Calculus: The Mathematics of Change (Part-II)

By Nosherwan Khan (Team Research)

This is a continuation of the post, Calculus: The Mathematics of Change

Differentiation

It’s easy to find the slope of a linear function. Suppose y = x + 2 we know that its slope is 1 and it’s inclined at an angle of 45 degrees. If you know the slope of a function you can find its steepness which is very useful but the problem began when the need was to find the slope of a curve. If you want to find the slope of a straight line, all you have to do is select two points on that line and make a right angled triangle and calculate rise/run. But how do you do that for a curve? You cannot find the slope with this method. This was the dominant mathematical question of the early seventeenth century and it is hard to deduce how badly the scientists of the day wanted to know the answer. In optics, the tangent determined the angle at which the ray of light entered a curved lens. In mechanics, the tangent determined the direction a body’s motion at every point along its path. In geometry, the tangents to two curves at a point of intersection determined the angle at which the curves intersected. Descartes went so far as to say that the problem of finding a tangent to a curve was “the most useful and most general problem not only that I know but even that I have any desire to know”.

Suppose you are going in car and you traveled from your city to your friend’s city. You want to calculate your average speed of the whole journey. What you do is that you take the total distance travelled by you and divide it by the total time it took you to reach there. But what if you wanted to know your speed at any instant during the journey? In your journey, you might have stopped at some places and at some places you might have travelled with very high speed or at very low speed. How will you know your speed at any instant?

If you’re a physicist and you want to know the speed of a ball hit by a bat; you can find the average speed very easily but how will you find its instantaneous speed? Or the speed at any instant along its motion?

The solution to above questions is differentiation.

Differentiation can be used in physics, economics, medicine, and computer science to calculate velocities and acceleration, to explain the behavior of machinery, to estimate the drop in water levels as water is pumped out of a tank and to predict the consequences of making errors in measurements.

Remember the problem of finding tangent to a curve? The answer is to find derivative at that point. What does it mean?

Suppose your function is y = x² .You want to find the slope at point (2,0). What  do you do?

Applying the differential operator;

Using the rules of differentiation we have,

So the slope is 2x or if you want to find the value of slope at (2,0) just plug in 2 for the value of x, You get 2(2) = 4.

Tangent to a curve gives the slope at that point. If you the know slope – that means you know the rate of change which can be very helpful in understanding about the curve.

Derivative has very important applications, not only in mathematics but also in science, engineering, physics, etc.

Derivatives can be very helpful in studying:

1. Rate of change of any quantity
2. Increasing or decreasing functions and their behavior
3. Tangent and normal to a curve
4. Minimum and maximum values

Another way to look at derivatives is by thinking of it as the rate of change of one quantity with respect to another quantity.

Example: velocity is derivative of displacement with respect to time

Acceleration is derivative of velocity with respect to time or we can say acceleration is double derivative of displacement with respect to time.

Now the most beautiful application of derivatives:

Maximum and Minimum Values

If you are asked to find the maximum profit your company will have or the minimum loss your company has to bear, what will you do?

Profit and loss vary with time. Hence we can use the concept of derivative and some other terms.

1. Maxima: The point on a graph where the graph gives the maximum output value
2. Minima: The point on a graph where the graph gives the minimum output value

So every company has some mathematical models with which they work. If you take those mathematical models or equations and you differentiate them and set that differentiated value equal to 0. You can find the values where you will get maximum or minimum value of your model. Beautiful, isn’t it? You can have infinite values for your functions, yet you found it so easily with few steps to find the maximum and minimum value for your model. That’s the power of calculus.

Still not impressed?

Go on and find the slope of the function y = – ln(cosx) at point (0.25π,0)

You have a negative sign involved, cosine function, natural log. With differentiation it’s a piece of cake to find slopes of such difficult graphs.

Integration

Integration, also called as anti-derivative is the reverse process of differentiation. If you differentiate or take derivative of a function f(x), you will get a function g(x). Now if you integrate the function g(x), you get the previous function f(x). What does this mean graphically? Well differentiation gives you the slope or rate of change of a quantity or function. Integration gives you the area under the curve made by that function. Let’s take the same function y = x². Now if someone asks you that what will be the area bounded by the curve y = x² from 0 to 2. All you have to do is integrate the function x² and you will get the function x³/ 3 ( by rules of integration) and then put in the limits and you will get the area in that region.

There are two types of integration.

1. Indefinite integration: where limits are not given

2. Definite Integration: where limits are given.

If you want to find the area under a curve, you use definite integration where you are given limits and at the end you get a numerical value. If you want to find the function, which when differentiated would give the given function, you do indefinite integration.

Example of indefinite integration:

What is the integration of x³? (What is the function which when differentiated would give x³.)

So integral x³ will give you x⁴/4.

Now how is it useful?

The early wonders of calculus were that we could predict the future position of a moving body from one of its known points and formula for its velocity function. Today this is viewed as one of a number of times on which we determine a function from one of its known values and a formula for its rate of change. It is a daily process today, thanks to calculus. To calculate how fast a space vehicle needs to be going at a certain point to escape the earth’s gravitational field or to predict the useful life of a sample of radioactive uranium-235 from its present level of activity and its rate of decay. Finding the area or volumes can be done very easily through integral calculus.

Integration is like filling a tank from a tap.

The input (before integration) is the flow rate from the tap.

Integrating the flow (adding up all the little bits of water) gives us the volume of water in the tank.

Integration: With a flow rate of 1, the tank volume increases by x

Derivative: If the tank volume increases by x, then the flow rate is 1

This shows that integrals and derivatives are opposites!

Now for an increasing flow rate;

Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap)

As the flow rate increases, the tank fills up faster and faster.

Integration: With a flow rate of 2x, the tank volume increases by x²

Derivative: If the tank volume increases by x², then the flow rate must be 2x.

Integration in Physics

We all know that if we differentiate velocity function, we get acceleration function. Now what if we wanted to find the area under acceleration time graph? That can be done easily with the help of integration.

Suppose we have a function, y = v(t) = t³. And we want to find the acceleration. We will differentiate velocity function which will give 3t². Now if we integrate 3t² we get the previous function t³. So we can find the area bounded by the curve 3t².

That’s why we say area under acceleration time graph gives us change in velocity or the area covered under that graph.

We can do the same to calculate area bounded by velocity time graph which would give us displacement.

These were just a few important concepts and applications of calculus. There are thousands of other uses of the beautiful field of mathematics; the language of the universe.