Limit Comparison Test
This test works only when given series is positive and we assume a positive series for comparison .This test says:
Find = c
If c > 0 and a finite number then both series either converge or diverge together.
Please look at the following examples to understand this test better.
1)
Here we have =
Lets assume =
So =
=
=
Using L Hospital’s rule for limit we get limit as 1
So = 1 (which is a finite value)
Now lets check convergence of =
Which is a geometric series with common ratio r= 1/3 < 1
So this series convergent by geometric series test.
And hence original series is also convergent by limit comparison test.
2)
Here we have =
Lets assume =
So, =
= *
=
= (a finite value)
That means both series will either converge or diverge together.
Lets check convergence of =
Since being a harmonic series , is always divergent,
Therefore original series is also divergent by Limit comparison test.
3)
Here we have =
Lets assume =
So, =
= *
=
= 2 ( a finite number)
That means both series will either converge or diverge together.
Lets check convergence of =
Using P series test here p= 2 >1 so this series is convergent.
Therefore original series is also convergent by Limit Comparison test.
Practice problems:
Check convergence of following series using Limit comparison test.
Simplify Exponential Expressions
To learn the simplification of expressions with exponents, we should be aware about the rules of exponents first.
- = powers get added when same bases multiplied
- = powers get subtracted when same bases divided
- (=
- = having same powers, different bases get multiplied
- negative powers can be written as fractions
- = 1
There may be many ways to simplify an expression using rules of exponents and still ending up with same result (answer). Let’s understand simplification of exponential expressions using following examples.
Examples:
1) Simplify (4
(4)(3)( (write numbers and variable terms together)
12 ( (multiply terms using rules of exponents)
12
2) Simplify:
(write numbers and variables terms together)
(8) (divide the terms using rule of exponents)
3) Simplify:
(power get distributed to numerical and variable parts
625
4) Simplify:
(distributing powers to each element)
()()() (similar bases multiplied )
(( (exponents of same bases combined)
()() =
5) Simplify: ()()
()() (cross cancelling z)
()()()
()() (simplifying the powers)
()() =
Practice Problems:
Simplify using rules of exponents:
Simplify the Radical Expressions
To learn simplification of radical expressions, we first need to know some basic rules. We can write radicals using exponents as follows:
=
=
= and so on……
So basically we can write = .
Some more rules :
- = or =
= or =
To simplify nth root radicals, first we write the numerical part in form of their prime factors and then we divide the exponents by nth root (index). Whole numbers go outside radical and remainder remain inside radical. To understand this process lets see few examples.
Examples:
- Simplify :
Step (i) write numerical part (24) in form of prime factors. And cube root as exponent 1/3
Step (ii) divide the exponents by3 write whole numbers outside brackets and remainders inside.
2xy Final answer
- Simplify :
Step (i) Write nth root in exponent form and numerical part as product of exponential prime factors
Step (ii) divide all the exponent by 4 . Here remainder is 0 for each one.
(2 Final answer
Relations and Functions
What is a relation ?
Before knowing about relation we should know about ordered pair. An ordered pair consists of two objects or elements in a given fixed order like (x,y) First element is always x coordinate and second element is always y coordinate. Some other examples of ordered pair are (4,9), (-2,3) , (-6,-7) etc. All these are points. A point is always written in ordered form.
Relation: Relation is a set of inputs and outputs ,written as ordered pair (x,y). Here x in input and y is output. Formal definition of relation is written as
Let A and B be two sets .Then a relation R from A to B is a subset of AXB.
Thus R is a relation from A to B <=> R
There are many ways to represent a relation. A relation can be represented in Roster form, Set builder form, by arrow diagram or by just plotting the points.
Domain and range of a relation: Let R be a relation from set A to set B. then the set of all first components or x coordinates of ordered pairs, is called domain of R, while the set of all second components or y coordinates of all ordered pairs of R is called range of relation R.
Example: Let A={1,3,5,7} and B={2,4,6,8} and let R={(1,8),(3,6),(5,2),(1,4)} be a relation from A to B . Find its domain and range.
Answer: Domain: {1,3,5} Range: {2,4,6,8}
To check whether it is a relation or not.
Example: If A={1,2,3}, B={4,5,6} which one of them is a relation from A to B. Give reason in support of your answer. R1= {(1,4) ,(1,5),(2,5),(3,6)} R2={(2,6),(5,1),(4,2)}
Answer: R1 is a relation from A to B because R1⊆AxB But R2 is not a relation from A to B because ordered pairs (4,2) ,(5,1) AxB So R2 is not a relation from A to B.
What is a function?
A function relates an input to an output. Formal definition of a function is that a function relates each element of a set with exactly one element of other set.
A function has two properties:
i) each element in X is related to some element in Y.
ii) A function is single valued. It will not give back two or more results for the same input.
One to many is not allowed but many to one is allowed.
Example: Find which of the following relations are functions?
A={(4,6),(1,4),(2,3),(1,6)} , B= {(3,5),(-2,5),(1,3),(2,3)}
Answer: Relation A is not a function because two points (1,4) and (1,6) have same x value 1. But relation B is a function as each pair has different x values.
