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calculus, Field of mathematics that analyzes aspects of change in processes or systems that can be modeled by functions. Through its two primary tools—the derivative and the integral—it allows precise calculation of rates of change and of the total amount of change in such a system. The derivative and the integral grew out of the idea of a limit, the logical extension of the concept of a function over smaller and smaller intervals. The relationship between differential calculus and integral calculus, known as the fundamental theorem of calculus, was discovered in the late 17th century independently by Isaac Newton and Gottfried Wilhelm Leibniz. Calculus was one of the major scientific breakthroughs of the modern era.
calculus, branch of mathematics concerned with the calculation of instantaneous rates of change (differential calculus) and the summation of infinitely many small factors to determine some whole (integral calculus). Two mathematicians, Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany, share credit for having independently developed the calculus in the 17th century. Calculus is now the basic entry point for anyone wishing to study physics, chemistry, biology, economics, finance, or actuarial science. Calculus makes it possible to solve problems as diverse as tracking the position of a space shuttle or predicting the pressure building up behind a dam as the water rises. Computers have become a valuable tool for solving calculus problems that were once considered impossibly difficult.
Calculating curves and areas under curves
The roots of calculus lie in some of the oldest geometry problems on record. The Egyptian Rhind papyrus (c. 1650 bce) gives rules for finding the area of a circle and the volume of a truncated pyramid. Ancient Greek geometers investigated finding tangents to curves, the centre of gravity of plane and solid figures, and the volumes of objects formed by revolving various curves about a fixed axis.
By 1635 the Italian mathematician Bonaventura Cavalieri had supplemented the rigorous tools of Greek geometry with heuristic methods that used the idea of infinitely small segments of lines, areas, and volumes. In 1637 the French mathematician-philosopher René Descartes published his invention of analytic geometry for giving algebraic descriptions of geometric figures. Descartes’s method, in combination with an ancient idea of curves being generated by a moving point, allowed mathematicians such as Newton to describe motion algebraically. Suddenly geometers could go beyond the single cases and ad hoc methods of previous times. They could see patterns of results, and so conjecture new results, that the older geometric language had obscured.
For example, the Greek geometer Archimedes (287–212/211 bce) discovered as an isolated result that the area of a segment of a parabola is equal to a certain triangle. But with algebraic notation, in which a parabola is written as y = x2, Cavalieri and other geometers soon noted that the area between this curve and the x-axis from 0 to a is a3/3 and that a similar rule holds for the curve y = x3—namely, that the corresponding area is a4/4. From here it was not difficult for them to guess that the general formula for the area under a curve y = xn is an + 1/(n + 1).
Calculating velocities and slopes
The problem of finding tangents to curves was closely related to an important problem that arose from the Italian scientist Galileo Galilei’s investigations of motion, that of finding the velocity at any instant of a particle moving according to some law. Galileo established that in t seconds a freely falling body falls a distance gt2/2, where g is a constant (later interpreted by Newton as the gravitational constant). With the definition of average velocity as the distance per time, the body’s average velocity over an interval from t to t + h is given by the expression [g(t + h)2/2 − gt2/2]/h. This simplifies to gt + gh/2 and is called the difference quotient of the functiongt2/2. As h approaches 0, this formula approaches gt, which is interpreted as the instantaneous velocity of a falling body at time t.
This expression for motion is identical to that obtained for the slope of the tangent to the parabola f(t) = y = gt2/2 at the point t. In this geometric context, the expression gt + gh/2 (or its equivalent [f(t + h) − f(t)]/h) denotes the slope of a secant line connecting the point (t, f(t)) to the nearby point (t + h, f(t + h)) (see figure). In the limit, with smaller and smaller intervals h, the secant line approaches the tangent line and its slope at the point t.
Thus, the difference quotient can be interpreted as instantaneous velocity or as the slope of a tangent to a curve. It was the calculus that established this deep connection between geometry and physics—in the process transforming physics and giving a new impetus to the study of geometry.
Differentiation and integration
Independently, Newton and Leibniz established simple rules for finding the formula for the slope of the tangent to a curve at any point on it, given only a formula for the curve. The rate of change of a function f (denoted by f′) is known as its derivative. Finding the formula of the derivative function is called differentiation, and the rules for doing so form the basis of differential calculus. Depending on the context, derivatives may be interpreted as slopes of tangent lines, velocities of moving particles, or other quantities, and therein lies the great power of the differential calculus.
