MTH101 Highlighted Handouts PDF

Calculus And Analytical Geometry MTH 101 Virtual University of Pakistan Knowledge beyond the boundaries 45- Planning Production Levels: Linear Programming VU TABLE OF CONTENTS : Lesson 1 :Coordinates, Graphs, Lines 3 Lesson 2 :Absolute Value 15 Lesson 3 :Coordinate Planes and Graphs 24 Lesson 4 :Lines 34 Lesson 5 :Distance; Circles, Quadratic Equations 45 Lesson 6 :Functions and Limits 57 Lesson 7 :Operations on Functions 63 Lesson 8 :Graphing Functions 69 Lesson 9 :Limits (Intuitive Introduction) 76 Lesson 10:Limits (Computational Techniques) 84 Lesson 11: Limits (Rigorous Approach) 93 Lesson 12 :Continuity 97 Lesson 13 :Limits and Continuity of Trigonometric Functions 104 Lesson 14 :Tangent Lines, Rates of Change 110 Lesson 15 :The Derivative 115 Lesson 16 :Techniques of Differentiation 123 Lesson 17 :Derivatives of Trigonometric Function 128 Lesson 18 :The chain Rule 132 Lesson 19 :Implicit Differentiation 136 Lesson 20 :Derivative of Logarithmic and Exponential Functions 139 Lesson 21 :Applications of Differentiation 145 Lesson 22 :Relative Extrema 151 Lesson 23 :Maximum and Minimum Values of Functions 158 Lesson 24 :Newton’s Method, Rolle’s Theorem and Mean Value Theorem 164 Lesson 25 :Integrations 169 Lesson 26 :Integration by Substitution 174 Lesson 27 :Sigma Notation 179 Lesson 28 :Area as Limit 183 Lesson 29 :Definite Integral 191 Lesson 30 :First Fundamental Theorem of Calculus 200 Lesson 31 :Evaluating Definite Integral by Subsitution 206 Lesson 32 :Second Fundamental Theorem of Calculus 210 Lesson 33 :Application of Definite Integral 214 Lesson 34 :Volume by slicing; Disks and Washers 221 Lesson 35 :Volume by Cylindrical Shells 230 Lesson 36 :Length of Plane Curves 237 Lesson 37 :Area of Surface of Revolution 240 Lesson 38:Work and Definite Integral 245 Lesson 39 :Improper Integral 252 Lesson 40 :L’Hopital’s Rule 258 Lesson 41 :Sequence 265 Lesson 42 :Infinite Series 276 Lesson 43 :Additional Convergence tests 285 Lesson 44 :Alternating Series; Conditional Convergence 290 Lesson 45 :Taylor and Maclaurin Series 296 1-Coordinates, Graphs and Lines VU Lecture 1 Coordinates, Graphs and Lines What is Calculus?? Well, it is the study of the continuous rates of the change of quantities. It is the study of how various quantities change with respect to other quantities. For example, one would like to know how distance changes with respect to (from now onwards we will use the abbreviation w.r.t) time, or how time changes w.r.t speed, or how water flow changes w.r.t time etc. You want to know how this happens continuously. We will see what continuously means as well. In this lecture, we will talk about the following topics: -Real Numbers -Set Theory -Intervals -Inequalities -Order Properties of Real Numbers Let's start talking about Real Numbers. We will not talk about the COMPLEX or IMANGINARY numbers, although your text has something about them which you can read on your own. We will go through the history of REAL numbers and how they popped into the realm of human intellect. We will look at the various types of REALS - as we will now call them. So Let's START. The simplest numbers are the natural numbers Natural Numbers 1, 2, 3, 4, 5,… They are called the natural numbers because they are the first to have crossed paths with human intellect. Think about it: these are the numbers we count things with. So our ancestors used these numbers first to count, and they came to us naturally! Hence the name NATURAL!!! The natural numbers form a subset of a larger class of numbers called the integers. I have used the word SUBSET. From now onwards we will just think of SET as a COLLECTION OF THINGS. This could be a collection of oranges, apples, cars, or politicians. For example, if I have the SET of politicians then a SUBSET will be just a part of the COLLECTION. In mathematical notation we say A is subset of B if x  A  x  B.Then we write A B. Set The collection of well defined objects is called a set. For example © Copyright Virtual University of Pakistan 3 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 1-Coordinates, Graphs and Lines VU {George Bush, Toney Blair, Ronald Reagoan} Subset A portion of a set B is a subset of A iff every member of B is a member of A. e.g. one subset of above set is {George Bush, Tony Blair} The curly brackets are always used for denoting SETS. We will get into the basic notations and ideas of sets later. Going back to the Integers. These are …, -4, -3, -2, -1, 0, 1, 2, 3, 4,… So these are just the natural numbers, plus a 0, and the NEGATIVES of the natural numbers. The reason we didn’t have 0 in the natural numbers is that this number itself has an interesting story, from being labeled as the concept of the DEVIL in ancient Greece, to being easily accepted in the Indian philosophy, to being promoted in the use of commerce and science by the Arabs and the Europeans. But here, we accept it with an open heart into the SET of INTEGERS. What about these NEGATIVE Naturals??? Well, they are an artificial construction. They also have a history of their own. For a long time, they would creep up in the solutions of simple equations like x+2 = 0. The solution is x = - 2 So now we have the Integers plus the naturals giving us things we will call REAL numbers. But that's not all. There is more. The integers in turn are a subset of a still larger class of numbers called the rational numbers. With the exception that division by zero is ruled out, the rational numbers are formed by taking ratios of integers. Examples are 2/3, 7/5, 6/1, -5/2 Observe that every integer is also a rational number because an integer p can be written as a ratio. So every integer is also a rational. Why not divide by 0? Well here is why: If x is different from zero, this equation is contradictory; and if x is equal to zero, this equation is satisfied by any number y, so the ratio does not have a unique value a situation that is mathematically unsatisfactory. x / 0  y  x  0.y  x  0 For these reasons such symbols are not assigned a value; they are said to be undefined. So we have some logical inconsistencies that we would like to avoid. I hope you see that!! Hence, no division by 0 allowed! Now we come to a very interesting story in the history of the development of Real numbers. The discovery of IRRATIONAL numbers. Pythagoras was an ancient Greek philosopher and mathematician. He studied the properties of numbers for its own sake, not necessarily for any applied problems. This was a major change in mathematical thinking as © Copyright Virtual University of Pakistan 4 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 1-Coordinates, Graphs and Lines VU math now took on a personality of its own. Now Pythagoras got carried away a little, and developed an almost religious thought based on math. He concluded that the size of a physical quantity must consist of a certain whole number of units plus some fraction m / n of an additional unit. Now rational numbers have a unique property that if you convert them to decimal notation, the numbers following the decimal either end quickly, or repeat in a pattern forever. Example: 1/2 =0.500000… =0.5 1/3= 0.33333… This fit in well with Pythagoras’ beliefs. All is well. But this idea was shattered in the fifth century B.C. by Hippasus of Metapontum who demonstrated the existence of irrational numbers, that is, numbers that cannot be expressed as the ratio of integers. Using geometric methods, he showed that the hypotenuse of the right triangle with base and opposite side equal to 1 cannot be expressed as the ratio of integers, thereby proving that 2 is an IRRATIONAL number. The hypotenuse of this right triangle can be expressed as the ratio of integers. Other examples of irrational numbers are Cos 190,1  2 The rational and irrational numbers together comprise a larger class of numbers, called REAL NUMBERS or sometimes the REAL NUMBER SYSTEM.So here is a pictorial summary of the hierarchy of REAL NUMBNERS. © Copyright Virtual University of Pakistan 5 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 1-Coordinates, Graphs and Lines VU Pictorial summary of the hierarchy of REAL NUMBNERS COORDINATE Line In the 1600’s, analytic geometry was “developed”. It gave a way of describing algebraic formulas by geometric curves and, conversely, geometric curves by algebraic formulas. So basically you could DRAW PICTURES OF THE EQUATIONS YOU WOULD COME ACROSS, AND WRITE DOWN EQUATIONS OF THE PICTURES YOU RAN INTO! The developer of this idea was the French mathematician, Descartes.The story goes that he