Example : find domain and range of following function.
f ={(0,5),(1,5),(0,4),(2,4),(1,4)}
Answer: Domain ={0,1,2}
Range = {4,5}
Vertical line test : If we draw a vertical line on the graph and it intersect the graph at more than one point then graph is not a function
Horizontal line test : If we draw a horizontal line on the graph and it intersect the graph more than once, then function is not one-one.
If a graph satisfy both vertical and horizontal line test then this is one-one function.
Domain and Range of Different Functions
Domain: Set of x values for which a function is defined, is called domain of that function. Domain of different type of functions is found differently.
While finding domain we need keep in mind the following important points:
1) There are some functions which are defined for all real numbers from – infinity to + infinity.
All polynomials , exponential functions, absolute value functions like f(x)=|x| and trigonometric functions ( sin(x) and cos(x)only) have domain from – to + .
2)Domain of logarithmic functions is (0, ) as log functions are never defined for 0 and negative values.
3) Denominator of rational functions can’t be zero.
4)Radical(sqrt) functions are never defined for negative values.
Range: Range is the output of the function. In simple words range is the resulting y values which we get after substituting all possible x values. We can substitute different x values into the function to see the kind of y values we get. Are y values always positive or always negative?
Drawing a sketch always help to know about range of a function.
1.Examples: Find domain and range of following functions.
f₁ = {(3,5),(-2,5),(1,3),(2,3)}
Domain: {-2,1,2,3}
Range: {3,5}
f₂= {(1,4),(2,3),(3,1),(0,2)}
Domain: {0,1,2,3}
Range:{1,2,3,4}
2. Find domain of polynomials.
F(x)=x^3-5x^2+2x+1
Domain : (-
Range : (-
Domain and Range of polynomials is all Real Numbers.
3)Find domain of Rational expression.
f(x)=
Rational functions are not defined when denominator is 0.So to find domain we just set denominator =0 and solve for x. Those x values are not included in domain.
X+2=0
X = -2
Domain : Interval form = (-
Set form = R-{-2} where R is the set of all real numbers.
Range : This function can never achieve 0 value as its numerator is never 0. So 0 is not included in its range.
Interval form = (-
Set form = R-{0}
4) Find domain of ration function.
f(x) =
If possible write the function in fully factored form.
Domain and Range of Different Functions
f(x) =
Set the denominator =0 and solve for x. Those x values will be excluded from its domain.
(x-4)(x-1)=0
X= 1,4
Domain: Interval form= (-
Set form = R- {1,4}
5) Find domain of given Rational function.
f(x)=
Radical functions are never defined for negative values. Therefore to find domain of radical functions, we always set radicand and solve for x.
x-2 0
x 2
Domain = [2,
Range: Output of a radical function is always positive so [0,
6) f(x)=
In this function we have two conditions , first radical can’t be negative and second denominator can’t be 0. So we just set radicand >0
(1-x)>0
1 > x
OR x < 1
Domain = (-
Range = (0, This function can never be 0 because of constant numerator and output always remain positive.
7) Find domain of given log function.
Log functions are defined for only non negative real numbers and range is set of all real numbers. So to find domain of log functions we set the function > 0 and solve for x.
f(x)=
3x+6 > 0
3x> -6
x > -2
Domain =(-2, )
Range = (-
Practice Worksheet:
Find domain of given functions.
1) f(x)= Ans: D =(-
2) f(x) = Ans:D=(
3) f(x) = Ans : D=[-3,3]
Find domain and range of given functions.
4) f(x)= Ans: D=(- R = -1
5) f(x)= 3x^2-5 Ans : D= ( R =[-5,
6) f(x)= ) Ans : D=(-4, R= ( )
Checking Convergence of Series
Let { } be the sequence of real numbers.
The expression + + +………..+ is called finite series and is denoted using sigma notation as
The expression + + +………..+ +…… is called infinite series and is denoted using sigma notation as
A series either converge or diverge and there are many tests to check convergence of different type of series.
Here we are going to study them one by one.
Applications of Derivatives.
Maxima and Minima : Differentiation is used to find maximum and minimum values of differentiable functions in their domains. There are two types of extrema(maxima & minima).
Relative (local) extrema: These are the turning points in the domain of function at which function has a value which is greater (for maxima) or smaller (for minima) then the values at its neighbouring points. There can be more than one relative maxima or minima for a function.
Absolute extrema: These extreme values are found on a closed interval. Absolute maxima is the greatest value of the function for given closed interval and absolute minima is the smallest value of function on given closed interval. It may be at critical points or at the end points of given closed interval. There can be only one abs. maxima or minima for given domain.
Difference between Relative and Absolute extrema : Difference between two types of extrema can be understood with a beautiful example here.
Assume a country having many states. Each state has an airport which serves domestic interstate flights. All these state airports are local (relative) extrema. But There is an international airport in the capital city which serves international flights connecting to other countries. This international airport is absolute extrema. There can be more than one one local maxima and minima(like many state airports) but there is only one absolute maxima and minima like international airport in capital city.