# Lecture notes on Calculus

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MATH 221 FIRST SEMESTER CALCULUS fall 2009 Typeset:June 8, 2010 1MATH 221 1st SEMESTER CALCULUS LECTURE NOTES VERSION 2.0 (fall 2009) This is a self contained set of lecture notes for Math 221. The notes were written by Sigurd Angenent, starting A from an extensive collection of notes and problems compiled by Joel Robbin. The LT X and Python les E which were used to produce these notes are available at the following web site http://www.math.wisc.edu/ angenent/Free-Lecture-Notes They are meant to be freely available in the sense that \free software" is free. More precisely: Copyright (c) 2006 Sigurd B. Angenent. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".Contents 3. Exercises 64 4. Finding sign changes of a function 65 5. Increasing and decreasing functions 66 6. Examples 67 Chapter 1. Numbers and Functions 5 7. Maxima and Minima 69 1. What is a number? 5 8. Must there always be a maximum? 71 2. Exercises 7 9. Examples functions with and without maxima or 3. Functions 8 minima 71 4. Inverse functions and Implicit functions 10 10. General method for sketching the graph of a 5. Exercises 13 function 72 11. Convexity, Concavity and the Second Derivative 74 Chapter 2. Derivatives (1) 15 12. Proofs of some of the theorems 75 1. The tangent to a curve 15 13. Exercises 76 2. An example tangent to a parabola 16 14. Optimization Problems 77 3. Instantaneous velocity 17 15. Exercises 78 4. Rates of change 17 5. Examples of rates of change 18 Chapter 6. Exponentials and Logarithms (naturally) 81 6. Exercises 18 1. Exponents 81 2. Logarithms 82 Chapter 3. Limits and Continuous Functions 21 3. Properties of logarithms 83 1. Informal de nition of limits 21 4. Graphs of exponential functions and logarithms 83 2. The formal, authoritative, de nition of limit 22 x 5. The derivative of a and the de nition of e 84 3. Exercises 25 6. Derivatives of Logarithms 85 4. Variations on the limit theme 25 7. Limits involving exponentials and logarithms 86 5. Properties of the Limit 27 8. Exponential growth and decay 86 6. Examples of limit computations 27 9. Exercises 87 7. When limits fail to exist 29 8. What's in a name? 32 Chapter 7. The Integral 91 9. Limits and Inequalities 33 1. Area under a Graph 91 10. Continuity 34 2. When f changes its sign 92 11. Substitution in Limits 35 3. The Fundamental Theorem of Calculus 93 12. Exercises 36 4. Exercises 94 13. Two Limits in Trigonometry 36 5. The inde nite integral 95 14. Exercises 38 6. Properties of the Integral 97 7. The de nite integral as a function of its integration Chapter 4. Derivatives (2) 41 bounds 98 1. Derivatives De ned 41 8. Method of substitution 99 2. Direct computation of derivatives 42 9. Exercises 100 3. Di erentiable implies Continuous 43 4. Some non-di erentiable functions 43 Chapter 8. Applications of the integral 105 5. Exercises 44 1. Areas between graphs 105 6. The Di erentiation Rules 45 2. Exercises 106 7. Di erentiating powers of functions 48 3. Cavalieri's principle and volumes of solids 106 8. Exercises 49 4. Examples of volumes of solids of revolution 109 9. Higher Derivatives 50 5. Volumes by cylindrical shells 111 10. Exercises 51 6. Exercises 113 11. Di erentiating Trigonometric functions 51 7. Distance from velocity, velocity from acceleration 113 12. Exercises 52 8. The length of a curve 116 13. The Chain Rule 52 9. Examples of length computations 117 14. Exercises 57 10. Exercises 118 15. Implicit di erentiation 58 11. Work done by a force 118 16. Exercises 60 12. Work done by an electric current 119 Chapter 5. Graph Sketching and Max-Min Problems 63 Chapter 9. Answers and Hints 121 1. Tangent and Normal lines to a graph 63 2. The Intermediate Value Theorem 63 GNU Free Documentation License 125 31. APPLICABILITY AND DEFINITIONS 125 2. VERBATIM COPYING 125 3. COPYING IN QUANTITY 125 4. MODIFICATIONS 125 5. COMBINING DOCUMENTS 126 6. COLLECTIONS OF DOCUMENTS 126 7. AGGREGATION WITH INDEPENDENT WORKS 126 8. TRANSLATION 126 9. TERMINATION 126 10. FUTURE REVISIONS OF THIS LICENSE 126 11. RELICENSING 126 4CHAPTER 1 Numbers and Functions The subject of this course is \functions of one real variable" so we begin by wondering what a real number \really" is, and then, in the next section, what a function is. 1. What is a number? 