wanted to find out as to what Made humans HUMANS?? Well, he is said to have seated himself in a 17th century furnace (it was not burning at the time!) and cut himself from the rest of the world. In this world of cold and darkness, he felt all his senses useless. But he could still think!!!! So he concluded that his ability to think is what made him human, and then he uttered the famous line : “ I THINK, THEREFORE I AM”.In analytic geometry , the key step is to establish a correspondence between real numbers and points on a line. We do this by arbitrarily designating one of the two directions along the line as the positive direction and the other as © Copyright Virtual University of Pakistan 6 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 1-Coordinates, Graphs and Lines VU the negative direction. So we draw a line, and call the RIGHT HAND SIDE as POSITIVE DIRECTION, and the LEFT HAND SIDE as NEGATIVE DIRECTION. We could have done it the other way around too. But, since what we just did is a cultural phenomenon where right is + and left is -, we do it this way. Moreover, this has now become a standard in doing math, so anything else will be awkward to deal with. The positive direction is usually marked with an arrowhead so we do that too. Then we choose an arbitrary point and take that as our point of reference. We call this the ORIGIN, and mark it with the number 0. So we have made our first correspondence between a real number and a point on the Line. Now we choose a unit of measurement, say 1 cm. It can be anything really. We use this unit of measurement to mark of the rest of the numbers on the line. Now this line, the origin, the positive direction, and the unit of measurement define what is called a coordinate line or sometimes a real line. With each real number we can now associate a point on the line as fo1lows:  Associate the origin with the number 0.  Associate with each positive number r the point that is a distance of r units (this is the unit we  chose, say 1 cm) in the positive direction from the origin.  Associate with each negative number ‘- r ’ the point that is a distance of r units in the negative direction from the origin. The real number corresponding to a point on the line is called the coordinate of the point. Example 1: In Figure we have marked the locations of the points with coordinates -4, -3, -1.75, -0.5,  , 2 and 4. The locations of  and 2 which are approximate, were obtained from their decimal approximations,  = 3.14 and 2 = 1.41 © Copyright Virtual University of Pakistan 7 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 1-Coordinates, Graphs and Lines VU It is evident from the way in which real numbers and points on a coordinate line are related that each real number corresponds to a single point and each point corresponds to a single real number. To describe this fact we say that the real numbers and the points on a coordinate line are in one-to-one correspondence. Order Properties In mathematics, there is an idea of ORDER of a SET. We won’t go into the general concept, since that involves SET THEORY and other high level stuff. But we will define the ORDER of the real number set as follows: For any two real numbers a and b, if b-a is positive, then we say that b > a or that a < b. Here I will assume that we are all comfortable working with the symbol “” which is read as “greater than.” I am assuming this because this stuff was covered in algebra before Calculus. So with this in mind we can write the above statement as If b - a is positive, then we say that b > a or that a < b.A statement involving < or > are called an INEQUALITY.Note that the inequality a < b can also be expressed as b > a. So ORDER of the real number set in a sense defines the SIZE of a real number relative to another real number in the set. The SIZE of a real number a makes sense only when it is compared with another real b. So the ORDER tells you how to “ORDER” the numbers in the SET and also on the COORDINATE LINE! A little more about inequalities. The inequality a  b is defined to mean that either a < b or a = b. So there are two conditions here. For example, the inequality 2  6 would be read as 2 is less than or it is equal to 6. We know that it’s less than 6, so the inequality is true. SO IF ONE OF THE CONDITIONS IS TRUE, THEN THE INEQUALITY WILL BE TRUE, We can say a similar thing about. The expression a < b < c is defined to mean that a < b and b < c. It is also read as “b is between a and c”. As one moves along the coordinate line in the positive direction, the real numbers increase in size. In other words, the real numbers are ordered in an ascending manner on the number line, just as they are in the SET of REAL NUMBERS. So that on a horizontal coordinate line the inequality a < b implies that a is to the left of b, and the inequality a < b < c implies that a is to the left of b and b is to the left of c. The symbol a < b < c means a < b and b < c. I will leave it to the reader to deduce the meanings of such symbols as  and . Here is an example of INEQUALITIES. abc abc abcd © Copyright Virtual University of Pakistan 8 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 1-Coordinates, Graphs and Lines VU Example: Correct Inequalities 3  8,  7  1.5, 12  x, 5  5 0  2  4,         12, 5  5, 3  0  1  3 Some incorrect inequalities are: 2  4,   0, 5  3 REMARK: To distinguish verbally between numbers that satisfy a  0 and those that satisfy a > 0, we shall call a nonnegative if a  0 and positive if a > 0. Thus, a nonnegative number is either positive or zero. The following properties of inequalities are frequently used in calculus. We omit the proofs, but will look at some examples that will make the point. THEOREM 1.1.1 a) If a  b and b  c, then a  c b) If a  b and a  c  b  c, then a  c  b  c c) If a  b and ac  bc, when c is positive and ac  bc when c is negative. d) ) If a  b and c  d , then a  c  b  d e) If a and b are both positive or both negative 1 1 and a  b then  a b REMARK These five properties remain true if < and > are replaced by  and  INTERVALS We saw a bit about sets earlier. Now we shall assume in this text that you are familiar with the concept of a set and fully understand the meaning of the following symbols. However, we will give a short explanation of each. Given two sets A and B a  A : a is an element of the set A, 2 {1, 2, 3, 4} © Copyright Virtual University of Pakistan 9 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 1-Coordinates, Graphs and Lines VU a  A : a is NOT an element of the set A 5{1, 2, 3, 4}  represents the Empty set, or the set that contains nothing. A  B represents the SET of all the elements of the Set A and the Set B taken together. Example: A = {1,2,3,4}, B = {1,2,3,4,5,6,7}, then, A  B = {1,2,3,4,5,6,7} A  B represents the SET of all those elements that are in Set A AND in Set B. Example: A = {1,2,3,4}, B = {1,2,3,4,5,6,7}, then A  B = {1,2,3,4} A = B means the A is exactly the same set as B Example: A = {1,2,3,4} and B = {1,2,3,4}, then A = C and A  B means that the Set A is contained in the Set B. Recall the example we did of the Set of all politicians! {George Bush, Tony Blair}  {George Bush, Toney Blair, Ronald Reagoan} One way to specify the idea of a set is to list its members between braces. Thus, the set of all positive integers less than 5 can be written as { 1, 2, 3, 4} and the set of all positive even integers can be written as { 2, 4, 6, …} where the dots are used to indicate that only some of the members are explicitly and the rest can be obtained by continuing the pattern. So here the pattern is that the set consists of the even numbers, and the next element must be 8, then 10, and then so on. When it is inconvenient or impossible to list the members of a set, as would be if the set is infinite, then one can use the set- builder notation. This is written as {x: } which is read as “the set of all x such that ” , In place of the line, one would state a property that specifies the set, Thus, { x : x is real number and 2 < x < 3} is read, "the set of all x such that x is a real number and 2 < x < 3," Now we know by now that © Copyright Virtual University of Pakistan 10 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 1-Coordinates, Graphs and Lines VU 2 < x < 3 means that all the x between 2 and 3. This specifies the “description of the elements of the set” This notation describes the set, without actually writing down all its elements. When it is clear that the members of a set are real numbers, we will omit the reference to this fact. So we will write the above set as