An important application of differential calculus is graphing a curve given its equation y = f(x). This involves, in particular, finding local maximum and minimum points on the graph, as well as changes in inflection (convex to concave, or vice versa). When examining a function used in a mathematical model, such geometric notions have physical interpretations that allow a scientist or engineer to quickly gain a feeling for the behaviour of a physical system.
The other great discovery of Newton and Leibniz was that finding the derivatives of functions was, in a precise sense, the inverse of the problem of finding areas under curves—a principle now known as the fundamental theorem of calculus. Specifically, Newton discovered that if there exists a function F(t) that denotes the area under the curve y = f(x) from, say, 0 to t, then this function’s derivative will equal the original curve over that interval, F′(t) = f(t). Hence, to find the area under the curve y = x2 from 0 to t, it is enough to find a function F so that F′(t) = t2. The differential calculus shows that the most general such function is x3/3 + C, where C is an arbitrary constant. This is called the (indefinite) integral of the function y = x2, and it is written as ∫x2dx. The initial symbol ∫ is an elongated S, which stands for sum, and dx indicates an infinitely small increment of the variable, or axis, over which the function is being summed. Leibniz introduced this because he thought of integration as finding the area under a curve by a summation of the areas of infinitely many infinitesimally thin rectangles between the x-axis and the curve. Newton and Leibniz discovered that integrating f(x) is equivalent to solving a differential equation—i.e., finding a function F(t) so that F′(t) = f(t). In physical terms, solving this equation can be interpreted as finding the distance F(t) traveled by an object whose velocity has a given expression f(t).
The branch of the calculus concerned with calculating integrals is the integral calculus, and among its many applications are finding work done by physical systems and calculating pressure behind a dam at a given depth.
What is Calculus?
Calculus focuses on some important topics covered in math such as differentiation, integration, limits, functions, and so on. Calculus, a branch of mathematics, deals with the study of the rate of change, was developed by Newton and Leibniz.
Calculus Definition: Calculus in Mathematics is generally used in mathematical models to obtain optimal solutions and thus helps in understanding the changes between the values related by a function. Calculus is broadly classified into two different sections:
- Differential Calculus
- Integral Calculus
Both differential and integral calculus serve as a foundation for the higher branch of Mathematics known as “Analysis”, dealing with the impact of a slight change in dependent variable, as it leads to zero, on the function.
Based on the complexity of the concepts covered under calculus, we classify the topics under different categories as listed below,
- Calculus 1
- Calculus 2
Functions in calculus represent the relationship between two variables, which are the independent variable and the dependent variable. Let’s consider the following diagram.
There is an INPUT, a black box, and an OUTPUT. For example, suppose we want to make a pizza. We would need the following basic ingredients.
- Pizza Base
- Pizza Sauce
The above real-life example can be represented in the form of a function as explained below,
The taste of our pizza depends on the quality of the ingredients. Let’s take another example
Suppose that: y = x2
|Value of x||Value of y|
Using the diagram given above, we get:
We can see that the value of y depends on the value of x. We can conclude that
- INPUT is independent of the OUTPUT
- OUTPUT depends on the INPUT
- Black Box is responsible for the transformation of the INPUT to the OUTPUT
- INPUT is an independent variable
- OUTPUT is a dependent variable
- Black Box is a function
Differential calculus focuses on solving the problem of finding the rate of change of a function with respect to the other variables. To find the optimal solution, derivatives are used to calculate the maxima and minima values of a function. Differential helps in the study of the limit of a quotient, dealing with variables such as x and y, functions f(x), and the corresponding changes in the variables x and y. The notations dy and dx are known as differentials. The process used to find the derivatives is called differentiation. The derivative of a function, y with respect to variable x, is represented by dy/dx or f’(x).
Limit helps in calculating the degree of closeness to any value or the approaching term. A limit is normally expressed using the limit formula as,
limx→cf(x) = A
This expression is read as “the limit of f of x as x approaches c equals A”.