1.1. Di erent kinds of numbers. The simplest numbers are the positive integers 1; 2; 3; 4; the number zero 0; and the negative integers  ;4;3;2;1: Together these form the integers or \whole numbers." Next, there are the numbers you get by dividing one whole number by another (nonzero) whole number. These are the so called fractions or rational numbers such as 1 1 2 1 2 3 4 ; ; ; ; ; ; ; 2 3 3 4 4 4 3 or 1 1 2 1 2 3 4 ; ; ; ; ; ; ; 2 3 3 4 4 4 3 By de nition, any whole number is a rational number (in particular zero is a rational number.) You can add, subtract, multiply and divide any pair of rational numbers and the result will again be a rational number (provided you don't try to divide by zero). One day in middle school you were told that there are other numbers besides the rational numbers, and the rst example of such a number is the square root of two. It has been known ever since the time of the m greeks that no rational number exists whose square is exactly 2, i.e. you can't nd a fraction such that n  m 2 2 2 = 2; i.e. m = 2n : n 2 2 Nevertheless, if you compute x for some values ofx between 1 and 2, and check if you x x get more or less than 2, then it looks like there should be some number x between 1:4 and 1:2 1:44 1:5 whose square is exactly 2. So, we assume that there is such a number, and we call it 1:3 1:69 p the square root of 2, written as 2. This raises several questions. How do we know there 1:4 1:96 2 2 really is a number between 1:4 and 1:5 for which x = 2? How many other such numbers 1:5 2:25 2 are we going to assume into existence? Do these new numbers obey the same algebra rules 1:6 2:56 (like a +b =b +a) as the rational numbers? If we knew precisely what these numbers (like p 2) were then we could perhaps answer such questions. It turns out to be rather dicult to give a precise description of what a number is, and in this course we won't try to get anywhere near the bottom of this issue. Instead we will think of numbers as \in nite decimal expansions" as follows. One can represent certain fractions as decimal fractions, e.g. 279 1116 = = 11:16: 25 100 51 Not all fractions can be represented as decimal fractions. For instance, expanding into a decimal fraction 3 leads to an unending decimal fraction 1 = 0:333 333 333 333 333 3 1 It is impossible to write the complete decimal expansion of because it contains in nitely many digits. 3 But we can describe the expansion: each digit is a three. An electronic calculator, which always represents 1 numbers as nite decimal numbers, can never hold the number exactly. 3 Every fraction can be written as a decimal fraction which may or may not be nite. If the decimal expansion doesn't end, then it must repeat. For instance, 1 = 0:142857 142857 142857 142857 ::: 7 Conversely, any in nite repeating decimal expansion represents a rational number. A real number is speci ed by a possibly unending decimal expansion. For instance, p 2 = 1:414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 376 9::: Of course you can never write all the digits in the decimal expansion, so you only write the rst few digits p and hide the others behind dots. To give a precise description of a real number (such as 2) you have to explain how you could in principle compute as many digits in the expansion as you would like. During the next three semesters of calculus we will not go into the details of how this should be done. p 1.2. A reason to believe in 2. The Pythagorean theorem says that the hy- p potenuse of a right triangle with sides 1 and 1 must be a line segment of length 2. In middle or high school you learned something similar to the following geometric construction p of a line segment whose length is 2. Take a square with side of length 1, and construct a new square one of whose sides is the diagonal of the rst square. The gure you get consists of 5 triangles of equal area and by counting triangles you see that the larger square has exactly twice the area of the smaller square. Therefore the diagonal of the smaller square, being p the side of the larger square, is 2 as long as the side of the smaller square. Why are real numbers called real? All the numbers we will use in this rst semester of calculus are \real numbers." At some point (in 2nd semester calculus) it becomes useful to assume that there is a number whose square is1. No real number has this property since the square of any real number is positive, so it was decided to call this new imagined number \imaginary" and to refer to the numbers we already have p (rationals, 2-like things) as \real." 1.3. The real number line and intervals. It is customary to visualize the real numbers as points on a straight line. We imagine a line, and choose one point on this line, which we call the origin. We also decide which direction we call \left" and hence which we call \right." Some draw the number line vertically and use the words \up" and \down." To plot any real number x one marks o a distance x from the origin, to the right (up) if x 0, to the left (down) if x 0. The distance along the number line between two numbers x and y isjxyj. In particular, the distance is never a negative number. 3 2 1 0 1 2 3 Figure 1. To draw the half open interval 1; 2) use a lled dot to mark the endpoint which is included and an open dot for an excluded endpoint. 