Intervals. We have had a short introduction of Sets. Now we look particular kind of sets that play a crucial role in Calculus and higher math. These sets are sets of real numbers called intervals. What is an interval? { x: 2< x< 3} Well, geometrically, an interval is a line segment on the co-ordinate line. S if a and b are real numbers such that a < b, then an interval will be just the line segment joining a and b. But if things were only this simple! Intervals are of various types. For example, the question might be raised whether a and b are part of the interval? Or if a is, but b is not?? Or maybe both are? Well, this is where we have to be technical and define the following: The closed interval from a to b is denoted by [a, b] and is defined as [a, b]  x : a  x  b} Geometrically this is the line segment So this includes the numbers a and b, a and b a are called the END- POINTS of the interval. The open interval from a to b is denoted by and is defined by (a, b) = { x: a < x < b} This excludes the numbers a and b. The square brackets indicate that the end points are included in the interval and the parentheses indicate that they are not. Here are various sorts of intervals that one finds in mathematics. In this picture, the geometric pictures use solid dots to denote endpoints that are included in the interval and open dots to denote endpoints that are not. © Copyright Virtual University of Pakistan 11 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 1-Coordinates, Graphs and Lines VU As shown in the table, an interval can extend indefinitely in either the positive direction, the negative direction, or both. The symbols  (read "negative infinity") and  (read , 'positive infinity' ') do not represent numbers: the  indicates that the interval extends indefinitely in the positive direction, and the  indicates that it extends indefinitely in the negative direction. An interval that goes on forever in either the positive or the negative directions, or both, on the coordinate line or in the set of real numbers is called an INFINITE interval. Such intervals have the symbol for infinity at either end points or both, as is shown in the table An interval that has finite real numbers as end points are called finite intervals. A finite interval that includes one endpoint but not the other is called half-open (or sometimes half-closed). [a,  ), (a,  ), (, b], (, b) Infinite intervals of the form [a,  ) and (, b] are considered to be closed because they contain their endpoint. Those of the form (a,  ) and (, b) has no endpoints; it is regarded to be both open and closed. As one of my Topology Instructors used to say: “A set is not a DOOR! It can be OPEN, it can be CLOSED, and it can be OPEN and CLOSED!! Let's remember this fact for good!” Let's look at the picture again for a few moments and digest the © Copyright Virtual University of Pakistan 12 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 1-Coordinates, Graphs and Lines VU information. PAUSE 10 seconds. SOLVING INEQUALITIES We have talked about Inequalities before. Let's talk some more. First Let's look at an inequality involving and unknown quantity, namely x. Here is one: x < 5,x = 1, is a solution of this inequality as 1 makes it true, but x = 7 is not. So the set of all solutions of an inequality is called its solution set. The solution set of x < 5 will be It is a fact, though we wont prove this that if one does not multiply both sides of an inequality by zero or an expression involving an unknown, then the operations in Theorem 1.1.1 will not change the solution set of the inequality. The process of finding the solution set of an inequality is called solving the Inequality. Let's do some fun stuff, like some concrete example to make things a bit more focused Example 4. Solve 3  7x  2x  9 Solution. We shall use the operations of Theorem 1.1.1 to isolate x on one side of the inequality 7x  2x 12 Subtracting 3 from both sides 5x  12 Subtracting 2x from both sides x  12 / 5 Dividing both sides by 5 Because we have not multiplied by any expressions involving the unknown x, the last inequality has the same solution set as the first. Thus, the solution set is the interval shown in Figure 1.1.6. Example Solve 7  2  5x  9 Solution ; The given inequality is actually a combination of the two inequalities 7  2  5x and 2  5x  9 We could solve the two inequalities separately, then determine the value of x that satisfy both by taking the intersection of the solution sets , however, it is possible to work with the combined inequality in this problem © Copyright Virtual University of Pakistan 13 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 1-Coordinates, Graphs and Lines VU 5  5x  7 Subtracting 2 from both sides 1  x  7 /5 Dividing by  5 inequality symbols reversed 7 / 5  x  1 Re writing with smaller number first      Thus the solution set is interval shown the figure Example Similarly, you can find © Copyright Virtual University of Pakistan 14 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 2-Absolute Values VU Lecture # 2 Absolute Value In this lecture we shall discuss the notation of Absolute Value. This concept plays an important role in algebraic computations involving radicals and in determining the distance between points on a coordinate line. Definition The absolute vale or magnitude of a real number a is denoted by |a| and is defined by a if a  0, that is, a is non  negative | a |   a if a  0, that is, a is positive. Technically, 0 is considered neither positive, nor negative in Mathematics. It is called a non-negative number. Hence whenever we want to talk about a real number a such that a  0, we call a non-negative, and positive if a > 0. Example 4 4 4 5 5 ,   ( )  , 0 0 7 7 7 since 5>0 since -4/7 < 0 since 0 0 Note that the effect of taking the absolute value of a number is to strip away the minus sign if the number is negative and to leave the number unchanged if it is non-negative. Thus, |a| is a non-negative number for all values of a and  a  a  a , if ‘a’ is itself is negative, then ‘–a’ is positive and ‘+a’ is negative. a + b  0 or a + b < 0 a +b= a+b ab  a  b a  b  (a  b) © Copyright Virtual University of Pakistan 15 2-Absolute Values VU Caution: Symbols such as +a and –a are deceptive, since it is tempting to conclude that +a is positive and –a is negative. However this need not be so, since a itself can represent either a positive or negative number. In fact , if a itself is negative, then –a is positive and +a is negative. Example: Solve x3 4 Solution: Depending on whether x-3 is positive or negative , the equation |x-3| = 4 can be written as x-3 = 4 or x-3 = -4 Solving these two equations give x=7 and x=-1 Example Solve 3x  2  5x  4 Because two numbers with the same absolute value are either equal or differ only in sign, the given equation will be satisfied if either 3x  2  5x  4 3x  5x  4  2 2x  6 x  3 Or 3x  2  (5x  4) 3x  2  5x  4 3x  5x  4  2 1 x 4 © Copyright Virtual University of Pakistan 16 2-Absolute Values VU Relationship between Square Roots and Absolute Values : Recall that a number whose square is a is called a square root of a. In algebra it is learned that every positive real number a has two real square roots, one positive and one negative. The positive square root is denoted by a For example, the number 9 has two square roots, -3 and 3. Since 3 is the positive square root, we have 9  3. In addition, we define 0  0. It is common error to write a2  a. Although this equality is correct when a is nonnegative, it is false for negative a. For example, if a=-4, then a2  (4)2  16  4  a The positive square root of the square of a number is equal to that number. A result that is correct for all a is given in the following theorem. Theorem: For any real number a, a2  a Proof : Since a2 = (+a)2 = (-a)2, the number +a and –a are square roots of a2. If a  0 , then +a is nonnegative square root of a2, and if a  0 , then -a is nonnegative square root of a2. Since a2 denotes the nonnegative square root of a2, we have a2  a if a  0 if a2  a if a  0 That is, a2  a Properties of Absolute Value Theorem If a and b are real numbers, then (a) |-a| = |a|, a number and its negative have the same absolute value. (b) |ab| = |a| |b|, the absolute value of a product is the product of absolute values. (c) |a/b| = |a|/|b|, the absolute value of the ratio is the ratio of the absolute values © Copyright Virtual University of Pakistan 17 2-Absolute Values VU Proof (a) : | a |  (a)2  a2 | a | Proof (b) : ab  (ab)2  a2b2  a2 b2  a b This result can be extended to three or more factors. More precisely, for any n real numbers, a1,a2,a3,……an, it follows that |a 1 a 2 …..a n | = |a 1 | |a 2 | …….