Calculus - Limits and Continuity

The concept of Limitlays the foundation for the popular branch of Mathematics called Calculus. Calculusinvolves the analysis of functions and their behaviour. To study the behaviour of functions one needs to have a good hold on the fundamental concepts of Limit and Continuity. Calculus Limits and Continuity is the language of science and engineering.Limits and Continuity in Calculus have led to the development of ideas like derivative, integration etc.

Consider a function f(x):R->R and ‘a’ is any point in the domain of the function, limit of the function f(x) exists at x = a, if f(x) = f(a), where x is arbitrarily close to a. The idea is that as we approach the point x = a on the real line, f(x) approaches f (a). The limiting value of the function when x is very close to ‘a’ is ‘f (a)’. In the language of mathematics it is commented as “lim x->a, f(x) =f(a)”.

Continuity is another important concept which is based on the concept of limits. The function f(x) is said to be continuous at x = a, if the limit of the function at x = a on approaching the point x = a from both sides is equal to the value of the function at x = a. Precisely this can be stated as – if  lim h->0, f(a+h) = f(a-h)=f(a), then f(x) is continuous at x = a. Mathematically this can be stated as – a function is said to be continuous at a point if the right hand limit and the left hand limit of the function at that point is equal to the value of the function at that point. Geometrically speaking, a continuousfunction is the one which can be sketched on paper without lifting the pen even once.


 Properties of Continuous Functions:There are two important results for continuous functions which are stated in the form of theorems-
• Intermediate value theorem: The intermediate value theoremstates that if f is a real valued continuousfunctionon the closed interval [a, b] andt is any number between f(a) and f(b), then there exists a numberc in [a, b] such that f(c) = t.Example: The height of a child increases from 1 m to 1.7 m between the ages of eight and sixteen years, then, at some point of timebetween eight and sixteen years of age, the child must have had a height of 1.5 m.
• Extreme value theorem: The extreme value theorem states that if f is continuousreal valued function on [a,b], then f has a maximumvalue in [a,b], i.e. there exists some c in [a,b] such that f(c) >= f(x)  for all x in  [a,b]. The same holds for the minimum of f(x). For example, consider the function sin(x) where x lies between [0,2*pi]. This function attains at maximum at x = pi/2 and a minimum at x = 3*pi/2.

Derivatives of Exponential Functions with Trignometric function power

What are exponential functions? A mathematical function which is in the form, f(x) = a^x is called an exponential function, here x is a variable and a is a constant which is the base of the function (a is greater than zero but does not equal one). The most commonly used exponential function is e which is the natural exponential function which is denoted as e^x. Some definitions of e are
• e=lim(n?inifinity)[1+1/n]^n
• Lim(h?0)[e^h-1]/h=1, where e is a unique positive number
• e=summation(n=0 to infinity)[1/n(factorial)]
We know that derivative is defined as, f’(x) = lim(h?0)[f(x+h) – f(x)]/h
Using the above, we can find the derivative of the natural exponential function f(x) = e^x
d[e^x]/dx = lim(h?0)[e^(h+x) – e^x]/h
    =lim(h?0)[e^h.e^x – e^x]/h  [using rule of exponents]
    =lim(h?0)e^x[e^h-1]/h            [taking e^x common]
    =e^x lim(h?0)[e^h-1]/h
    =e^x .1
    = e^x
As we observe, e^x is its own derivative. Let us find the derivative of the exponential function with trigonometric function as the power.

E-sinx
We have, derivative of e^u is given by e^u.du/dx. So, the derivative of E-sinx written as e^-sinx would be d[e^-sinx]/dx. Taking –sinx = u which gives du/dx = -cosx, we get, e^u. du/dx which equals e^-sinx.(-cosx)
Finally we get the derivative of E-sinx as –cosx.e^-sinx

E –sinx
We have, y= e^-sinx
     y= e^u where u= -sinx, derivative of u= du/dx = -cosx
taking derivative, we get,
dy/dx= dy/du. du/dx
          = e^-sinx. [-cosx]
          = -cosx.e^-sinx

E^-sinx
Let us find the derivative of the exponential function with the trigonometric function –sinx as the power.
Using the chain rule, we get,
d[e^(-sinx)]/dx = d[e^(-sinx)/dx. d[-sinx]/dx
             = e^(-sinx). (-cosx)
             = -cosx.e^(-sinx)

Basics of exponential functions

An exponential function is a function of the form y = a^x where a belongs to positive real numbers and x is any real number. We shall first try to understand such numbers with the help of graphs.