6p 2 1 0 1 2 2 p Figure 2. To nd 2 on the real line you draw a square of sides 1 and drop the diagonal onto the real line. Almost every equation involving variables x, y, etc. we write down in this course will be true for some values of x but not for others. In modern abstract mathematics a collection of real numbers (or any other kind of mathematical objects) is called a set. Below are some examples of sets of real numbers. We will use the notation from these examples throughout this course. The collection of all real numbers between two given real numbers form an interval. The following notation is used  (a;b) is the set of all real numbers x which satisfy axb.  a;b) is the set of all real numbers x which satisfy axb.  (a;b is the set of all real numbers x which satisfy axb.  a;b is the set of all real numbers x which satisfy axb. If the endpoint is not included then it may be1 or1. E.g. (1; 2 is the interval of all real numbers (both positive and negative) which are 2. 1.4. Set notation. A common way of describing a set is to say it is the collection of all real numbers which satisfy a certain condition. One uses this notation  A = xjx satis es this or that condition Most of the time we will use upper case letters in a calligraphic font to denote sets. (A,B,C,D, . . . ) For instance, the interval (a;b) can be described as  (a;b) = xjaxb The set  2 B = xjx 1 0 2 consists of all real numbers x for which x 1 0, i.e. it consists of all real numbers x for which either x 1 or x1 holds. This set consists of two parts: the interval (1;1) and the interval (1;1). You can try to draw a set of real numbers by drawing the number line and coloring the points belonging to that set red, or by marking them in some other way. Some sets can be very dicult to draw. For instance,  C = xjx is a rational number can't be accurately drawn. In this course we will try to avoid such sets. Sets can also contain just a few numbers, like D =f1; 2; 3g which is the set containing the numbers one, two and three. Or the set  3 2 E = xjx 4x + 1 = 0 3 2 which consists of the solutions of the equation x 4x + 1 = 0. (There are three of them, but it is not easy to give a formula for the solutions.) IfA andB are two sets then the union ofA andB is the set which contains all numbers that belong either toA or toB. The following notation is used  AB = xjx belongs toA or toB or both. 7Similarly, the intersection of two setsA andB is the set of numbers which belong to both sets. This notation is used:  A\B = xjx belongs to bothA andB. 2. Exercises th 1. What is the 2007 digit after the period in the expan- 4. SupposeA andB are intervals. Is it always true that 1 sion of ? A\B is an interval? How aboutAB? 7 5. Consider the sets 2. Which of the following fractions have nite decimal   expansions? M = xjx 0 andN = yjy 0 : 2 3 276937 Are these sets the same? a = ; b = ; c = : 3 25 15625 6. Group Problem. 3. Draw the following sets of real numbers. Each of these Write the numbers sets is the union of one or more intervals. Find those x = 0:3131313131:::; y = 0:273273273273::: intervals. Which of thee sets are nite?  2 and z = 0:21541541541541541::: A = xjx 3x + 2 0  2 m B = xjx 3x + 2 0 as fractions (i.e. write them as , specifying m and n.) n  2 C = xjx 3x 3 (Hint: show that 100x = x + 31. A similar trick  2 D = xjx 5 2x works for y, but z is a little harder.)  2 E = tjt 3t + 2 0 7. Group Problem.  2 F = j 3 + 2 0 Is the number whose decimal expansion after the G = (0; 1) (5; 7 p  period consists only of nines, i.e. H = f1gf2; 3g \ (0; 2 2)  1 x = 0:99999999999999999::: Q = j sin = 2  R = 'j cos' 0 an integer? 3. Functions Wherein we meet the main characters of this semester 3.1. De nition. To specify a function f you must (1) give a rule which tells you how to compute the value f(x) of the function for a given real number x, and: (2) say for which real numbers x the rule may be applied. The set of numbers for which a function is de ned is called its domain. The set of all possible numbers f(x) as x runs over the domain is called the range of the function. The rule must be unambiguous: the same xmust always lead to the same f(x). p For instance, one can de ne a function f by putting f(x) = x for all x 0. Here the rule de ning f is \take the square root of whatever number you're given", and the function f will accept all nonnegative real numbers. The rule which speci es a function can come in many di erent forms. Most often it is a formula, as in the square root example of the previous paragraph. Sometimes you need a few formulas, as in ( 2x for x 0 g(x) = domain of g = all real numbers. 2 x for x 0 Functions which are de ned by di erent formulas on di erent intervals are sometimes called piecewise de ned functions. 3.2. Graphing a function. You get the graph of a function f by drawing all points whose coordi- nates are (x;y) where x must be in the domain of f and y =f(x). 