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Calculus
In math, calculus is used for studying and understanding changes. We use calculus in the fields of engineering, telecommunication systems, space exploration, physics, data modeling, etc.
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What is Calculus?
Calculus is used to study and understand changes. Changes in quantities like speed or time. The important functions of calculus are differentiation, integration, and limits. It was developed by Isaac Newton and Gottfried Wilhelm Leibniz.
History of Calculus
Calculus is a branch of mathematics that studies motion. It includes derivatives, integrals, limits, etc. The need to calculate the area of complex shapes like spirals led to the development of calculus.
In earlier times, people used calculus to find the area and volume of an object. For that, they used exhaustion and inverse heuristic, a precursor to limits and similar to integrals. Archimedes, Ibn al-Haytham, Bhaskara II, and Cavalieri were some ancient mathematicians who used techniques related to calculus.
The attribution for the development of calculus is given to Isaac Newton and Gottfried Wilhelm Leibniz. They used it to find the curvature and solve complex problems. Newton's work was based on derivatives for calculating the rate of change. During the same time, to calculate the area under curves, Leibniz used integral calculus.
Today, we use it in the fields of economics, music, weather forecasting, and many more.
Branches of Calculus
Based on the change and mathematical analysis, calculus can be divided into four parts:
- Differential Calculus
- Integral Calculus
- Multivariable Calculus
- Vector Calculus
Begin your journey into Calculus by exploring key concepts. Understand important calculus topics in detail by selecting from the list below:
Differential Calculus
Differential calculus is used to find the rate of change of a quantity with respect to another. The key concepts are limits, continuity, and slope of a curve.
Integral Calculus
To find the quantity when the rate is known, integral calculus is used. Integral is the inverse process of differentiation. We use it to calculate the area and volume.
Multivariable Calculus
It is used when the function of two or more variables is calculated. It is an extension of differential and integral calculus.
Vector Calculus
Vector calculus is about the use of differentiation and integration of vector fields in Euclidean space.
Fundamental Theorems of Calculus
The integral is the inverse process of the differential. The fundamental theorem of calculus links derivatives and integrals. There are two theorems that are the first and the second fundamental theorem of calculus.
Basic Calculus Formulas
In calculus, for each function, we have different formulas. And students should remember and use the correct formula. For a better understanding of calculus, it's important to remember all the basic formulas.
Derivative Formulas
d / dx(xn) = nxn-1
d / dx(C) = 0 → C is the constant
d / dx(Cf) = C.d / dx(f)
d / dx(f±g) = d / dx(f) ± d / dx(g)
d / dx(fg) = d / dx(fg) + d / dx(gf)
d / dx(fg) = g(d / dx)(f) - fd / dx(g)
g2
d / dx(sin x) = cos x
d / dx(cos x) = -sin x
d / dx(tan x) = sec2x
d / dx(cot x) = -cosec2x
d / dx(secx) = sec x tan x
d / dx(cosec x) = -cosec x cot x
d / dx(ax) = ax In a
d / dx(ex) = ex
d / dx(In x) = 1 / x
d / dx(logax) = 1
x In a
Integral formula
∫1dx = x + C
∫a dx = ax + C
∫(1 / x) dx = In |x| + C
∫1 / √ 1 - x2dx = sin-1x + C
∫1 / √1 + x2dx = tan-1x + C
∫1 / |x| √1 - x2dx = sec-1x + C
∫sin x dx = -cos x + C
∫cos x dx = sin x + C
∫sec x (tan x) dx = sec x + C
∫csc x (cot x) dx = -csc x + C
∫sec2 x dx = tan x + C
∫csc2 x dx = -cot x + C
∫e2 dx = ex + C
∫a2 dx = ax/ In a + C
∫1x dx = In |x| + C
∫loga x dx = x In x -x + C
In a
Applications of Calculus
We have talked a lot about calculus. We know that we use it in our daily life. Now let’s see the applications of calculus.
Physics: In Physics, calculus is used to calculate motion, force, change, energy and many more. The speed of a vehicle, rocket, or even the light can be calculated.
Engineering: From building bridges to developing circuits, we use calculus in different fields of engineering.
