Function, Limit, Continuity PDF 2020

Departmental Library I Department of Applied Mathematics Semester The Maharaja Sayajirao University of Baroda,Vadodara...

Departmental Library I Department of Applied Mathematics Semester The Maharaja Sayajirao University of Baroda,Vadodara 2020 Polytechnic Since 2011 Subject: Applied Mathematics-I Function, Limit and Continuity  INTRODUCTION  Some key concepts: 1. Variables and constants: Variables:. Quantities which change their values are called variables. eg: temperature of a city, the sale and profit of a firm etc Constants: Quantities which do not change their values are called constant. eg: Volume of a solid, weight of a machines etc. 2. Intervals: Every variable takes value lying between two certain values 𝑎 and 𝑏 Closed If the variables take the value between 𝑎 and 𝑏 including 𝑎 and Interval 𝑏 then it is called closed interval denoted by [𝑎, 𝑏]. Open If the variables take the value between 𝑎 and 𝑏 but not the Interval: endpoints 𝑎 and 𝑏 then it is called open interval denoted by (𝑎, 𝑏).  Function:- Suppose there are two variables 𝑥 and 𝑦. If the variable 𝑦 is related to the variable 𝑥 such that for a given value of 𝑥 in its interval we are able to determine the corresponding value of 𝑦 then we say that 𝑦 is a function of 𝑥. Here 𝑥 can be given any value within an interval it is called independent variable. Since 𝑦 depends upon the value of 𝑥 it is called dependent variable. It is denoted by 𝑓: 𝐴 → 𝐵 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑦 = 𝑓(𝑥) where 𝑓(𝑥) is function of x.  Range and domain:- For a function 𝑓: 𝐴 → 𝐵 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑦 = 𝑓(𝑥)  Departmental Library I Department of Applied Mathematics Semester The Maharaja Sayajirao University of Baroda,Vadodara 2020 Polytechnic Since 2011 Subject: Applied Mathematics-I Domain The set from where 𝑥 can take values is called Domain. In the above function A is the domain. Range: The set of values which 𝑦 can take is called Range.  TYPES OF FUNCTIONS: 1. Explicit Function The function in which 𝑦 can be expressed in 𝑥 explicitly in th form 𝑦 = 𝑓(𝑥) eg: 𝑦 = 𝑥 − 2𝑥 + 3 2. Implicit Function The function where 𝑦 cannot be separated from 𝑥 𝑎nd is of the form 𝑓(𝑥, 𝑦) = 0 eg: 𝑦𝑠𝑖𝑛𝑥 + 𝑦 𝑥 = 0 3. Parametric Function Sometimes it is more convenient to express 𝑦 and 𝑥 both in terms of a third variable 𝑡. Here 𝑡 is called a parameter.Such a function is of the form 𝑥 = ∅(𝑡), 𝑦 = ∅(𝑡). eg: 𝑦 = 𝑠𝑖𝑛𝑡 and 𝑥 = 𝑡 4. Even Function If 𝑓(−𝑥) = 𝑓(𝑥) then 𝑓(𝑥) is called the even function eg:𝑓(𝑥) = 𝑥 5. Odd Function If 𝑓(−𝑥) = −𝑓(𝑥) then 𝑓(𝑥) is called the odd function eg:𝑓(𝑥) = 𝑥 6. Composite Function If 𝑦 is not