|a n | In special case where a1, a2,…….,an have the same value, a, it follows from above equation that |an|=|a|n Example (a) |-4|=|4|=4 (b) |(2)(-3)|=|-6|=6=|2||-3|=(2)(3)=6 (c) |5/4|=5/4=|5|/|4|=5/4 Geometric Interpretation Of Absolute Value The notation of absolute value arises naturally in distance problems, since distance is always nonnegative. On a coordinate line, let A and B be points with coordinates a and b, the distance d between A and B is b  a if a  b  d  a  b if a  b  if a  b 0 © Copyright Virtual University of Pakistan 18 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 2-Absolute Values VU As shown in figure b-a is positive, so b-a=|b-a| ; in the second case b-a is negative, so a-b = -(b-a) = |b-a|. Thus, in all cases we have the following result: Theorem (Distance Formula) If A and B are points on a coordinate line with coordinates a and b, respectively, then the distance d between A and B is d = |b-a| This formula provides a useful geometric interpretation Of some common mathematical expressions given in table here Table EXPRESSION GEOMETRIC INTERPRETATION ON A COORDINATE LINE |x-a| The distance between x and a |x+a| The distance between x and -a |x| The distance between x and origin Inequalities of the form |x-a|k arise often, so we have summarized the key facts about them here in following table © Copyright Virtual University of Pakistan 19 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 2-Absolute Values VU Example Solve |x-3| < 4 Solution: This inequality can be written as -4 < x-3 < 4 adding 3 throughout we get -1 < x < 7 This can be written in interval notation as (-1,7) © Copyright Virtual University of Pakistan 20 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 2-Absolute Values VU Example Solve x  4  2 Solution: The given inequality can be written as x  4  2  or x  4  2  or simply x  6  or x  2  Which can be written in set notation as , 6U 2,                    The Triangle Inequality © Copyright Virtual University of Pakistan 21 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 2-Absolute Values VU   It is not generally true that a  b  a  b For example , if a  2 and b  3, then a  b  1 so that a  b  1  1 whereas a  b  2  3  2  3  5 so a  b  a  b It is true, however, that the absolute value of a sum is always less than or equal to the sum of the absolute values. This is the content of the following very important theorem, known as the triangle inequality. This TRIANGLE INEQUALITY is the essence of the famous HISENBURG UNCERTAINITY PRINCIPLE IN QUANTUM PHYSICS, so make sure you understand it fully. THEOREM 1.2.5 (Triangle Inequality) If a and b are any real numbers, then a b  a  b PROOF Remember the following inequalities we saw earlier.  a  a  a and  b  b  b Let's add these two together. We get a aa   b b b  (  a )  ( b )  a  b  a  b (B) Since a and b are real numbers, adding them will also result in a real number. Well, there are two types of real numbers. What are they?? Remember!!!!! They are either > = 0, or they are < 0! Ok!! SO we have a + b  0 or a + b < 0 In the first of these cases where a+b 0 certainly a+b= a +b by definition of absolute value. so the right-hand inequality in (B) gives a b  a  b In the second case © Copyright Virtual University of Pakistan 22 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 2-Absolute Values VU a  b  (a  b) But this is the same as a b   a b So the left-hand inequality in (B) can be written as ( a  b )   a  b Multiplying both sides of this inequality by - 1 give a b  a  b © Copyright Virtual University of Pakistan 23 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 3- Coordinate Planes and Graphs VU Lecture # 3 In this lecture we will discuss  Graphs in the coordinate plane.  Intercepts.  Symmetry Plane. We begin with the Coordinate plane. Just as points on a line can be placed in one-to-one correspondence with the real numbers, so points in the PLANE can be placed in one-to-one correspondence with pairs of real numbers. What is a plane? A PLANE is just the intersection of two COORDINATE lines at 90 degrees. It is technically called the COORDINATE PLANE, but we will call it the plane also whenever it is convenient. Each line is a line with numbers on it, so to define a point in the PLANE, we just read of the corresponding points on each line. For example I pick a point in the plane By an ordered pair of real numbers we mean two real numbers in an assigned order. Every point P in a coordinate plane can be associated with a unique ordered pair of real numbers by drawing two lines through P, one perpendicular to the x-axis and the other to the y-axis. © Copyright Virtual University of Pakistan 24 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 3- Coordinate Planes and Graphs VU For example if we take (a,b)=(4,3), then on coordinate plane To plot a point P(a, b) means to locate the point with coordinates (a, b) in a coordinate plane. For example, In the figure below we have plotted the points P(2,5), Q(-4,3), R(-5,-2), and S(4,-3).Now this idea will enable us to visualise algebraic equations as geometric curves and, conversely, to represent geometric curves by algebraic equations. Labelling the axes with letters x and y is a common convention, but any letters may be used. If the letters x and y are used to label the coordinate axes, then the resulting plane is also called an xy-plane. In applications it is common to use letters other than x and y to label coordinate axes. Figure below shows a uv-plane and a ts- plane. The first letter in the name of the plane refers to the horizontal axis and the second to the vertical axis. © Copyright Virtual University of Pakistan 25 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 3- Coordinate Planes and Graphs VU Here is another terminology. The COORDINATE PLANE and the ordered pairs we just discussed is together known as the RECTANGULAR COORDINATE SYSTEM. In a rectangular coordinate system the coordinate axes divide the plane into four regions called quadrants. These are numbered counter clockwise with Roman numerals as shown in the Figure below. Consider the equations 5xy  2 x2  2 y2  7 y  x3  7 We define a solution of such an equation to be an ordered pair of real numbers(a,b) so that the equation is satisfactory when we substitute x=a and y=b. Example 1 The pair (3,2) is a solution of 6x  4 y = 10 © Copyright Virtual University of Pakistan 26 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 3- Coordinate Planes and Graphs VU since this equation is satisfied when we substitute x = 3 and y = 2. That is 6(3)  4(2)  10 which is true!! However, the pair (2,0) is not a solution, since 6(2)  4(0)  18  10 We make the following definition in order to start seeing algebraic objects geometrically. Definition. The GRAPH of an equation in two variables x and y is the set of all points in the xy-plane whose coordinates are members of the solution set of the equation. Example 2 Sketch the graph of y = x2 When we plot these on the xy-plane and connect them, we get this picture of the graph © Copyright Virtual University of Pakistan 27 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 3- Coordinate Planes and Graphs VU IMPORTANT REMARK. It should be kept in mind that the curve in above is only an approximation to the graph of y = x2. When a graph is obtained by plotting points, whether by hand, calculator, or computer, there is no guarantee that the resulting curve has the correct shape. For example, the curve in the Figure here pass through the points tabulated in above table. © Copyright Virtual University of Pakistan 28 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 3- Coordinate Planes and Graphs VU INTERCEPTS Points where a graph intersects the coordinate axes are of special interest in many problems. As illustrated before, intersections of a graph with the x-axis have the form (a, 0) and intersections with the y-axis have the form (0, b). The number a is called an x-intercept of the graph and the number b a y-intercept. Example: Find all intercepts of Solution is the required x-intercept is the required y-intercept Similarly you can solve part (b), the part (c) is solved here In the following figure, the points (x,y),(-x,y),(x,-y) and (-x,-y) form the corners of a rectangle. © Copyright Virtual University of Pakistan 29 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 3- Coordinate Planes and Graphs VU SYMMETRY Symmetry is at the heart of many mathematical arguments concerning the structure