Exponential function graph:
Let us try to graph exponential function y = 2^x. So here we see that a = 2 (which is a positive real number and x is any real number). To obtain certain number of points on the graph we construct the following table:

Note that both the above functions are not inverse of each other. With this understanding, let us now define an exponential function.

Definition: Let a belongs to R+. Then the function f: R ->R+, f(x) = a^x is called an exponential function. a is the base of the function. The corresponding exponential function equation would be: y = a^x
For example:
f(x) = 3^x, g(x) = (1/4)^x (where x belongs to R), h(x) = 1^x, (where x belongs to R), are all exponential functions.

Natural exponential function:
We are familiar with the irrational number pi which we come across in connection with the area and circumference of a circle. There is another important irrational number, which is denoted by e and which lies between 2 and 3. Its approximate value is 2.71828. Exact value of e is given by the sequence: e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + ....An exponential function to the base e, f:R->R+, f(x) = e^x is called the natural exponential function. This function is very often used in study of various branches of science and math.

Inverse of exponential function:
Inverse of an exponential function is a logarithmic function. In other words, the exponential function and the logarithmic function are inverses of each other. Thus, if f:R->R+, f(x) = a^x, a belongs to R+ -{1}, then f^(-1):R+->R, f^(-1)(x) = log(a)x [read that as log of x base a]

Derivative of exponential function:
Based on limit definition of derivatives, the derivative of an exponential function can be shown as follows:
If y= a^x, then dy/dx = (a^x)Ln(a)

Trigonometry Made Simple

Introduction to Trigonometry:
Trigonometry is a derived from a Greed word ‘tri’ (meaning three) and ‘gon’ (meaning sides and ‘metron’ (meaning measure). Trigonometry is the study of relationships between the sides and angles of a triangle. The earliest known work on Trigonometry was recorded in Egypt and Babylon. Early Astronomers used it to find out the distances of the stars and planets from the Earth. Even today, most of the technologically advanced methods used in Engineering and Physical Sciences are based on Trigonometric concepts.

The Trigonometric Ratios of the angle A in right triangle ABC are defined as:
Sine of angle A = (side opposite to angle A)/hypotenuse; cosec = 1/Sine
Cosine of angle A = (side adjacent to angle A)/hypotenuse; sec = 1/cosine
Tangent of angle A = (side opposite to angle A)/(side opposite to angle A); cot =1/tan
The trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the lengths of its sides

Trigonometry Problems and Answers:
Problem: Given angle A=51 degrees, Adjacent side length = x and opposite side length = 10. Find x and H, hypotenuse of the triangle
Answer: tan(A) = opposite side/adjacent side
tan(51)= 10/x
X = 10/(tan 51) = 8.1 (two significant digits)
Sin(A) = opposite side/hypotenuse
Sin(51) = 10/H
H = 10/ Sin(51) = 13 (two significant digits)
x=8.1 and H=13

Problem: If sin 3A = cos (A- 26 degrees), where 3A is an acute angle, find the value of A
Answer:  sin 3A can be written as cos (90-3A)
      So, we get            cos (90-3A) = cos(A-26degrees)
Since both 90-3A and A-26 are both acute angles,
90 – 3A = A- 26
4A = 116
A = 29 degrees

Problem:  An observer 1.5m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from his eyes is 45 degrees. Calculate the height of the chimney.
Answer: Let us draw a rough triangle ABC with the right angle at B. let us draw a line DE parallel to BC such that AB (AE+EB) will be the height of the chimney,  CD (equal to BE) the observer and Angle ADE, the angle of elevation.  Here, ADE is the right triangle, right-angled at E
We have, AB = AE +EB = AE +1.5 DE = CB = 28.5 (distance from the chimney)
Let us use the tangent of the angle of elevation
tan(45degrees) = AE/DE
1 = AE/28.5
AE= 28.5
Height of the chimney = AE +EB =28.5 +1.5 = 30 m

Parallelogram Definition

Let us learn about the Parallelogram Definition general, the diagonals of a parallelogram are having different lengths. The two diagonals in the figure which intersects at a particular point and lie in the interior part of parallelogram. When two pairs of the sides are opposite and they are parallel to each other.Then it is called as parallelogram .Now let us see about the parallelograms sides introduction.In parallelograms introduction, we can draw a pair of parallel lines. Draw another pair of parallel lines intersecting the former.Thus the parallelogram can be formed.Thus we can say that the pair of opposite sides of parallelogram is of equal length. Similarly we can also learn about other topics such as types of lines.Hope you like the above example of Parallelogram Definition.Please leave your comments, if you have any doubts.