8range of f y =f(x) (x;f(x)) x domain of f Figure 3. The graph of a function f. The domain of f consists of all x values at which the function is de ned, and the range consists of all possible values f can have. m 1 P 1 y 1 y y 1 0 P 0 y 0 x x 1 0 n x x 0 1 Figure 4. A straight line and its slope. The line is the graph of f(x) =mx +n. It intersects the y-axis at height n, and the ratio between the amounts by which y and x increase as you move from one point y y 1 0 to another on the line is =m: x x 1 0 3.3. Linear functions. A function which is given by the formula f(x) =mx +n where m and n are constants is called a linear function. Its graph is a straight line. The constants m and n are the slope and y-intercept of the line. Conversely, any straight line which is not vertical (i.e. not parallel to the y-axis) is the graph of a linear function. If you know two points (x ;y ) and (x ;y ) on the 0 0 1 1 line, then then one can compute the slope m from the \rise-over-run" formula y y 1 0 m = : x x 1 0 This formula actually contains a theorem from Euclidean geometry, namely it says that the ratio (y y ) : 1 0 (x x ) is the same for every pair of points (x ;y ) and (x ;y ) that you could pick on the line. 1 0 0 0 1 1 3.4. Domain and \biggest possible domain. " In this course we will usually not be careful about specifying the domain of the function. When this happens the domain is understood to be the set of all x for which the rule which tells you how to compute f(x) is meaningful. For instance, if we say that h is the function p h(x) = x 93 y =x x 3 Figure 5. The graph of y =x x fails the \horizontal line test," but it passes the \vertical line test." The circle fails both tests. then the domain of h is understood to be the set of all nonnegative real numbers domain of h = 0;1) p since x is well-de ned for all x 0 and unde ned for x 0. A systematic way of nding the domain and range of a function for which you are only given a formula is as follows:  The domain of f consists of all x for which f(x) is well-de ned (\makes sense")  The range of f consists of all y for which you can solve the equation f(x) =y. 2 2 3.5. Example nd the domain and range of f(x) = 1=x . The expression 1=x can be computed for all real numbers x except x = 0 since this leads to division by zero. Hence the domain of the function 2 f(x) = 1=x is  \all real numbers except 0" = xjx =6 0 = (1; 0) (0;1): To nd the range we ask \for which y can we solve the equation y =f(x) for x," i.e. we for which y can you 2 solve y = 1=x for x? 2 2 If y = 1=x then we must have x = 1=y, so rst of all, since we have to divide by y, y can't be zero. 2 2 Furthermore, 1=y =x says thaty must be positive. On the other hand, ify 0 theny = 1=x has a solution p (in fact two solutions), namely x =1= y. This shows that the range of f is \all positive real numbers" =fxjx 0g = (0;1): 3.6. Functions in \real life. " One can describe the motion of an object using a function. If some object is moving along a straight line, then you can de ne the following function: Let x(t) be the distance from the object to a xed marker on the line, at the time t. Here the domain of the function is the set of all times t for which we know the position of the object, and the rule is Given t, measure the distance between the object and the marker at time t. There are many examples of this kind. For instance, a biologist could describe the growth of a cell by de ning m(t) to be the mass of the cell at time t (measured since the birth of the cell). Here the domain is the interval 0;T , where T is the life time of the cell, and the rule that describes the function is Given t, weigh the cell at time t. 3.7. The Vertical Line Property. Generally speaking graphs of functions are curves in the plane but they distinguish themselves from arbitrary curves by the way they intersect vertical lines: The graph of a function cannot intersect a vertical line \x = constant" in more than one point. The reason why this is true is very simple: if two points lie on a vertical line, then they have the same x coordinate, so if they also lie on the graph of a function f, then their y-coordinates must also be equal, namely f(x). 103 3.8. Examples. The graph off(x) =x x \goes up and down," and, even though it intersects several horizontal lines in more than one point, it intersects every vertical line in exactly one point. 2 2 The collection of points determined by the equation x +y = 1 is a circle. It is not the graph of a function since the vertical line x = 0 (the y-axis) intersects the graph in two points P (0; 1) and P (0;1). 1 2 See Figure 6. 4. Inverse functions and Implicit functions For many functions the rule which tells you how to compute it is not an explicit formula, but instead an equation which you still must solve. A function which is de ned in this way is called an \implicit function." 4.1. Example. One can de ne a functionf by saying that for eachx the value off(x) is the solutiony of the equation 2 x + 2y 3 = 0: In this example you can solve the equation for y, 2 3x y = : 2 2 Thus we see that the function we have de ned is f(x) = (3x )=2. Here we have two de nitions of the same function, namely 2 (i) \y =f(x) is de ned by x + 2y 3 = 0," and 2 (ii) \f is de ned by f(x) = (3x )=2." The rst de nition is the implicit de nition, the second is explicit. You see that with an \implicit function" it isn't the function itself, but rather the way it was de ned that's implicit. 