of the universe, and certainly symmetry plays an important role in applied mathematics and engineering fields. Here is what it is. As illustrated in Figure the points (x, y), (  x, y), (x,  y) and (  x,  y) form the corners of a rectangle. © Copyright Virtual University of Pakistan 30 Join VU Group: https://chat.whatsapp.com/GeDlA1sU1ZF2k12A4ZyaGw 3- Coordinate Planes and Graphs VU For obvious reasons, the points (x,y) and (x,-y) are said to be symmetric about the x-axis and the points ( x, y) and ( -x, y) are symmetric about the y-axis and the points (x, y) and ( -x, -y) symmetric about the origin. SYMMETRY AS A TOOL FOR GRAPHING By taking advantage of symmetries when they exist, the work required to obtain a graph can be reduced considerably. Example 9 Sketch the graph of the equation 1 y x4  x2 8 1 Solution. The graph is symmetric about the y-axis since substituting  x for x yields y  (x)4  (x)2 8 which simplifies to the original equation. As a consequence of this symmetry, we need only calculate points on the graph that lies in the right half of the xy-plane ( x >= 0). The corresponding points in the left half of the xy-plane ( x a f(x), I will assume that f(x) will have a limit that matches from both sides and so the LIMIT EXISTS for f (x). So I won’t distinguish between left and right hand limits. We begin with a table of LIMITS of two basic functions The functions are f (x)  k g(x)  x Here is the table of the limits and the same information from the graph f (x)  k g(x)  x © Copyright Virtual University of Pakistan 84 10-Limits and Computational Techniques VU Here we have a theorem that will help with computing limits. Won’t prove this theorem, but some of the parts of this theorem are proved in Appendix C of your text book. THEOREM 2.5.1 Let Lim stand for one of the limits lim, lim, lim, lim , lim xa xa xa x x  if L1  lim f (x) and L2  lim g(x) both exists, then a) lim[ f (x)  g(x)]  lim f (x)  lim g(x)  L1  L b) lim[ f (x)  g(x)]  lim f (x)  lim g(x)  L1  L c) lim[ f (x)  g(x)]  lim f (x). lim g(x)  L1L f (x) lim f (x) d) ) ] lim[ g(x) lim g(x) L  1 (L   L  For the Last theorem, say things like “Limit of the SUM is the SUM of the LIMITS etc. Parts a) and c) of the theorem apply to as many functions as you want Part a) gives lim[ f1 (x)  f2 (x) ...  f (xn )]  lim f1 (x)  lim f2 (x) ...lim f (xn ) Part c) gives © Copyright Virtual University of Pakistan 85 10-Limits and Computational Techniques VU lim[ f1 (x)  f2 (x) ... fn (x)]  lim f1 (x)  lim f2 (x) ... lim fn (x) Also if   f1  f2 ...  f n then lim[ f (x)]n  [lim f (x)]n From this last result we can say that lim(xn )  [lim xn ]  an xa xa This is a useful result and we can use it later. Another useful result follows from part c) of the theorem. Let f(x) = k in part c), where k is a constant (number). lim[kg(x)]  lim(k )  lim g(x)  k  lim g(x) So a constant factor can be moved through a limit sign LIMITS OF POLYNOMIAL Polynomials are functions of the form f (x)  b n xn  b n1 xn1 ....  b1x  b0 Where the a’s are all real numbers Let's find the Limits of polynomials and x approaches a numbers a Example lim(x2  4x  3) x5  lim x2  lim 4x  lim 3 x5 x5 x5  lim x  4 lim x  lim 3 2 x5 x5 x5  (5)  4(5)  3  8 2 Theorem 2.5.2 Proof lim p(x)  lim(c  c x ...  c xn ) 0 1 n xa xa  lim c  lim c x ...  lim c xn 0 1 n xa xa xa  lim c  c lim x ...  c lim xn 0 1 n xa xa xa  c  ca ...  c a  p(a) n 0 1 n © Copyright Virtual University of Pakistan 86 10-Limits and Computational Techniques VU 1 Limits Involving x 1 Let's look at the graph of f (x)  x Then by looking at the graph AND by looking at the TABLE of values we get the following Results 1 lim   x0 x 1 lim   x0 x 1 lim 0 x x 1 lim 0 x x © Copyright Virtual University of Pakistan 87 10-Limits and Computational Techniques VU For every real number a, the function 1 1 g(x)  is a translation of f (x) . xa x So we an say the following about this function LIMITS OF POLYNOMIALS AS X GOES TO +INF AND –INF From the graphs given here we can say the following about polynomials of the form  lim x   n  1, 2, 3,... n x  lim xn   n  2, 4, 6,...  x  n  1, 3, 5,... © Copyright Virtual University of Pakistan 88 10-Limits and Computational Techniques VU EXAMPLE lim 2x5   x lim  7x6   x  n 1  1  lim   lim n   0 x xn  x x n 1  1  lim   lim n   0 x xn  x x   HERE are the graphs of the functions y  f (x)  1/ xn ( n is positive integer) Limit as x goes to +inf or –inf of a polynomial is like the Limits of the highest power of x lim (c  c x ...  c xn )  lim c xn 0 1 n n x x Motivation © Copyright Virtual University of Pakistan 89 10-Limits and Computational Techniques VU c0 c1 (c  c x ...  c xn )  xn (  ...  c ) 0 1 n n xn xn1 Factor out xn , and then from what we just saw about 1 the limit of n , everything goes to 0 as x + x or x  except cn Limits of Rational Functions as x goes to a A rational function is a function defined by the ratio of two polynomials Example 5x3  4 Find lim x2 x  3 Sol: lim 5x3  4  x2 lim x  3 x2 5(2)3  4     44 23  We used d) of theorem 2.5.1 to evaluate this limit. We would not be able to use it if the denominator turned out to be 0 as that is not allowed in Mathematics. If both top and bottom approach 0 as x approaches a, then the top and bottom will have a common factor of x – a. In this case the factors can be cancelled and the limit works out. Example x2  4 lim x2 x2 (x  2)(x  2)  lim x2 (x  2)  lim(x  2)  4 x2 Note that x is not equal to two after Simplification for the two functions to be the same. Nonetheless, we calculated the limit as if we were substituting x = 2 using rule for polynomials That’s ok since REALLY LIMIT means you are getting close to 2, but not equaling it!! What happens if in a rational functions, the bottom limit is 0, but top is not?? It’s like the limit as x goes to 0 of f(x) = 1/x. © Copyright Virtual University of Pakistan 90 10-Limits and Computational Techniques VU The limit may be + inf The limits may be –inf +inf from one side and –inf from another Example 2 x Find lim x4 (x  4)(x  2) The top is –2 as x goes to 4 from right side. The bottom goes to 0, so the limit will be inf of some type. To get the sign on inf, Let's analyze the sign of the bottom for various values of real numbers Break the number line into 4 intervals as in The important numbers are the ones that make the top and bottom zero. As x approaches 4 from the right, the ratio stays negative and the result is –inf. You can say something about what happens from the left. Check yourselves by looking at the pic. 2 x So lim   x4  (x  4)(x  2)  LIMITS of Rational Functions as x goes to +inf and -inf Algebraic manipulations simplify finding limits in rational functions involving +inf and –inf. Example 4x2  x lim x 2x3  5 Divide the top and the bottom by the highest power of x 4x2 x 4 1 4 1 3  3  2  ) x x  x x lim (  x x x 2 lim lim 3 5 5 lim (2  ) x 2x x 5 2  3 3  3 x x x x x3 1 1 4 lim  lim x x x x2  1 2  5 lim x x3 © Copyright Virtual University of Pakistan 91 10-Limits and Computational Techniques VU 4(0)  0  0 2  5(0) Quick Rule for finding Limits of Rational Functions as x goes to +inf or –inf c0  c1 x ...  c nxn c xn lim  lim n d 0  d1 x ...  d n x  n x x d n xn c0  c1 x ...  c nxn c xn lim  lim n d 0  d1 x ...  d n x  n x x d xn n Not true if x goes to a finite number a. Example 2 lim 4x  x  lim 4x  lim  0 2 2 x 2x3  5 x 2x3 x x Same answer as the one we got earlier from algebraic manipulations © Copyright Virtual University of Pakistan 92 11-Limits: A Rigorous Approach VU Lecture # 11 Limits: A Rigorous Approach In this section we will talk about -Formal Definition of Limit - Left-hand and Right-hand Limits So far we have been talking about limits informally. We haven't given FORMAL mathematical definitions of limit yet. We will give a formal definition of a limit. It will include the idea of left hand and right hand limits. We intuitively said that lim f (x)  L xa means that as x approaches a, f (x) approaches L. The concept of “approaches” is intuitive. The concept of “approaches” is intuitive so far, and does not use any of the concepts and theory of Real numbers we have been using so far. So let’s formalize LIMIT Note that when we talked about “f (x) approaches L” as “x approaches a” from left and right, we are saying that we want f( x) to get as close to L as we want provided we can get x as close to a as we want as well, but maybe not equal to a since f (a) maybe undefined and f (a) may not equal L. So naturally we see the idea of INTERVALS involved here. I will rephrase the statement above in intervals as For any number   0 if we can find an open interval (x0 , x1) on the x  axis containing a po int a such that L    f (x)  L   for each x in (x0 , x1 ) except possibly x  a.Then we say lim f (x)  L xa So, f (x) is in the interval ( L   , L   ) Now you may ask, what is this  all about?? Well, it is the number that signifies the idea of “f (x) being as close to L as we want to be” could be a very small positive number, and that why it Let's us get as close to f (x) as we want. Imagine it to be something like the number at the bottom is called a GOOGOLPLEX!! 