directly expressed in terms of x but if y is expressed in terms of some variable 𝑢 and 𝑢 is expressed in terms of 𝑥. Then it is called composite function and is of the form 𝑦 = ∅(𝑢) and 𝑢 = ∅(𝑥) eg: 𝑦 = 𝑢 and 𝑢 = 𝑠𝑖𝑛𝑥 7. Algebraic Functions A function in which only algebraic operations such as addition, subtraction, multiplication, division, taking roots etc are involved is called Algebraic Function. eg:𝑦 = 𝑥 + 2𝑥 + 3  Departmental Library I Department of Applied Mathematics Semester The Maharaja Sayajirao University of Baroda,Vadodara 2020 Polytechnic Since 2011 Subject: Applied Mathematics-I 8. Transcendental A function along with algebraic operations involves Functions trigonometric ,logarithmic,exponential etc functions are called transcendental functions eg: 𝑦 = 𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥  Value of function: If 𝑦 = 𝑓(𝑥) is a given function then the value of 𝑓(𝑥) obtained by replacing 𝑥 by 𝑎 is called the value of the function at 𝑥 = 𝑎 and is denoted by 𝑓(𝑎). eg: 𝐼𝑓 𝑓(𝑥) = 2𝑥 + 3 then 𝑓(2) = 2(2) + 3 = 7 Examples: 1. If 𝑓(𝑥) = 𝑥 − 4𝑥 + 7 then find 𝑓(0), 𝑓(−1) and 𝑓(2). Soln: Given 𝑓(𝑥) = 𝑥 − 4𝑥 + 7, Substitute 𝑥 = 0, 𝑓(0) = 0 − 4(0) + 7 = 7 Substitute 𝑥 = −1, 𝑓(−1) = (−1) − 4(−1) + 7 = 12 Substitute 𝑥 = 2, 𝑓(2) = 2 − 4(2) + 7 = 3 2. If 𝑓(𝑡) = 50sin (100𝜋𝑡 + 0.4) show that 𝑓 + 𝑡 = 𝑓(𝑡). Proof: Given 𝑓(𝑡) = 50sin (100𝜋𝑡 + 0.4), ∴𝑓 + 𝑡 = 50 sin 100𝜋 + 𝑡 + 0.4 ∴𝑓 + 𝑡 = 50 sin(2𝜋 + 100𝜋𝑡 + 0.4) = 50 sin 2𝜋 + (100𝜋𝑡 + 0.4) = 50 sin(100𝜋𝑡 + 0.4) = 𝑓(𝑡) [∵ sin(2𝜋 + 𝜃) = 𝑠𝑖𝑛𝜃] 3. Determine the range of the function 𝑓(𝑥) = 𝑥 + 1 having domain (−5,2). Soln: For 𝑥 = −5 we get 𝑓(−5) = (−5) + 1 = 26 𝑥 = 2 we get 𝑓(2) = (2) + 1 = 5 ∴ The range of given function is (5,26)  Departmental Library I Department of Applied Mathematics Semester The Maharaja Sayajirao University of Baroda,Vadodara 2020 Polytechnic Since 2011 Subject: Applied Mathematics-I  LIMIT Definition: If 𝑓(𝑥) goes nearer and nearer to 𝑙 as 𝑥 goes nearer aand nearer to 𝑎 then we can say that the limit of 𝑓(𝑥) is 𝑙 as 𝑥 tends to 𝑎 and is written as lim → 𝑓(𝑥) = 𝑙  Some Standard results: 1. lim 𝑥 = 𝑎 7. 𝑒 −1 → lim =1 → 𝑥 2. 𝑥 −𝑎 8. 𝑠𝑖𝑛𝑥 lim = 𝑛𝑎 lim =1 → 𝑥−𝑎 → 𝑥 lim → = 𝑎 3. 1 9. 𝑡𝑎𝑛𝑥 lim =0 lim =1 → 𝑛 → 𝑥 4. 1 10. lim 𝑐𝑜𝑠𝑥 = 1 lim 1 + =𝑒 → → 𝑛 5. 11. lim 𝑡𝑎𝑛𝑥 = 0 lim (1 + 𝑛) = 𝑒 → → 6. 𝑎 −1 12. lim 𝑟 = 0 lim = log 𝑎 → → 𝑥  Algebra of limits If lim → 𝑓(𝑥) = 𝑙 and lim → 𝑔(𝑥) = 𝑚 1. lim → 𝑓(𝑥) + 𝑔(𝑥) = lim → 𝑓(𝑥) + lim → 𝑔(𝑥) = 𝑙 + 𝑚 2. lim → 𝑓(𝑥) − 𝑔(𝑥) = lim → 𝑓(𝑥) − lim → 𝑔(𝑥) = 𝑙 − 𝑚 3. lim → 𝑓(𝑥). 𝑔(𝑥) = lim → 𝑓(𝑥). lim → 𝑔(𝑥) = 𝑙. 