Derivatives represent the instantaneous rate of change of a quantity with respect to the other. The derivative of a function is represented as:
limx→h[f(x + h) − f(x)]/h = A
A function f(x) is said to be continuous at a particular point x = a, if the following three conditions are satisfied –
- f(a) is defined
- limx→af(x) exists
- limx→a− f(x) = limx→a+ f(x) = f(a)
Continuity and Differentiability
A function is always continuous if it is differentiable at any point, whereas the vice-versa for this condition is not always true.
Integral calculus is the study of integrals and the properties associated to them. It is helpful in:
- calculating f from f’ (i.e. from its derivative). If a function, say f is differentiable in any given interval, then f’ is defined in that interval.
- calculating the area under a curve for any function.
Integration is the reciprocal of differentiation. As differentiation can be understood as dividing a part into many small parts, integration can be said as a collection of small parts in order to form a whole. It is generally used for calculating areas.
A definite integral has a specific boundary or limit for the calculation of the function. The upper and lower limits of the independent variable of a function are specified. A definite integral is given mathematically as,
∫ab f(x).dx = F(x)
An indefinite integral does not have a specific boundary, i.e. no upper and lower limit is defined. Thus the integration value is always accompanied by a constant value (C). It is denoted as:
∫ f(x).dx = F(x) + C
Calculus Formulas can be broadly divided into the following six broad sets of formulas. The six broad formulas are limits, differentiation, integration, definite integrals, application of differentiation, and differential equations. All of these formulas are complementary to each other.
Limits Formulas: Limits formulas help in approximating the values to a defined number, and are defined either to zero or to infinity.
Ltx→0 (xn – an)(x – a) = na(n – 1)
Ltx→0 (sin x)/x = 1
Ltx→0 (tan x)/x = 1
Ltx→0 (ex – 1)/x = 1
Ltx→0 (ax – 1)/x = logea
Ltx→0 (1 + (1/x))x = e
Ltx→0 (1 + x)1/x = e
Ltx→0 (1 + (a/x))x = ea
Differentiation Formula: Differentiation Formulas are applicable to basic algebraic expressions, trigonometric ratios, inverse trigonometry, and exponential terms.
Integration Formula: Integrals Formulas can be derived from differentiation formulas, and are complimentary to differentiation formulas.
∫ xn.dx = xn + 1/(n + 1) + C
∫ 1.dx = x + C
∫ ex.dx = ex + C
∫(1/x).dx = log|x| + C
∫ ax.dx = (ax/log a) + C
∫ cos x.dx = sin x + C
∫ sin x.dx = -cos x + C
∫ sec2x.dx = tan x + C
∫ cosec2x.dx = -cot x + C
∫ sec x.tan x.dx = sec x + C
∫ cosec x.cotx.dx = -cosec x + C
Definite Integrals Formulas: Definite Integrals are the basic integral formulas and are additionally having limits. There is an upper and lower limit, and definite integrals, that are helpful in finding the area within these limits.
∫ba f'(x).dx = f(b) – f(a)
∫ba f(x).dx = ∫ba f(t).dt
∫ba f(x).dx = – ∫ab f(x).dx
∫ba f(x).dx = ∫ca f(x).dx + ∫bc f(x).dx
∫ba f(x).dx = ∫ba f(a + b – x).dx
∫a0 f(x).dx = ∫a0 f(a – x).dx
∫2a0 f(x).dx = 2∫a0 f(x).dx
∫a-a f(x).dx = 2∫a0 f(x).dx, f is an even function
∫a-a f(x).dx = 0 , f is an odd function
Application of Differentiation Formulas: The application of differentiation formulas is useful for approximation, estimation of values, equations of tangent and normals, maxima and minima, and for finding the changes of numerous physical events.
dy/dx = (dy/dt)/(dx/dt)
Equation of a Tangent: y – y1 = dy/dx.(x – x1)
Equation of a Normal: y – y1 = -1/(dy/dx).(x – x1)
Differential Equations Formula: Differential equations are higher-order derivatives and can be comparable to general equations. In the general equation, we have the unknown variable ‘x’ and here we have the differentiation dy/dx as the variable of the equation.