1 100 1010 © Copyright Virtual University of Pakistan 93 11-Limits: A Rigorous Approach VU So   0 but very close to it, and for ANY such  we can find an interval on the x-axis that can confine a. Let's pin down some details. Notice that When we said L    f (x)  L   holds for every x in the interval (x0, x1) ( except possibly at x= a), it is the same as saying that the same inequality hold of all x in the interval set (x0 , a)  (a, x1). But then the inequality L    f (x)  L   holds in any subset of this interval, namely (x0 , a)  (a, x1)  is any positive real number smaller than a – x0 and x1- a. Look at figure below L    f (x)  L   can be written as | f (x)  L |  and ( a   , a)  (a, a   ) as 0  | x  a |  . Are same sets by picking numbers close to a and a-delta. Have them look at definition again and talk. Let's use this definition to justify some GUESSES we made about limits in the previous lecture. Example Find lim(3x  5)  1 x2 Given any positive number  we can find an  such that (3x  5) 1   if x satisfies 0  x  2   In this example we have f (x)  3x  5 L  1 a2 © Copyright Virtual University of Pakistan 94 11-Limits: A Rigorous Approach VU So our task is to find out the  for which will work for any  we can say the following (3x  5) 1   if 0  x  2    3x  6   if 0  x  2    3 x  2   if 0  x  2     x  2   if 0  x  2   3 Now we find our  that makes our statement true. Note that the first part of the statement depends on the second part for being true. So our CHOICE of  will determine the trueness of the first part.  I let in the second part which makes the first part true. So we have 3   x2   if 0  x  2   3 3 Hence we have proved that lim(3x  5)  1 x2 Example   1 if x  0 f (x)   1 if x  0 Show that lim f (x) does not exit. x0 Suppose that the Limit exists and its L. So lim f (x)  L. Then for any   0 we can find   0 x0 such that   f (x)  L   if 0  x  0   In particular, if we take   1 there is a   0 such that f (x)  L  1 if 0  x  0      But x  and x   both satisfy requirement above, so 2 2 © Copyright Virtual University of Pakistan 95 11-Limits: A Rigorous Approach VU    f ( )  L  1 and f ( )  L  1 2 2   But is positive and   is negative, so 2 2   f ( )  1 and f ( )  1 2 2 So we get 1 L  1 and 1 L  1 0L2 and 2 L  0 But this is a contradiction since L cannot be between these two bounds at the same time. © Copyright Virtual University of Pakistan 96 12-Continuity VU LECTURE # 12 Continuity Develop the concept of CONTINUITY by examples Give a mathematical definition of continuity of functions Properties of continuous functions Continuity of polynomials and rational functions Continuity of compositions of functions The Intermediate values theorem CONTINUITY of a function becomes obvious from its graph at certain points in the plane.We will say CONTINUITY of a function or graph of a function interchangeably. DISCONTINUITY The above given curve is discontinuous at point c since f(x) is not defined there. when the following things happens then there is a break or discontinuity in the graph of a function f(x) at x =c f is undefined at c The lim f (x) does not exist. xc The function is defined at c and the lim f (x) exists, but the values of f(x) and the values of the xc limit differ at the point c So we get the following definition for continuity © Copyright Virtual University of Pakistan 97 12-Continuity VU Definition2.7.1 (a) f( c) is defined (b) lim f (x) exists xc (c ) lim f (x)  f (c) xc If any of these conditions in this definition fail to hold for a function f(x) at a point c, then f is called discontinuous at c c is called the point of discontinuity If f(x) is continuous at all points in an interval (a, b), then we say that f is continuous on (a,b) A function continuous on the interval ,  is called a continuous function Example x2  4 f (x)    x2  x2  4  if x  2 g(x)   x  2 3 if x  2 f is discontinuous at x = 2 because f(2) is undefined. g is discontinuous because g(2)=3 and x2  4 lim g(x)  lim  lim(x  2)  4 x2 x2 x2 x2 So lim g(x)  g(2) x2 © Copyright Virtual University of Pakistan 98 12-Continuity VU The last equation does not satisfy the condition of continuity. Condition (3) of the definition is enough to determine whether a function is continuous or not. This is so because if (3) is true, then (1) and (2) have to be true Example Show that f (x)  x2  2x 1 is a continuous function. CONTINUOUS means continuous at all real numbers. Show that part (3) of definition is met for all real number c. By what we know about polynomials so far, we have lim f (x)  f (c) xc So lim(x2  2x 1)  c2  2c 1 xc Part (3) is met and f(x) is continuous Theorem 2.7.2 Polynomials are continuous functions. Proof: If P is polynomial and c is any real number then by theorem 2.5.2 lim p(x)  p(c) xc Where p is a polynomial, and c is any real number. Since c is any real number, it follows that p(x) is continuous. Example Show that f (x)  x is continuous Rewrite f(x) as x if x  0 f (x)  x   x if x  0 Show lim f (x)  f (c) for any real number c. xc Let c  0. Then f (c)  c by definition of f (x). Also lim f (x)  lim x  lim x  c Since c  0 xc xc xc x may be negative to begin with, but since it approaches c which is positive or 0, we use the first part of the definition of f(x) to evaluate the limit That is just f(x) = x which is a polynomial and hence we get the desired result. © Copyright Virtual University of Pakistan 99 12-Continuity VU Now let c < 0. Then again f (c)  c by definition of f (x) and lim f (x)  lim x  lim x  c xc xc xc x may be Positive or 0 to begin with, but since it approaches c which is negative, we use the Second part of the definition of f(x) to evaluate the limit. That is just f(x) = -x which is a polynomial and hence we get the desired result. Properties of Continuous Functions Theorem 2.7.3 If the function f and g are continuous at c, then a) f + g is continuous at c; b) f – g is continuous at c; c) f. g is continuous at c; d) f/g is continuous at c if g(c)  0 and is discontinuous at c if g(c )=0 PROOF Let f and g be continuous function at the number c Then lim f (x)  f (c) xc lim g(x)  g(c) xc So lim f (x).g(x)  lim f (x).lim g(x) by Limit Rules xc xc xc  f (c).g(c) by continuity of f and g Continuity of Rational Functions Example x2  9 Where is h(x)  continuous? x2  5x  6 Since the top and the bottom functions in h are polynomials, they are continuous everywhere Hence, by property (d) of theorem 2.7.3, h will be continuous at all points c as long as g(c)  0. x2  5x  6  0 © Copyright Virtual University of Pakistan 100 12-Continuity VU Will give us all the x values where h will be discontinuous. These are x  2 x  3 which you get after solving the above equation for x. Continuity of Composition of functions Theorem 2.7.5 Let limit stand for one of the limits lim , lim , lim , lim , or lim. If lim g(x)  L and if the xc xc xc x x function f is continuous at L, Then lim f (g(x))  f (L). that is lim f (g(x))  f (lim g(x)). Example f (x)  5  x2 Here, f (x)  x , g(x)  5  x2 SO by theorem 2.7.5 lim 5  x2  lim 5  x2  4  4 x3 x3 Theorem 2.7.6 If the function g is continuous at the point c and the function f is continuous at the point g (c ), then the composition f o g is continuous at c. Continuity from the left and right Definition we use does not incorporate end points as at end points only left hand or right hand limits make sense (a) (b) © Copyright Virtual University of Pakistan 101 12-Continuity VU (c ) Graph of function in a) shows that f is discontinuous at a Graph of function in b) shows that f is discontinuous at b Graph of function in c) shows that f is continuous at a and b Definition 2.7.7 A function f is called continuous from the left at point c if the conditions in the left column below are satisfied, and is called continuous from the right at the point c if the conditions in the right column are satisfied. 1. f(c ) is defined. 1`. f(c ) is defined 2. lim f (x) exists. 2`. lim f (x) exists. xc xc  3. lim f (x)  f (c). 3`. lim f (x)  f (c). xc xc  Definition 2.7.8 A function f is said to be continuous on a closed interval [a, b] if the following conditions are satisfied: 1. f is continuous on (a, b). 