4.2. Another example: domain of an implicitly de ned function. De ne g by saying that for any x the value y =g(x) is the solution of 2 x +xy 3 = 0: Just as in the previous example one can then solve for y, and one nds that 2 3x g(x) =y = : x Unlike the previous example this formula does not make sense when x = 0, and indeed, for x = 0 our rule for g says that g(0) =y is the solution of 2 0 + 0y 3 = 0; i.e. y is the solution of 3 = 0: That equation has no solution and hence x = 0 does not belong to the domain of our function g. 2 2 p x +y = 1 2 y = + 1x p 2 y = 1x 2 2 Figure 6. The circle determined byx +y = 1 is not the graph of a function, but it contains the graphs p p 2 2 of the two functions h (x) = 1x and h (x) = 1x . 1 2 114.3. Example: the equation alone does not determine the function. De ney =h(x) to be the solution of 2 2 x +y = 1: 2 If x 1 or x1 then x 1 and there is no solution, so h(x) is at most de ned when1x 1. But when1x 1 there is another problem: not only does the equation have a solution, but it even has two solutions: p p 2 2 2 2 x +y = 1 () y = 1x or y = 1x : The rule which de nes a function must be unambiguous, and since we have not speci ed which of these two solutions is h(x) the function is not de ned for1x 1. One can x this by making a choice, but there are many possible choices. Here are three possibilities: 2 2 h (x) = the nonnegative solution y of x +y = 1 1 2 2 h (x) = the nonpositive solution y of x +y = 1 2 ( h (x) when x 0 1 h (x) = 3 h (x) when x 0 2 4.4. Why use implicit functions? In all the examples we have done so far we could replace the implicit description of the function with an explicit formula. This is not always possible or if it is possible the implicit description is much simpler than the explicit formula. For instance, you can de ne a function f by saying that y =f(x) if and only if 3 (1) y + 3y + 2x = 0: 3 This means that the recipe for computing f(x) for any given x is \solve the equation y + 3y + 2x = 0." 3 E.g. to compute f(0) you set x = 0 and solve y + 3y = 0. The only solution is y = 0, so f(0) = 0. To 3 compute f(1) you have to solve y + 3y + 2 1 = 0, and if you're lucky you see that y =1 is the solution, and f(1) =1. In general, no matter whatx is, the equation (1) turns out to have exactly one solution y (which depends on x, this is how you get the function f). Solving (1) is not easy. In the early 1500s Cardano and Tartaglia 1 discovered a formula for the solution. Here it is: q q p p 3 3 2 2 y =f(x) = x + 1 +x x + 1 +x : The implicit description looks a lot simpler, and when we try to di erentiate this function later on, it will be much easier to use \implicit di erentiation" than to use the Cardano-Tartaglia formula directly. 1 4.5. Inverse functions. If you have a function f, then you can try to de ne a new function f , the so-called inverse function of f, by the following prescription: 1 (2) For any given x we say that y =f (x) if y is the solution to the equation f(y) =x. 1 So to nd y =f (x) you solve the equation x =f(y). If this is to de ne a function then the prescription (2) must be unambiguous and the equation f(y) =x has to have a solution and cannot have more than one solution. 1 To see the solution and its history visit http://www.gap-system.org/ history/HistTopics/Quadratic_etc_equations.html 12The graph of f f(c) 1 The graph of f c b f(b) a f(a) a b c f(a) f(b) f(c) Figure 7. The graph of a function and its inverse are mirror images of each other. 4.6. Examples. Consider the function f with f(x) = 2x + 3. Then the equation f(y) =x works out to be 2y + 3 =x and this has the solution x 3 y = : 2 1 1 So f (x) is de ned for all x, and it is given by f (x) = (x 3)=2. 2 Next we consider the function g(x) =x with domain all positive real numbers. To see for which x the 1 2 inverse g (x) is de ned we try to solve the equation g(y) =x, i.e. we try to solve y =x. If x 0 then this p  = equation has no solutions since y 0 for all y. But if x 0 then y x does have a solution, namely y = x. p 1 1 So we see that g (x) is de ned for all nonnegative real numbers x, and that it is given by g (x) = x. 4.7. Inverse trigonometric functions. The familiar trigonometric functions Sine, Cosine and Tangent have inverses which are called arcsine, arccosine and arctangent. 1 y =f(x) x =f (y) y = sinx (=2x=2) x = arcsin(y) (1y 1) y = cosx (0x) x = arccos(y) (1y 1) y = tanx (=2x=2) x = arctan(y) 1 1 1 The notations arcsiny = sin y, arccosx = cos x, and arctanu = tan u are also commonly used for 1 the inverse trigonometric functions. We will avoid the sin y notation because it is ambiguous. Namely, everybody writes the square of siny as  2 2 siny = sin y: Replacing the 2's by1's would lead to  1 1 ?? 