Calculus is the study of how things change. It provides a framework to study the change and to deduce predictions for such change. To understand calculus you need to have an understanding of two things - numbers and functions! Calculus helps us understand changes between values that are related by a function.
For example, in the study of the spread of infectious disease, we heavily rely on calculus. Three main factors are taken into account,
With these three variables, calculus can be used to determine how far and fast a disease is spreading, where it has originated from, and what is the best possible way to treat it. As rates of infection and recovery change over time, so the equations must be dynamic enough to respond to the new models evolving every day. Many of these formulas are functions of time, and one way to think of calculus is to see it as a study of functions of time.
To tackle the problem of changing quantities with respect to time, calculus has three tools:
(1) Limits, \(\lim\limits_{x \to a} f(x)\): A limit gives the value that a function approaches as that function's inputs get closer and closer to some number. Limits are tools to describe how a function approaches a value
(2) Derivatives, \(\frac{d}{dx} f(x)\): It is the rate of change of a function with respect to a variable. Derivative describes how a function changes
(3) Integral, \(\int f(x)dx\): Corresponds to summing infinitesimal pieces to find the area, volume of a continuous region. Integral derive area underneath a curve of a function
All these tools are related to one another. Derivatives are built from limits and an integral is the inverse of a derivative.
The formal study of calculus started in the 17th century by well-known scientists and mathematicians like Isaac Newton and Gottfried Leibni. It's a mathematical discipline that is primarily concerned with functions, limits, derivatives, and integrals. There are 2 different fields of calculus. The first subfield is called differential calculus. Using the concept of function derivatives, it studies the behaviour and rate of how different quantities change. Using the process of differentiation, the graph of a function can actually be computed, analyzed, and predicted. The second subfield is called integral calculus. Integration is actually the reverse process of differentiation, concerned with the concept of the anti-derivative.
When do you use calculus in the real world? It is used to create mathematical models in order to arrive at an optimal solution. For example,
- In physics, the concept of calculus is used in motion, electricity, heat, light, harmonics, acoustics, astronomy, dynamics, electromagnetism and Einstein's theory of relativity use calculus.
- In chemistry, calculus can be used to predict functions such as reaction rates and radioactive decay.
- In biology, it is utilized to formulate rates such as birth and death rates.
- In economics, calculus is used to compute marginal cost and marginal revenue, enabling economists to predict maximum profit in a specific setting.
Let us try to understand calculus using a few examples:
One of the scenarios where the solution is only in calculus is to know the rate of change of the volume of a cube with respect to the change in its sides. If \(dy\) represents the change of volume of a cube and dx represents the change of sides of the cube, then we can use the derivative form \(^{dy}/_{dx}\).Let us plot the car displacement with respect to time. The x-axis represents time and y is the displacement. Now can you find what was the speed at point (t1,y1)?
Check figure 2, if the car covers distance y2 − y1 at time interval x2 − x1 then, \(tan\theta = \ ^p/_b = (y2-y1) / (x2 -x1) = \Delta y / \Delta x\) This can also be written as a change in distance/change in time, which is speed. So any slope of a line in this graph gives speed. as \(\Delta t \) reduce we get closer to finding the instantaneous speed at a point in this graph. To find speed we need two points as speed is equal to the change in distance ∕ change in time. If you are trying to find the instantaneous speed using this formula by reducing the time interval to almost 0, then we are deriving the derivative of this function. You will learn how to derive a derivative of a function in the Derivative lesson.
So if this graph is defined as y = t2 + 2 then speed at any point of time will be 2t(derived using derivative formula). Now you can find the instantaneous speed at any point in time.
The process of finding the derivatives is called differentiation. Let, the derivative of a function be y = f(x). It is the measure of the rate at which the value of y changes with respect to the change of the variable x. It is known as the derivative of the function “f”, with respect to the variable x.
If an infinitesimal change in x is denoted as dx, then the derivative of y with respect to x is written as dy∕dx.
A car that travels at 30 km per hour. If it drives for 4 hours then the distance travelled is 30 × 4 = 120 km. But here the question is, can a car run at a constant speed of 30km∕hour? No, considering that the road will have traffic signals, bumps, and turns the speed is going to vary. So now the same problem becomes complex, for how to determine the distance travelled by car at a particular instant that was running at varying speeds?
This problem has a solution in calculus! The total displacement of the car can be found by taking the integral of the car's velocity with respect to time.
Let us consider another graph where speed is plotted with respect to time. If we want to find how much distance the car travelled in time interval t2− t1, then the distance is speed × time, which is the area below the curve between two points t1 and t2.
To derive the area we use integral calculus. If speed s is a function of time t, i.e. S = F(t) then using integral we can find the area of this portion as \(F(t) = \int s\cdot dt\) . To find the area underneath this curve we derive the integration of a function. How to do that you will learn in the integral lesson. If this graph plots the function y = x2 then the area under the curve for time t1= 1 to t2= 2 is \(\int _1^2{x^2} = \frac{x^3}{3} + C\) (where C is a constant) = 7/3
Integration is a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation, where we reduce the functions into parts. Integral is used to find the summation under a vast scale. Calculation of small addition problems can be done manually or by calculators, but for big addition problems, where the limits could reach to even infinity, integration methods are used.
A limit allows us to examine the tendency of a function around a given point even when the function is not defined at the point. Let us look at the function below.
\(f(x)=\frac{x^2−1}{x−1}\)
Since its denominator is zero when x=1, f(1) is undefined, however, its limit at x=1 exists and indicates that the function value approaches 2 there.
\(\lim\limits_{x \to1} \frac{ x^2 - 1}{x-1} =\lim\limits_{x \to1} \frac{(x+1)(x-1)}{x-1} =2\)