2. f is continuous from the right at a. 3. f is continuous from the left at b. EXAMPLE Show that f (x)  9  x2 is continuous on the interval [3,-3].By definition 2.7.8 and theorem 2.5.1(e), for c in (3,-3) lim f (x)  lim 9  x2  lim(9  x2 )  9  c2  f (c) xc xc xc So f is continuous on (3,-3) Also lim 9  x2  lim (9  x2 )  f (3)  0 x3 x3 © Copyright Virtual University of Pakistan 102 12-Continuity VU lim 9  x2  lim (9  x2 )  f (3)  0 x3 x3 Why approach 3 from the left and –3 from the right? Well, draw the graph of this function and you will see WHY!?? So f is continuous on [-3, 3]. Intermediate Value Theorem(Theorem 2.7.9) If f is continuous on a closed interval [a, b] and C is any number between f(a) and f(b), inclusive, then there is at least one number x in the interval [a, b] such that f(x) = C. Theorem 2.7.10 If f is continuous on [a, b], and if f(a) and f(b) have opposite signs, then there is at least one solution of the equation f(x) = 0 in the interval (a, b). Example x3  x 1  0 Cannot be solved easily by factoring. However, by the MVT, f(1) = -1 and f(2) = 5 implies that the equation has one solution in the interval (1,2). © Copyright Virtual University of Pakistan 103 13-Limits and continuity of Trigonometric functions VU LECTURE # 13 Limits and continuity of Trigonometric functions Continuity of Sine and Cosine functions Continuity of other trigonometric functions Squeeze Theorem Limits of Sine and Cosine as x goes to +- infinity You will have to recall some trigonometry. Refer to Appendix B of your textbook. Continuity of Sine and Cosine Sin and Cos are ratios defined in terms of the acute angle of a right angle triangle and the sides of the triangle. Namely, adjacent side Opposite side cos  sin  Hypoteneous Hypoteneous We look at these ratios now as functions. We consider our angles in radians Instead of  we will use x Here is a picture that shows the graph of f(x) = sin (x). Put the circle picture here, and then unravel it and get the standard picture. From the graph of Sin and cosine, its obvious that © Copyright Virtual University of Pakistan 104 13-Limits and continuity of Trigonometric functions VU lim sin(x)  0 lim cos(x)  1 x0 x0 This is the intuitive approach. Prove this using the Delta Epsilon definitions!! Note that sin(0) = 0 and cos(0) = 1 Well, the values of the functions match with those of the limits as x goes to 0!! So we have this theorem THEOREM 2.8.1 The functions sin(x) and cos(x) are continuous. As clear from figure Here is the definition of continuity we saw earlier. A function f is said to be continuous at c if the following are satisfied (a) f( c) is defined (b) lim f (x) exists xc (c) lim f (x)  f (c) xc Let h = x - c. So x = h + c. Then x c is equivalent to the requirement that h 0. So we have Definition A function is continuous at c if the following are met (a) f( c) is defined (b) lim f (h  c) exists h0 (c) lim f (h  c)  f (c) h0 We will use this new definition of Continuity to prove Theorem 2.8.1 The functions sin(x) and cos(x) are continuous. © Copyright Virtual University of Pakistan 105 13-Limits and continuity of Trigonometric functions VU Proof We will assume that lim sin(x)  0 and lim cos(x)  1 x0 x0 From the above, we see that the first two conditions of our continuity definition are met. So just have to show by part 3) that limsin(c  h)  sin(c) h0 limsin(c  h)  lim [sin(c) cos(h)  cos(c) sin(h)] h0 h0  limsin(c) cos(h)  lim cos(c) sin(h) h0 h0  sin(c) lim cos(h)  cos(c) limsin(h) h0 h0  sin(c)(1)  cos(c)(0)  sin(c) The continuity of cos(x) is also proved in a similar way, and I invite you to try do that! Continuity of other trigonometric functions f (x) Remember by theorem 2.7.3 that if f(x) and g(x) are continuous , then so is h(x) . Except where g(x) g(x) = 0. So tan(x) is continuous everywhere except at cos(x) = 0 which gives sin(x) tan(x)  cos(x) 3 5 x ,  ,  ,... 2 2 2 Likewise, since cos(x) 1 1 cot(x)  , sec(x)  cosec(x)  sin(x) cos(x) sin(x) We can see that they are all continuous on appropriate intervals using the continuity of sin(x) and cos(x) and theorem 2.7.3 Squeeze Theorem for finding Limits We will show that.These is important results which will be used later. If you remember, the very first example of limits we saw was sin(x) 1 cos(x) lim  1 and lim 0 x0 x x0 x Now we prove this sin(x) lim x0 x Here are the graphs of the functions. © Copyright Virtual University of Pakistan 106 13-Limits and continuity of Trigonometric functions VU They suggest that the limits are what we want them to be! We need to prove this PROBLEM. As x goes to 0, both the top and the bottom functions go to 0. Sin(x) goes to 0 means that the fraction as a whole goes to 0. x goes to zero means that the fraction as a whole goes to +inf!. There is a tug of war between the Dark Side and the Good Side of the Force. So there is a tug-of-war between top and bottom. To find the limit we confine our function between two simpler functions, and then use their limits to get the one we want. SQUEZZING THEOREM  Let fm g and h be functions satisfying g(x)  f (x)  h(x) for all x in some open interval containing the point a, with the possible exception that the inequality need not to hold at a. We g and h have the same limits as x approaches to a, say lim g(x)  lim h(x)  L xa xa Then f also has this limit as x approaches to a, that is lim f (x)  L. xa Example 2 12 lim x sin ( ) x0 x Remember that the 0  sin(x)  1. So certainly 0  sin2 (x)  1. And so 0  sin 2 ( 1 )  1. x Multiply throughout this last inequality by x2. We get 0 2 2  x , 1 2 x sin ( ) x © Copyright Virtual University of Pakistan 107 13-Limits and continuity of Trigonometric functions VU But lim 0  lim x2  0 x0 x0 So by the Squeezing theorem 2 2 1 lim x sin ( )0 x0 x Now Let's use this theorem to prove our original claims. The proof will use basic facts about circles and areas of SECTORS with center angle of  radians and radius r. 1 2 The area of a sector is given by A  r. 2 Theorem 2.8.3 sin(x) lim 1 x0 x Proof  Let x be such that 0  x . Construct the angle x in the standard position starting from the center of a 2 unit circle. We have the following scenario From the figure we have © Copyright Virtual University of Pakistan 108 13-Limits and continuity of Trigonometric functions VU 0  area of  OBP  area of sector OBP  area of OBQ Now 1 1 1 area of  OBP  base.height  (1).sin(x)  sin(x) 2 2 2 1 2 1 area of sec tor OBP  (1).x  x 2 2 1 1 1 area of  OBQ  base.height  (1) tan(x)  tan(x) 2 2 2 So 1 1 1 0  sin(x)  x  tan(x) 2 2 2 2 Multiplying through by gives sin(x) x 1 1  sin(x) cos(x) Taking reciprocals gives sin(x) cos(x)  1 x  We had made the assumption that 0  x . 2  Also works when   x  0 You can check when you do exercise 4.9 So our last equation holds for all 2 angles x except for x = 0. Remember that lim cos(x)  1 and lim1  1 x0 x0 Taking limit now and using squeezing theorem gives sin(x) lim cos(x)  lim  lim1 x0 x0 x0 x sin(x)  1  lim 1 x0 x Since the middle thing is between 1 and 1, it must be 1!! Prove yourself that 1 cos(x) lim 0 x0 x Limits of sin(x) and cos(x) as x goes to +inf or –inf By looking at the graphs of these two functions its obvious that the y-values oscillate btw 1 and –1 as x goes to +inf or –inf and so the limits DNE!! © Copyright Virtual University of Pakistan 109 14-Rate of Change VU LECTURE # 14 Tangent Lines and Rates of Change In lecture 9, we saw that a Secant line between two points was turned into a tangent line. This was done by moving one of the points towards the other one. The secant line rotated into a LIMITING position which we regarded as a TANGENT line. For now just consider Secant lines joining two points on a curve (graph) if a function of the form y = f (x). If P(x0 , y0 ) and Q(x1 , y1 ) are distinct points on a curve y = f(x), then secant line connecting them has slope f (x1 )  f (x0 ) msec  x1  x0 © Copyright Virtual University of Pakistan 110 14-Rate of Change VU If we let x1 x0 then Q will approach P along the graph of the function y = f(x), and the secant line will approach the tangent line at P. This will mean that the slope of the Secant line will approach that of the Tangent line at P as x1 x0 , So we have the following f (x1 )  f (x0 ) mtan  lim x1 x0 x1  x0 We just saw how to find the slope of a tangent line. This was a geometric problem. In the 17th century, mathematicians wanted to define the idea of Instantaneous velocity. This was a theoretical idea. But they realized that this could be defined using the geometric idea of tangents. Let's define Average velocity formally