1 arcsiny = sin y = siny = ; which is not true siny 5. Exercises 8. The functions f and g are de ned by 10. Find a formula for the function f which is de ned by 2 2 2 f(x) =x and g(s) =s : y =f(x) () x yy = 6: Are f and g the same functions or are they di erent? What is the domain of f? 9. Find a formula for the function f which is de ned by 11. Let f be the function de ned by y =f(x) () y is 2 the largest solution of y =f(x) () x y +y = 7: 2 2 What is the domain of f? y = 3x 2xy: 13Find a formula for f. What are the domain and range of for all real numbers x. f? Compute 12. Find a formula for the function f which is de ned by (a) f(1) (b) f(0) (c) f(x) 2 y =f(x) () 2x + 2xy +y = 5 and yx: (d) f(t) (e) f(f(2)) Find the domain of f. where x and t are arbitrary real numbers. 13. Use a calculator to compute f(1:2) in three deci- What are the range and domain of f? mals where f is the implicitly de ned function fromx4.4. 21. Does there exist a function f which satis es (There are (at least) two di erent ways of nding f(1:2)) 2 f(x ) =x + 1 14. Group Problem. (a) True or false: for all real numbers x?  for all x one has sin arcsinx =x? (b) True or false:     for all x one has arcsin sinx =x? The following exercises review precalculus material in- 15. On a graphing calculator plot the graphs of the follow- 2 volving quadratic expressions ax +bx +c in one way or ing functions, and explain the results. (Hint: rst do the another. previous exercise.) 22. Explain how you \complete the square" in a quadratic f(x) = arcsin(sinx); 2x 2 2 expression like ax +bx. g(x) = arcsin(x) + arccos(x); 0x 1 23. Find the range of the following functions: sinx h(x) = arctan ; jxj=2 2 cosx f(x) = 2x + 3 cosx 2 k(x) = arctan ; jxj=2 g(x) =2x + 4x sinx 2 l(x) = arcsin(cosx); x h(x) = 4x +x 2 m(x) = cos(arcsinx); 1x 1 k(x) = 4 sinx + sin x 2 (x) = 1=(1 +x ) 16. Find the inverse of the function f which is given by f(x) = sinx and whose domain is x 2. Sketch 2 m(x) = 1=(3 + 2x +x ): 1 the graphs of both f and f . 17. Find a number a such that the function f(x) = 24. Group Problem. sin(x +=4) with domain axa + has an inverse. For each real number a we de ne a line  with a 1 Give a formula for f (x) using the arcsine function. 2 equation y =ax +a . 18. Draw the graph of the function h fromx4.3. 3 (a) Draw the lines corresponding to a = 1 1 2;1; ; 0; ; 1; 2. 19. A function f is given which satis es 2 2 2 (b) Does the point with coordinates (3; 2) lie on one f(2x + 3) =x or more of the lines  (where a can be any number, not a for all real numbers x. just the ve values from part (a))? If so, for which values Compute of a does (3; 2) lie on  ? a (a) f(0) (b) f(3) (c) f(x) (c) Which points in the plane lie on at least one of the lines ` ?. a (d) f(y) (e) f(f(2)) where x and y are arbitrary real numbers. 25. For which values of m and n does the graph of f(x) = mx +n intersect the graph of g(x) = 1=x in What are the range and domain of f? exactly one point and also contain the point (1; 1)? 20. A function f is given which satis es 26. For which values of m and n does the graph of  1 f = 2x 12: f(x) =mx +n not intersect the graph of g(x) = 1=x? x + 1 14CHAPTER 2 Derivatives (1) To work with derivatives you have to know what a limit is, but to motivate why we are going to study limits let's rst look at the two classical problems that gave rise to the notion of a derivative: the tangent to a curve, and the instantaneous velocity of a moving object. 1. The tangent to a curve Suppose you have a function y =f(x) and you draw its graph. If you want to nd the tangent to the graph of f at some given point on the graph of f, how would you do that? a secant Q tangent P Figure 1. Constructing the tangent by letting QP Let P be the point on the graph at which want to draw the tangent. If you are making a real paper and ink drawing you would take a ruler, make sure it goes through P and then turn it until it doesn't cross the graph anywhere else. If you are using equations to describe the curve and lines, then you could pick a point Q on the graph and construct the line through P and Q (\construct" means \ nd an equation for"). This line is called a \secant," and it is of course not the tangent that you're looking for. But if you choose Q to be very close toP then the secant will be close to the tangent. 15So this is our recipe for constructing the tangent through P : pick another point Q on the graph, nd the line through P and Q, and see what happens to this line as you take Q closer and closer to P . The resulting secants will then get closer and closer to some line, and that line is the tangent. We'll write this in formulas in a moment, but rst let's worry about how close Q should be to P. We can't set Q equal to P, because then P and Q don't determine a line (you need two points to determine a line). If you choose Q di erent from P then you don't get the tangent, but at best something that is \close" to it. Some people have suggested that one should take Q \in nitely close" to P , but it isn't clear what that would mean. The concept of a limit is meant to solve this confusing problem. 