Homogeneous Differential Equation: f(λx, λy) = λnf(x, y)
Linear Differential Equation: dy/dx + Py = Q
General solution of Linear Differential Equation is y.e– ∫P.dx = ∫(Q.e∫P.dx).dx + C
Applications of Calculus
Calculus is a very important branch, a mathematical model that helps in:
- Analyzing a system to find an optimal solution to predict the future of any given condition for a function.
- Concepts of calculus play a major role in real life, either it is related to solve the area of complicated shapes, evaluating survey data, the safety of vehicles, business planning, credit card payment records, or finding the changing conditions of a system that affect us, etc.
- Calculus is a language of economists, biologists, architects, medical experts, statisticians. For example, Architects and engineers use different concepts of calculus in determining the size and shape of the construction structures.
- Calculus is used in modeling concepts like birth and death rates, radioactive decay, reaction rates, heat and light, motion, electricity, etc.
Solved Examples on Calculus
Example 1: Let f(y) = y2 and g(y) = ey. Use the chain rule in calculus to calculate h′(y) where h(y) = f(g(y)).
Given, f(y) = y2 and g(y) = ey. First derivative above functions are f'(y) = 2y and g'(y) = ey
To find: h′(y)
Now, h(y) = f(g(y))
h'(y) = f'(g(y))g'(y)
h'(y) = f'(ey)ey
By substituting the values.
h'(y) = 2ey × ey
or h'(y) = 2e2y
Example 2: y is a function of x, and the function definition is given as: y = f(x) = 1/(1 + x2). Find the output values of the function for x = 0, x = −1, and x = √2.
f(0) = 1/[1 + (0)2] = 1/1 = 1
f(−1) = 1/[1 + (−1)2] = 1/2
f(√2) = 1/[1 + (√2)2] = 1/3
Example 3: Find the slope of the curve y = 2x2 + 3x + 1, at the point (-1, 0)
The given equation of the curve is y = 2x2 + 3x + 1.
The aim is to find the slope of the curve and hence we need to differentiate the equation.
(d/dx).y = d/dx. (2x2 + 3x + 1)
dy/dx = 4x + 3
Slope at at point (-1, 0) is m = 4(-1) + 3 = -1
Answer: Hence the slope of the curve is -1
Example 4: Differentiate x tan x.
d/dx. x tanx = x.( d/dx) tan x + tan x.(d/dx) x
Using calculus formula,
d/dx. x tan x = x.sec2x + tan x.1
= x sec2x + tan x
Answer: (d/dx) x tan x = x sec2x + tan x
FAQs on Calculus
What is Calculus?
Calculus is one of the most important branches of mathematics, that deals with continuous change. Calculus is also referred to as infinitesimal calculus or “the calculus of infinitesimals”. Infinitesimal numbers are the quantities that have values nearly equal to zero, but not exactly zero.
What are Important Calculus Formulas?
A few of the important formulas used in calculus to solve complex problems are as listed below,
- Ltx→0 (xn – an)(x – a) = na(n – 1)
- ∫ xn.dx = xn + 1/(n + 1) + C
- ∫ ex.dx = ex + C
What is Differential Calculus?
Differential calculus focuses on solving the problem of finding the rate of change of a function with respect to the other variables. The derivative of a function, y with respect to variable x, is represented by dy/dx or f’(x).
What do you Use Calculus for in Real Life?
Concepts of calculus play a major role in real life, either it is related to solve the area of complicated shapes, evaluating survey data, the safety of vehicles, business planning, credit card payment records, or finding the changing conditions of a system that affect us, etc. Calculus is a language of economists, biologists, architects, medical experts, statisticians. For example, architects and engineers use different concepts of calculus in determining the size and shape of construction structures.
How to Find the Maxima and Minima of a Function in Calculus?
To find the optimal solution, derivatives are used to calculate the maxima and minima values of a function. Maxima and minima are the highest and lowest points of a function respectively, which could be determined by finding the derivative of the function.
What are the Four Main Concepts of Calculus?
The four main concepts covered under calculus are as given below,
- Differential Calculus
- Integral Calculus
- Multivariable Calculus
What is Integral Calculus?
Integral calculus is the study of integrals and the properties associated with them. It is helpful in calculating f from f’ (i.e. from its derivative). If a function, say f is differentiable in any given interval, then f’ is defined in that interval. It enables us to calculate the area under a curve for any function.