Economics: Calculus is used to study the market and its behavior. To understand the minimum cost and maximum profit
Medicine: To study the growth of bacteria and infectious diseases, calculus is used. This data is used to treat cancer and diagnose patients.
Solved Examples on Calculus
Problem 1
Find the derivatives of 3x² + 2x + 1
The derivatives of 3x2 + 2x + 1 = 6x + 2
Explanation
Apply the differentiation rule to find the derivatives of 3x2 + 2x + 1
Differentiation rule: d/dx(f+g) = d/dx(f) + d/dx(g)
d/dx(3x2 + 2x +1) = d/dx(3x2) + d/dx(2x) + d/dx(1)
Derivatives of x2 = 2x
Derivatives of 2x = 2
Derivatives of 1 = 0
Add the derivatives: d/dx(3x2) + d/dx(2x) + d/dx(1)
= 3 × 2x + 2 + 0
= 6x + 2.
Therefore, the derivatives of 3x2 + 2x + 1 = 6x + 2
Problem 2
Calculate ∫2x dx
∫2x dx = x2 + c
Explanation
The power rule of integration = ∫xn dx = x(n + 1) / (n + 1) + c
∫2x dx = 2(x1+1 / 1 + 1) + c
= 2(x2 / 2) + c
= x2 + c
Therefore, ∫2x dx = x2 + c
Problem 3
Solve ∫(3x² - 4x + 5)dx
∫(3x2 - 4x + 5)dx = x3 - 2x2 + x + c
Explanation
According to the properties of integral, ∫f(x) ± g(x).dx = ∫f(x)dx ± ∫g(x).dx
∫(3x2 - 4x + 5)dx = ∫3x2dx - ∫4xdx +∫5dx
Using the power rule, ∫xn dx = x(n+1) / (n+1) + c
∫(3x2) = x3
∫(-4x) = -2x2
∫(5) = x
Adding these values to ∫3x2 dx - ∫4x dx +∫5 dx,
We get, x3 - 2x2 + x + c, where ‘c’ is the constant.
Tips and Tricks to Learn Calculus
We have already learned what is calculus in math. Calculus is a broad topic that includes limits, derivatives, integrals, and many more. Let’s explore tips and tricks to make it easier for students to learn.
Understand fundamental concepts like limits and continuity
Understanding the fundamental concepts is the best way to learn calculus. Derivatives, integrals, their relationship, limits, and continuity are the few fundamental concepts.
Practice derivatives and integral rules repeatedly
Students can learn calculus by remembering the basic rules. Regular practice will help students remember the rules.
Visualize problems using graphs
Students can use visual aids like graphs to understand the problem. To calculate the change, slope, area, cures, etc., students can use graphs.
Verify using reverse operations
When working on derivatives and integrals, students can verify the answer using reverse operations. As integration is reverse operation of differential.
Common Mistakes and How to Avoid Them in Calculus
Mistakes are common in calculus, moreover students repeat the same mistake. To master calculus, let’s learn some common mistakes.
Mistake 1
Confusing with derivative and integral notation
Students when introduced to calculus will easily get confused with integrals and derivatives. To avoid such errors, students should understand the concept well. Derivatives are the rate of change of a quantity, whereas integrals are finding the quantity when the rate is known.
Mistake 2
Forgetting the limits of integration
Students forget to specify the limits when doing integration. This can lead to an error in the final answer. So students should double-check to verify the answer.
Mistake 3
Confusing with integral formulas
For each type of integral, there are different formulas. So students need to remember the formulas and should apply the correct formula based on the type of integrals.
FAQs on Calculus
1.What is calculus?
2.What are the uses of calculus?
3.Who is the father of calculus?
4.What are the branches of calculus?
5.What is the basic concept of calculus?
Explore More Math Topics
From Numbers to Geometry and beyond, you can explore all the important Math topics by selecting from the list below:
| Numbers | Multiplication Tables |
| Geometry | Algebra |
| Measurement | Trigonometry |
| Commercial Math | Data |
| Math Formulas | Math Questions |
| Math Calculators | Math Worksheets |
