Calculus

Calculus is all about understanding how things change over time. It focuses on derivatives and integrals, which essentially involve adding up tiny changes step by step.

Derivatives help us know how one quantity changes in relation to another.

At the same time, integrals enable us to measure the total effect of those changes, such as finding the area under a curve or the distance traveled.

• Ancient Times
  
  In mathematics, the foundation of calculus dates back thousands of years. In the fifth century BC, Democritus worked with ideas based on infinitesimals during the ancient Greek period. If backed by a geometric proof, the Greeks would consider a theorem accurate.
  
  Greek philosophers also viewed ideas based on infinitesimals as paradoxes, as it is always possible to divide a number again, no matter how small it becomes.
  
  In the 3rd century BC, Archimedes developed the method of exhaustion, which is used for calculating the area of circles.

• The Middle East and India
  
  Like many other areas of science and math, the growth of calculus slowed down in the Western world during the Middle Ages. At the same time, on the other side of the world, people were discovering and studying both integrals and derivatives.
  
  An Arab mathematician, Ibn al-Haytham, used formulas he created to find the volume of a paraboloid. A solid shape made by the spinning part of a parabola (a curve) around a line.
  
  During this time, ideas moved between the Middle East and India, since some of them were also found in the Kerala School of Astronomy and Mathematics. By the start of the fourteenth century, many pieces of calculus were already known, but differentiation and integration had not yet been joined together as one subject.

• The Fourteenth Century
  
  In the 14th century, a group of mathematical scholars revived the study of mathematics. The group was known as the Oxford Calculators. They explored philosophical questions through the lens of math.

• The Seventeenth Century
  
  The early seventeenth century was the most critical time in the history of calculus. During this period, René Descartes created analytical geometry, and Pierre de Fermat studied the highest and lowest points of curves, as well as their tangents. Some of Fermat’s formulas are similar to the ones we still use today, even after almost 400 years.

• The Eighteenth Century and Beyond
  
  The debate surrounding the invention of calculus became more and more heated as time wore on, with Newton’s supporters openly accusing Leibniz of copying. A British firm still claims that calculus was Newton’s discovery.

Based on the change and mathematical analysis, calculus can be divided into four parts.

1. Differential Calculus - It 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.
   
    
2. Integral Calculus - To find the quality when the rate is known, integral calculus is used. An integral is the inverse process of differentiation. We use it to calculate the area and volume.
   
    
3. Multivariable Calculus - It is used when the function of two or more variables is calculated. It is an extension of differential and integral calculus.
   
    
4. Vector Calculus - Vector calculus is about the use of differentiation and integration of vector fields in Euclidean space.

Differential calculus determines the change of a function f(x) with respect to an independent variable. If the function f(x) defines a curve, then its derivatives give the value of its slope. 

The derivative of the function is expressed by \({dy \over dx} \text {or f'(x)}\). Here, 'dy' and 'dx' are known as the differentials. 

Some important concepts of differential calculus are: 

• Limits: The closeness of a function towards any values is defined by the limits. It is given by the formula: 
   

• Derivatives: The rate of change of a function at any instant is defined by the derivatives. It is expressed as:
   

• Continuity: If a function, satisfies the given condition, then it is said to be a continuous function at a point x = a. The conditions are: 
   

• Continuity and Differentiability:  If a function is differentiable at a given point, then it is a continuous function. However, the vice versa is not always followed. 

Integrals are the process of calculating the area under the given curve. They are also referred to as the antiderivative of the function. The method of finding the integrals is known as integration. 

Integral are further divided into two categories:

1. Definite Integrals: These integrals are bound with limits (upper and lower). These limits lie on the real axis. The formula for definite integral is given as:
   
   \(\int_{a}^{b} f(x) dx = f(b) - f(a)\)
   
   
   
   Here, 'a' is the lower limit and 'b' is the upper limit.
   
    
2. Indefinite Integrals: The integrals which are not bounded by any limits are called indefinite integrals. Indefinite integrals are expressed as: 

   
   \(\int f(x) dx = {F(x)} + c\)
   
   
   
   Here,

• F(x) is the antiderivative of function.
• f(x) is called the integrand.
• dx is called the integrating agent.
• C is called the constant of integration.
• x is the variable of integration.

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 theorems of calculus.

 

• First fundamental theorem of integral calculus
  
  The first fundamental theorem of integral calculus states that one of the antiderivatives (also known as the indefinite integral) of a function \(f\), can be found by integrating \(f\) with a variable as the upper limit of integration. This shows that every continuous function has an antiderivative.
  
  Statement: Let f be a continuous function on the closed interval [a, b] and let A(x) be the area function.
  
  Then \(A'(x)=f(x)\), for all \(x ∈ [a, b]\).
  
  Then F is uniformly continuous on the closed interval [a, b] and differentiable on the open interval (a, b).
  
  \(F'(x) = f(x) \ ∀ \ x ∈(a, b)\)
  
  Here, \(F'(x)\) is a derivative function of \(F(x)\).

