] + [0]
= 5 [2 * h1] + 2 [3 * h2] – 8 [4 * h3] + 3 [h0] + [0]
= 10 * h + 6 * h2 – 32 * h3 + 3 [1] + [0]
= 10 * h + 6h2 – 32h3 + 3
Conclusion
In this post, we have covered some wellknown types of calculus along with definitions, types, and solved examples. Now you can solve any problem of differential, integral, and limit just by following the above basics of these branches.