Applications of Calculus in Real Life and Problem Solving

Calculus Apps

The educators at Vedantu explain the key concepts in a way the students can easily grasp it well enough to solve complex sums. The educators use creative 3D graphics, short animated educational videos for better and effective learning. 

Nowadays, apps have been introduced to solve advanced problems of calculus with maximum accuracy. The apps can easily be installed in the Google play store.

With the advent of technology, solving complex mathematical equations has become a matter of seconds. These math problems solving apps can help solve anything from basic problems to the most complex problems. The apps also provide a step-by-step process of how it solved the answer.

In some of these apps, you can simply click a picture of that equation and the app will provide you with the answer in the most accurate way.

Uses of Calculus Apps

1. Limit of a function

2. Differentiation 

3. Integration (Definite and indefinite)

4. Maxima and minima

5. Approximate area of circles with the use of rectangles.

6. Taylor polynomial

One can easily find some of the best apps for Calculus on Google Play Store and iOS Apps Store.  

Calculus Explained

Integration is defined as the summation of all discrete elements. Integration is used to find volume, displacement and, the area which cannot be measured otherwise since it's very small. 

Basically, it is exactly the opposite of differentiation.

There are two types of integration:

• Differential Calculus

• Integral Calculus

Integral Calculus can be two types that are Indefinite integral and definite integral.

Integration is used to find the antiderivative of a function whereas as we already know differentiation helps in finding the derivative of a function.

Therefore, it is exactly the opposite of the way differentiation is solved.

For example: for a function f(x) = sin x

The differentiation of f(x):

d f(x)/ dx = cos x

To find the antiderivative of a function ∫, this is introduced. 

Therefore, it can be shown as: 

∫f (x) dx= F(x) + c

Here, f(x) is called the integrand of the function, and c is the constant. 

Indefinite Integration

Integration without the use of limits. 

It is represented as follows: 

∫(2x²-3x)dx 

Definite Integral

Integration with the use of lower and upper limits is called a definite integral.

For example: 

\[\int_{a}^{b}\] (2x²- 3x) 

Some basic formulae one must know in order to solve integration: 

• ∫  dx = x + C.

• ∫ a dx = ax + C.

• ∫ xn dx = ((xn+1)/(n+1))+C ; n≠1.

• ∫ sin x dx = – cos x + C.

• ∫ cos x dx = sin x + C.

• ∫ sec2 dx = tan x + C.

• ∫ cosec2 dx = -cot x + C.

• ∫ sec x (tan x) dx = sec x + C.

• ∫ (1/x) dx = ln |x| + C

• ∫ ex dx = ex+ C∫ ax dx = (ax/ln a) + C ; a>0,  a≠1

Example:

∫ 2x² + 3x + 7

Solution: 

∫(2x² + 3x + 7)dx

= 2 ∫1x² dx + 3 ∫ x dx + ∫7 dx   

= 2 x³/3 + 3x² /3 + 7x

Applications of Calculus

• Software Developer: 

They may use these apps to determine and meet a user's software needs or for preparing models for programmers. 

• Economist: 

An economist studies and analyses several economic issues using advanced mathematics. These apps can also help the economist for research purposes.

• Aerospace Engineer: 

An aerospace engineer uses advanced mathematical tools to develop or build different aerospace products, like satellites, missiles, spacecraft, aircraft. 

• Chemists: 

In the field of chemistry, calculus is used to determine the rate of change in a chemical equation.