𝑚 4. lim ( ) = → ( ) = → ( ) ( ) →  Sums based on Limit:  Departmental Library I Department of Applied Mathematics Semester The Maharaja Sayajirao University of Baroda,Vadodara 2020 Polytechnic Since 2011 Subject: Applied Mathematics-I 1. Evaluate → Soln → = → = → ( ) ( ) ( ) = → ( )( ) ( ) ( ) ( ) ( )( ) = → ( )( ) = → =0 2. Evaluate lim √ √ → Soln: lim √ √ = lim √ √ × √ √ → → √ √ ( ) = lim → ( ) √ √ = lim → ( ) √ √ = = = √ √ √ √ √ 3. If 𝑓(𝑥) = √7 − 2𝑥 then evaluate: → ( ) ( ) Soln: 𝑓(−1 + ℎ) − 𝑓(−1) 7 − 2(−1 + ℎ) − 7 − 2(−1) → = → ℎ ℎ √ = → √ √ = →. √ = → √ = → =− =− √ 4. Evaluate →  Departmental Library I Department of Applied Mathematics Semester The Maharaja Sayajirao University of Baroda,Vadodara 2020 Polytechnic Since 2011 Subject: Applied Mathematics-I Soln = ( ) → → ( ) = (64) = (64) = (2 ) = × = 5. Evaluate: → Soln = → → = → = → = = −1 6. Evaluate: → Soln → = → ( ) ( ) = → ( ) = → = → (Put − 𝑥 = ℎ ∴ As 𝑥 → 𝜋 2 , ℎ → 0) = → ( ) = → ( ) = → ( ) =. 7. Evaluate: → ∑( ) Soln → ∑( ) = → ∑ ∑ ∑  Departmental Library I Department of Applied Mathematics Semester The Maharaja Sayajirao University of Baroda,Vadodara 2020 Polytechnic Since 2011 Subject: Applied Mathematics-I ( ) ( )( ) = → = → = → = 8. Evaluate: → √𝑛 + 𝑛 + 3 − 𝑛 + 1 Soln √𝑛 + 𝑛 + 3 − 𝑛 + 1 = √𝑛 + 𝑛 + 3 − 𝑛 + 1 √ → → √ = → √ = → √ = → = 9. Evaluate: → Soln = → → = → = → − (Put − 𝑥 = ℎ, As 𝑥 → , ℎ → 0) = −4 → = −4 10. Evaluate: → Soln = → →  Departmental Library I Department of Applied Mathematics Semester The Maharaja Sayajirao University of Baroda,Vadodara 2020 Polytechnic Since 2011 Subject: Applied Mathematics-I =𝑒 → (Put 𝑥 − 1 = ℎ ,As 𝑥 → 1 , ℎ → 0 ) = 𝑒 log 𝑒 = 𝑒 11. Evaluate: (1 − 4𝑥) → Soln (1 − 4𝑥) = (1 − 4𝑥) → → = → { (1 − 4𝑥). } = → 1 + (−4𝑥) = =𝑒 12. Evaluate: → Soln = → → = → + = → +. = log 𝑒 + →. → =1+ = 13. Evaluate: → ( ) ( ) Soln → ( ) ( ) = → log(1 + 𝑥) − log(1 − 𝑥) = → log(1 + 𝑥) − log(1 − 𝑥) = log 𝑒 + log 𝑒 = 2  Departmental Library I Department of Applied Mathematics Semester The Maharaja Sayajirao University of Baroda,Vadodara 2020 Polytechnic Since 2011 Subject: Applied Mathematics-I Self Practice Questions: 1. √ √ Prove that → = √ √ 2. Prove that ∑ = 1 → 3. Prove that = log → 4. Prove that =. → ( ) 5. Prove that → =𝑒  CONTINUITY: Definition: A function 𝑓(𝑥) is said to be continuous at 𝑥 = 𝑐 if it staifies the following three conditions: (i)That 𝑓(𝑥) is defined at c i.e 𝑓(𝑐) is a real number. (ii)The limit as 𝑥 approaches 𝑐 exists i.e. lim → 𝑓(𝑥) = lim → 𝑓(𝑥) (iii) lim → 𝑓(𝑥) = lim → 𝑓(𝑥) = 𝑓(𝑐) Sums based on continuity: 1. Examine the continuity of the function 𝑓(𝑥) = 2𝑥 + 1 0≤𝑥≤2 = 4𝑥 + 1 2 2 Discuss the continuity of 𝑓(𝑥) at the points 𝑥 =1 and 𝑥 = 2. 3. Find 𝑎 & 𝑏 if 𝑓(𝑥) is continuous at 𝑥 = 2, 𝑥 = 4, Where, 𝑓(𝑥) = 𝑥 + 𝑎 when 0 ≤ 𝑥 ≤ 2 = 4𝑥 when 2 < 𝑥 < 4 =𝑥−𝑏 when 4 ≤ 𝑥 ≤ 6