8. Limits, Differentiation(1).

After discussing variables, functions, and graphs we proceed to talk about a somewhat esoteric mathematical topic namely ‘Calculus‘. Calculus is regarded as a difficult concept to comprehend. This is because the study of calculus requires imagining abstract concepts. We are going to learn the basics of this subject in a simplified way.

In Chemistry, we come across derivatives and their integrals several times (e.g. while studying thermodynamics, chemical kinetics), but most of the times we fail to understand the purpose of using these mathematical methods in our study.From this post onwards , we shall discuss the concept of calculus and how it help us in our study of Chemistry.

CALCULUS.

Calculus is the greatest achievement of the human mind!

Richard Feynman.

What is calculus?

Calculus is a mathematical concept used to study the continuous change in a parameter. It deals with rates of change. Thus, calculus is a concept of movement.

Till now we studied variables, functions, graphs, and equations- all of which are static mathematical concepts. However, in any science, we have quantities that keep varying. For example, the rate of a reaction, radioactive decay etc. Calculus helps us study these changes.

Thus basically, calculus is the study of how quantities change. Sometimes our experimental data values do not change regularly. They might increase or decrease abruptly. In such cases, studying the changes over small intervals of time gives us a more accurate understanding of the variation of the data. Calculus helps us study these ‘instantaneous’ changes – changes over small intervals of time.

Let us consider an example. Suppose we are studying the stock/equity market trends. In the stock market, the prices of the shares go up and down quite often. Thus, there is no regular change in the prices. The concept of calculus helps us study such fluctuating data.

Calculus helps us study two questions –

Q1. How steep a function is at a point?
Q2. What is the area on a graph under that region?

The first question is answered by a concept of ‘Differentiation’ and the second by ‘Integration’.

Thus calculus are of the following two types –

i) DIFFERENTIAL CALCULUS– Differentiation, finding Derivatives – this tells us the steepness of the graph at a point/rate of change of function at a point.

ii) INTEGRAL CALCULUS – Integration – gives area under a particular region.

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Differentiation

The process of finding the derivative or rate of change of a function is called differentiation. While differentiating, the measurements are broken down into small time intervals. A differential equation represents the relationship between a continuously varying quantity and its rate of change.

Derivative ⇒ instantaneous rate of change of a quantity ⇒ measurements in small time intervals.

Let us try to understand this concept with a hypothetical example.

Consider a point A on the graph below-

If we keep picking closer and closer points and computing the slopes of these lines, we start getting closer to the tangent line at point A. The values of the approximations for the slopes start to approach a number. This number is called a ‘LIMIT’.

I found this video helpful in understanding the concept of derivatives – https://youtu.be/w3GV9pumczQ?si=g4yMXNM-LdkVSOxw

LIMITS

Learning the concept of limits is crucial to the understanding of calculus. ‘Limits’ is a simple concept.

LIMITS give us an idea of how a function behaves near a point, instead of at that point.

As the example above shows, we choose points near point A on the curve and draw lines joining the two points. The slope of these lines is a good approximation of the steepness of the slope at point A. As we keep moving closer to point A, the calculated values get closer to the exact value of ‘the slope of the tangent‘ at A. Thus, we study different slope values near point A and check where the output leads us. The number at which the values approach is the LIMIT. This is the procedure for finding the limit of any function.

We designate the limit of a function f(x) as –

This is read as the ‘limit of f of x as x approaches a‘.

Let us consider a few examples to elucidate this concept better.

EXAMPLE 1 –

To find the limit for this function, let us choose values close to 2 and solve the function.

x	f(x) = x2
1.6	2.56
1.7	2.89
1.9	3.61
1.97	3.8809
1.99	3.9601
1.9999	3.99960001

As seen in the table above, as the value of x approaches 2, the value of the function x² approaches 4. Thus, the limit of this function is 4. We donate this as follows-

In this case, we can directly substitute x=2 in the function and find its limit.

f(x)= x² .
When x=2, x² = 4

However, this substitution method cannot be used for all functions. Let us consider an example where this method will NOT work.

EXAMPLE 2 – Consider the following function –

For the above function, any value of x other than 1, will give us the answer one. However, for x=1, the denominator will be zero (1-1=0). Any number divided by zero is an undefined quantity. Thus the function would be undefined at x=1.

Thus, in this kind of function, the substitution method will NOT work for all values of x. Thus, for this function, we can get infinitely close to 1 but the value of x cannot be 1. The limit for this function can be denoted as –

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A differential equation contains one or more terms involving derivatives of one variable (dependent variable, y) with respect to another variable (independent variable, x).

e.g. dy/dx = 2x is a differential equation.

What is the difference between an algebraic equation and a differential equation?
The solution/s for an algebraic equation are numbers or values. The solution for differential equations is a function or set of functions.

e.g.- y= x² is a function relating x & y^.The differential equation for this function is

y’ = f’(y)=  dy/dx = 2x

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Notation for differentiation

The first derivative of a function denotes the rate of change of the function. The second derivative denotes the rate of change of the first derivative.

e.g.- If x is the displacement. Then,

First Order Derivatives are denoted as  ⇒ y’ , f’(y) or  dy/dx       
Second order Derivatives are denoted as  ⇒ y”, f”(y) or  d²y/dx² 

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We can calculate derivatives with a derivative calculator available online. The link is –http://www.derivative-calculator.net.  One can just plug in the value to find the first, and second-order derivative and see the corresponding graph. Students of Chemistry can very well use this for our convenience as we are not expected to know the mathematical details of this concept.

We need to know some basic derivative rules though and they are as follows –

1) The derivative of a constant is zero.

   e.g. If c is a constant then dc/dt  = 0.                                          

2) If y is a function of the type y = xⁿ then,

     y’ = f’(y) = dy/dx = n x ^(n-1).

   e.g.  If y = x³ then y’ = 3x² .
If y= x ^-3 then y’ = -3x^-4.

3) Addition rule –

If y=f(x) + g(x) then y’=f’(x) + g’(x) i.e  d/dx  [f(x) +g(x)] = df/dx + dg/dx .
                                                                           

4)Multiplication rule –

If y=f(x).g(x) then y’=f’(x).g(x) + f(x).g’(x) and
If y= c. f(x) where c= constant then,
    y’= c. f’(x)  i.e    y’=dy/dx = c .df/dx.

5) If y = ln x then dy/dx = 1/x

6) If y = e^x then dy/dx =  e^{x .}

 7)If y = sin x then y’ = cos x
     If y = cosx then y’ = – sinx
If y= tan x then y’ = 1/cos2x .

In my next post we shall try to understand in greater depth the concept of derivatives. Till then ,

Be a perpetual student of life and keep learning …

Good day!

References and Further Reading –

1)http://www.maths.surrey.ac.uk/explore/vithyaspages/differential.html

2)http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/slope.html

3)http://www-math.mit.edu/~djk/calculus_beginners/chapter01/section02.html

4)https://www.math10.com/en/algebra/derivative-function.html

5)https://math.dartmouth.edu/opencalc2/cole/lecture8.pdf

6)https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/differential-equations-intro/v/differential-equation-introduction