2. An example tangent to a parabola 2 To make things more concrete, suppose that the function we had was f(x) =x , and that the point was (1; 1). The graph of f is of course a parabola. Any line through the point P (1; 1) has equation y 1 =m(x 1) where m is the slope of the line. So instead of nding the equation of the secant and tangent lines we will nd their slopes. 2 Let Q be the other point on the parabola, with coordinates (x;x ). We can 2 Q x \move Q around on the graph" by changing x. Whatever x we choose, it must be di erent from 1, for otherwise P and Q would be the same point. What we want to y nd out is how the line throughP andQ changes ifx is changed (and in particular, if P 1 x is chosen very close to a). Now, as one changes x one thing stays the same, namely, x the secant still goes through P. So to describe the secant we only need to know its slope. By the \rise over run" formula, the slope of the secant line joining P and Q is 1 x y 2 m = where y =x 1 and x =x 1: PQ x 2 By factoring x 1 we can rewrite the formula for the slope as follows 2 y x 1 (x 1)(x + 1) (3) m = = = =x + 1: PQ x x 1 x 1 As x gets closer to 1, the slope m , being x + 1, gets closer to the value 1 + 1 = 2. We say that PQ the limit of the slope m as Q approaches P is 2. PQ In symbols, lim m = 2; PQ QP or, since Q approaching P is the same as x approaching 1, (4) limm = 2: PQ x1 2 So we nd that the tangent line to the parabola y =x at the point (1; 1) has equation y 1 = 2(x 1); i.e. y = 2x 1: A warning: you cannot substitute x = 1 in equation (3) to get (4) even though it looks like that's what we did. The reason why you can't do that is that when x = 1 the point Q coincides with the point P so \the line through P and Q" is not de ned; also, if x = 1 then x = y = 0 so that the rise-over-run formula for the slope gives x 0 m = = = unde ned. PQ y 0 It is only after the algebra trick in (3) that setting x = 1 gives something that is well de ned. But if the intermediate steps leading to m =x + 1 aren't valid for x = 1 why should the nal result mean anything PQ for x = 1? 16Something more complicated has happened. We did a calculation which is valid for all x6= 1, and later looked at what happens if x gets \very close to 1." This is the concept of a limit and we'll study it in more detail later in this section, but rst another example. 3. Instantaneous velocity If you try to de ne \instantaneous velocity" you will again end up trying to divide zero by zero. Here is how it goes: When you are driving in your car the speedometer tells you how fast your are going, i.e. what your velocity is. What is this velocity? What does it mean if the speedometer says \50mph"? s = 0 Time =t Time =t + t s(t) s =s(t + t)s(t) We all know what average velocity is. Namely, if it takes you two hours to cover 100 miles, then your average velocity was distance traveled = 50 miles per hour: time it took This is not the number the speedometer provides you it doesn't wait two hours, measure how far you went and compute distance=time. If the speedometer in your car tells you that you are driving 50mph, then that should be your velocity at the moment that you look at your speedometer, i.e. \distance traveled over time it took" at the moment you look at the speedometer. But during the moment you look at your speedometer no time goes by (because a moment has no length) and you didn't cover any distance, so your velocity at that 0 moment is , i.e. unde ned. Your velocity at any moment is unde ned. But then what is the speedometer 0 telling you? To put all this into formulas we need to introduce some notation. Let t be the time (in hours) that has passed since we got onto the road, and let s(t) be the distance we have covered since then. Instead of trying to nd the velocity exactly at time t, we nd a formula for the average velocity during some (short) time interval beginning at time t. We'll write t for the length of the time interval. At time t we have traveled s(t) miles. A little later, at time t + t we have traveled s(t + t). Therefore during the time interval from t to t + t we have moved s(t + t)s(t) miles. Our average velocity in that time interval is therefore s(t + t)s(t) miles per hour. t The shorter you make the time interval, i.e. the smaller you choose t, the closer this number should be to the instantaneous velocity at time t. So we have the following formula (de nition, really) for the velocity at time t s(t + t)s(t) (5) v(t) = lim : t0 t 4. Rates of change The two previous examples have much in common. If we ignore all the details about geometry, graphs, highways and motion, the following happened in both examples: We had a function y =f(x), and we wanted to know how much f(x) changes if x changes. If you change x to x + x, then y will change from f(x) to f(x + x). The change in y is therefore y =f(x + x)f(x); and the average rate of change is y f(x + x)f(x) (6) = : x x 17This is the average rate of change of f over the interval from x tox + x. To de ne the rate of change of the function f at x we let the length x of the interval become smaller and smaller, in the hope that the average rate of change over the shorter and shorter time intervals will get closer and closer to some number. If that happens then that \limiting number" is called the rate of change of f at x, or, the derivative off at x. It is written as f(x + x)f(x) 0 (7) f (x) = lim : x0 x Derivatives and what you can do with them are what the rst half of this semester is about. The description we just went through shows that to understand what a derivative is you need to know what a limit is. In the next chapter we'll study limits so that we get a less vague understanding of formulas like (7). 