• Second fundamental theorem of integral calculus
  
  The second fundamental theorem of calculus states that if a function \(f\) is continuous on the closed interval [a, b] and \(F\) is an antiderivative, that is an indefinite integral of \(f\) on [a, b], then it can be expressed as follows:
  
  \(F(b)- F(a) = ∫_a ^b f(x) dx\)
  
  On the right-hand side, the integral of \(f(x)\) with respect to \(x\) is shown, where \(f(x)\) is the integrand, \(dx\) is the differential, 'a' is the upper limit, and 'b' is the lower limit.
  
  A definite integral of a function always has a unique value and can be interpreted as the limit of a sum. If the function has an antiderivative \(F\) on the interval [a, b], then the definite integral is given by the difference \(F(b) - F(a)\).

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. 

The derivatives of functions are calculated using different formulas. Children need to memorize these in order to solve complex problems. The formulas are mentioned below: 

\( \frac{d}{dx}(x^n) = n x^{n-1} \)
 

\( \frac{d}{dx}(C) = 0 \quad \text{where $C$ is a constant} \)

\( \frac{d}{dx}(C f) = C \frac{d}{dx}(f) \)

\( \frac{d}{dx}(f \pm g) = \frac{d}{dx}(f) \pm \frac{d}{dx}(g) \)

\( \frac{d}{dx}(f g) = f' g + f g' \)\( \frac{d}{dx}\left(\frac{f}{g}\right) = \frac{g \frac{d}{dx}(f) - f \frac{d}{dx}(g)}{g^2} \)

\( \frac{d}{dx}(\sin x) = \cos x \)

\( \frac{d}{dx}(\cos x) = -\sin x \)

\( \frac{d}{dx}(\tan x) = \sec^2 x \)

\( \frac{d}{dx}(\cot x) = -\csc^2 x \)

\( \frac{d}{dx}(\sec x) = \sec x \tan x \)

\( \frac{d}{dx}(\csc x) = -\csc x \cot x \)

\( \frac{d}{dx}(a^x) = a^x \ln a \)

\( \frac{d}{dx}(e^x) = e^x \)

\( \frac{d}{dx}(\log_a x) = \frac{1}{x \ln a} \)

Integrals are complementary to derivative, hence they are derived from the formula of derivative. The integrals formulas for different functions are as follows:  

\( \begin{aligned} &\int 1 \, dx = x + C \\ &\int a \, dx = a x + C \\ &\int \frac{1}{x} \, dx = \ln|x| + C \\ &\int \frac{1}{\sqrt{1 - x^2}} \, dx = \sin^{-1} x + C \\ &\int \frac{1}{\sqrt{1 + x^2}} \, dx = \tan^{-1} x + C \\ &\int \frac{1}{|x| \sqrt{x^2 - 1}} \, dx = \sec^{-1} x + C \\ &\int \sin x \, dx = -\cos x + C \\ &\int \cos x \, dx = \sin x + C \\ &\int \sec x \tan x \, dx = \sec x + C \\ &\int \csc x \cot x \, dx = -\csc x + C \\ &\int \sec^2 x \, dx = \tan x + C \\ &\int \csc^2 x \, dx = -\cot x + C \\ &\int e^x \, dx = e^x + C \\ &\int a^x \, dx = \frac{a^x}{\ln a} + C, \quad a>0, a\neq 1 \\ &\int \log x \, dx = x \ln x - x + C \\ &\int \log_a x \, dx = \frac{x \ln x - x}{\ln a} + C \end{aligned} \)

Calculus is a broad topic that includes limits, derivatives, integrals, and many more. Let’s learn some tips and tricks to make it easier for us to learn.

 

1. Solve calculus problems in different ways - Performance in math is often based on solving problems. However, with calculus, it is also important to understand the underlying principles behind each solution. 
   
    
2. Seek visual aids - For better understanding of the concept, use simple, compelling, and effective visualizations from other sources like YouTube. These are highly revered for their clear, computer generated visuals. 
   
    
3. Keep a dictionary of calculus notations and terms - This is the best way to keep track of all the definitions and notations. 
   
    
4. Don’t hesitate to ask for help - When you run into trouble while understanding a part of a problem or concept, reach out to your teacher for help.
   
    
5. Memorize the basic formulas - Practice all basic formulas to remember it accurately. It will help you solve problems effectively. 
    

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.

 

1. 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. Equations such as the Maxwell's, laws of thermodynamics, and Schrödinger are also expressed using calculus.
   
    
2. Engineering: From building bridges to developing circuits, we use calculus in different fields of engineering. For example, the change of voltage and current due to capacitor and inductor is given using differential equations.
   
    
3. Economics: Calculus is used to study the market and its behavior. To understand the minimum cost and maximum profit. Economic models such as Solow-Swan also uses differential equations.
   
    
4. Medicine: To study the growth of bacteria and infectious diseases, calculus is used. This data is used to treat cancer and diagnose patients. Different medical imaging technologies like CT scans and MRI also works on the algorithms based on calculus.
   
    
5. Computer Graphics: Calculus is widely used in making of animations, graphics, and video games. They are used to model, render curve surfaces, to program motion of objects and to determine orientation. Techniques such as Bézier curve, Non-Uniform Rational B-Splines applies calculus.

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