distance travelled Avergae Velocity  Time Elapsed This formula tells us that the average velocity is the velocity at which one travels on average during some interval of time!! More interesting than Average Velocity is the idea of Instantaneous velocity. This is the velocity that an object is traveling at a given INSTANT in time. When a car hits a tree, the damage is determined by the INSTANTANEOUS velocity at the moment of impact, not on the average speed during some time interval before the impact. To define the concept of instant velocity, we will first look at distance as a function of time, d = f(t). After all, distance covered is a physical phenomenon which is always measured with respect to time. Going from New York to San Francisco (km) takes about 6 hours, if your average speed is 800 km/h. This will give us a way to plot the position versus time curve for motion. Now we will give a geometric meaning to the concept of Average Velocity. Average velocity is defined as the distance traveled over a given time of period. So if your curve for f(t) looks like as given below © Copyright Virtual University of Pakistan 111 14-Rate of Change VU then the average velocity over the time interval [t0 , t1] is defined as distance traveled during the interval Avergae Velocity  Time Elapsed d1  d0 f (t )  f (t ) t  t  vave  1t  t 0 1 0 1 0 d1  d0 is the distance traveled in the interval. (t0 , d0 ) and (t1 , d1 ). So average velocity is just the slope of the Secant line joining the points t t Say we want to know the instantaneous velocity at the point 0. We can find this by letting t1 approach 0. When this happens, the interval over which the average velocity is measured shrinks and we can approximate instant velocity. t t As 1 gets very close to 0 , our approximate instantaneous velocity will get better and better. As this continues, we can see that the average velocity over the interval gets closer to instantaneous velocity t0. So we can say at f (t1 )  f (t0 ) vinst  lim vave  lim t1 t 0 t1 t 0 t1  t0 (t0 , d0 ) But this is just the slope of the tangent line at the point Remember that the limit here means that the two sided limits exist. Average and Instantaneous rates of change Let's make the idea of average and instantaneous velocity more general. Velocity is the rate of change of position with respect to time. Algebraically we could say: Rate of change of d with respect to t.Where d = f(t). Rate of change of bacteria w.r.t time. Rate of change a length of a metal rod w.r.t to temperature Rate of change of production cost w.r.t quantity produced. All of these have the idea of the rate of change of one quantity w.r.t another quantity. We will look at quantities related by a functional relationship y = f(x) So we consider the rate of change of y w.r.t x or in other words, the rate of change of the dependant variable (quantity) w.r.t the Independent variable (quantity). Average rate of change will be represented by the slope of a certain Secant Line. Instantaneous rate of change will be represented by the slope of a certain tangent Line. © Copyright Virtual University of Pakistan 112 14-Rate of Change VU Definition 3.1.1 If y = f(x), then the average rate of change of y with respect to x over the interval [x 0, x 1] is the slope m sec of the secant line joining the points ( x 0, f(x 0)) and ( x 1, f(x 1)) on the graph of f f (x1 )  f (x0 ) msec  x1  x0 If y= f(x), then the Average rate of Change of y with respect to x over the interval [x0 , x1] is the slope of the secant line joining the points [x0 , f (x0 )] and [x1 , f (x1 )]. That is f (x1 )  f (x0 ) msec  x1  x0 And on the graph of f. Definition 3.1.1 If y= f(x), then the instantaneous rate of Change of y with respect to x at the point x 0 is the slope mtan of the tangent line to graph of f at the point x0, that is f (x1 )  f (x0 ) mtan  lim x1 x0 x1  x0 © Copyright Virtual University of Pakistan 113 14-Rate of Change VU Example Let y  f (x)  x2 1 a) Find the average rate of y w..r.t to x over the interval [3,5] b) Find the instantaneous rate of change of y w.r.t x at the x  x0 point x0 = -4 c) Find the instantaneous rate of change of y w.r.t x at a general point Solution: We use the formula in definition of Average rate with jab instanteous a gaya tu hum na tan lim wala formula lagna ha aur agr average agya tu hum na sec wala formula y  f (x)  x2 1, x  3 and x  5 lagna ha 0 1 f (x1 )  f (x0 ) f (5)  f (3) 26 10 msec  x x  53  53  8 1 0 So y increase 8 units for each unit increases in x over the interval [3,5] b) Applying the formula with y  f (x)  x2 1and x0  4 gives f (x1)  f (x 0) (x 2 1) 17 m  lim  lim 1 tan x1 x0 x x x1 4 x 4 1 0 1 x1 2 16  lim  lim (x1  4)  8 x1 4 x1  4 x1 4 Negative inst rate of change means its DECREASING c) Here we have f (x1)  f (x 0) (x12 1)  (x 02 1) m  lim  lim tan x1 x0 x x x1 x0 x x 1 0 1 0 x  x02 2  lim 1  lim (x  x )  2x x1 x0 x  x x1 x0 1 0 0 1 0 The result of part b) can be obtained from this general result by letting. © Copyright Virtual University of Pakistan 114 15-Derivative VU Lecture # 15 The Derivative In the previous lecture we saw that the slope of a tangent line to the graph of y = f (x) is given by m  lim f (x1)  f (x0 ) tan x1 x0 x x 1 0 Let's do some algebraic manipulations. Let h  x1  x0 so that x1  x0  h and h 0 as x1 x0. So we can rewrite the above tangent formula as f (x0  h)  f (x0 ) mtan  lim h0 h Definition 3.2.1 If P(x 0, y 0) is a point on the graph of a function f then the tangent line to the graph of f at P is defined to be the line through P with slope f (x0  h)  f (x0 ) m tan  lim h0 h Tangent line at P(x0 , y0 ) is just called the tangent line at x0 for brevity. Also a point P(x0 , y0 ) make here that the Equation. We make this definition provided that the LIMIT in the definition exists! Equation of the tangent line at the point P(x0 , y0 ) is y  y0  mtan (x  x0 ) © Copyright Virtual University of Pakistan 115 15-Derivative VU Example Find the slope and an equation of the tangent line to the graph of f (x)  x2 at the point P(3,9). Here is the We use the formula given in the above definition with x0  3 and y0  9. First we find the slope of the tangent line at x0  3 mtan  lim f (3  h)  f (3) lim (3  h)  9 2 h0 h h0 h (9  6h  h2 )  9 6h  h2  lim lim h0 h h0 h h(6  h)  lim  lim(6  h)  6 h0 h0 h Now we find the equation of the tangent line y  9  6(x  3)  y  6x  9 Now notice that mtan is a function of x0 because since it depends on where along the curve is being computed. Also, from the formula for it, it should be clear that h eventually shrinks to 0 and whatever is left will be in terms of x0. This can be further modified by saying that we will call x0 is x. Then we have mtan as a function of x and this is nice. Since now we can say that we have associated a new function mtan to any given function. We can rewrite the formula for mtan as © Copyright Virtual University of Pakistan 116 15-Derivative VU f (x  h)  f (x) mtan  lim h0 h This is a function of x and its very important. Its called the Derivative function with respect to x for the function y = f (x) Definition 3.2.2 The function f defined by the formula f (x  h)  f (x) f   lim h0 h is called the derivative with respect to x of the function f. The domain of f’ consists of all x for which limit exists. We can interpret this derivative in 2 ways Geometric interpretation of the Derivative f’ is the function whose value at x is the slope of the tangent line to the graph of the function f at x Rate of Change is an interpretation of Derivative. If y = f(x), then f’ is the function whose value at x is the instantaneous rate of change of y with respect to x at the point x. Example Let f (x)  x2 1 Find f’ (x). Use the result from part a) to find the slope of the tangent line to y  f (x)  x2 1 f (x  h)  f (x) [(x  h)2 1] [x 2 1] f (x)  lim lim h0 h h0 h x2  2xh  h2 1 x2 1 2xh  h2  lim lim h0 h h0 h  lim(2x  h)  2x h0 we show that the slope of the tangent line at ANY point x is f '(x)  2x , So at point x = 2 we have slope f '(2)  2(2)  4 at point x = 0 we have slope f '(0)  2(0)  0 at point x = -2 we have slope f '(2)  2(2)  4 © Copyright Virtual University of Pakistan 117 15-Derivative VU Example 3 It should be clear that at each point on a straight line y  mx  b the tangent line coincides with the line itself. So the slope of the tangent line must be the same as that of the original line, namely m. We can prove this here f (x  h)  f (x) [m(x  h)  b]  (mx  b) f '(x)  lim  lim h0 h h0 h mx ?

MTH101 Highlighted Handouts PDF

Document Details

Tags

Related

Summary

Full Transcript

Upgrade to continue