5. Examples of rates of change 5.1. Acceleration as the rate at which velocity changes. As you are driving in your car your velocity does not stay constant, it changes with time. Suppose v(t) is your velocity at time t (measured in miles per hour). You could try to gure out how fast your velocity is changing by measuring it at one moment in time (you get v(t)), then measuring it a little later (you get v(t))). You conclude that your velocity increased by v =v(t + t)v(t) during a time interval of length t, and hence   v v(t + t)v(t) average rate at which = = : your velocity changed  t This rate of change is called your average acceleration (over the time interval from t to t + t). Your instantaneous acceleration at time t is the limit of your average acceleration as you make the time interval shorter and shorter: v(t + t)v(t) facceleration at time tg =a = lim : t0 t th the average and instantaneous accelerations are measured in \miles per hour per hour," i.e. in 2 (mi=h)=h = mi=h : Or, if you had measured distances in meters and time in seconds then velocities would be measured in meters 2 per second, and acceleration in meters per second per second, which is the same as meters per second , i.e. \meters per squared second." 5.2. Reaction rates. Think of a chemical reaction in which two substances A and B react to form AB according to the reaction 2 A + 2B AB : 2 If the reaction is taking place in a closed reactor, then the \amounts" of A and B will be decreasing, while the amount of AB will increase. Chemists write A for the amount of \A" in the chemical reactor (measured in 2 moles). Clearly A changes with time so it de nes a function. We're mathematicians so we will write \A(t)" for the number of moles of A present at time t. To describe how fast the amount of A is changing we consider the derivative of A with respect to time, i.e. A(t + t) A(t) 0 A (t) = lim : t0 t 0 This quantity is the rate of change of A. The notation \A (t)" is really only used by calculus professors. If you open a paper on chemistry you will nd that the derivative is written in Leibniz notation: dA dt More on this inx1.2 How fast does the reaction take place? If you add more A or more B to the reactor then you would expect that the reaction would go faster, i.e. that more AB is being produced per second. The law of mass-action 2 18kinetics from chemistry states this more precisely. For our particular reaction it would say that the rate at which A is consumed is given by dA 2 =k A B ; dt in which the constant k is called the reaction constant. It's a constant that you could try to measure by timing how fast the reaction goes. 6. Exercises 27. Repeat the reasoning inx2 to nd the slope at the 31. Look ahead at Figure 3 in the next chapter. What is 1 1 2  point ( ; ), or more generally at any point (a;a ) on the derivative of f(x) =x cos at the points A and B 2 4 x 2 on the graph? the parabola with equation y =x . 28. Repeat the reasoning inx2 to nd the slope at the 32. Suppose that some quantity y is a function of some 1 1 3 point ( ; ), or more generally at any point (a;a ) on other quantity x, and suppose that y is a mass, i.e. y 2 8 3 the curve with equation y =x . is measured in pounds, and x is a length, measured in feet. What units do the increments y and x, and the 29. Group Problem. derivative dy=dx have? Should you trust your calculator? 33. A tank is lling with water. The volume (in gallons) 2 Find the slope of the tangent to the parabolay =x of water in the tank at time t (seconds) is V (t). What 1 1 at the point ( ; ) (You have already done this: see 0 3 9 units does the derivative V (t) have? exercise 27). 34. Group Problem. Instead of doing the algebra you could try to compute the slope by using a calculator. This exercise is about LetA(x) be the area of an equilateral triangle whose how you do that and what happens if you try (too hard). sides measure x inches. y dA Compute for various values of x: x (a) Show that has the units of a length. dx 6 12 dA x = 0:1; 0:01; 0:001; 10 ; 10 : (b) Which length does represent geometrically? dx y Hint: draw two equilateral triangles, one with side x and As you choose x smaller your computed ought to x another with side x + x. Arrange the triangles so that get closer to the actual slope. Use at least 10 decimals they both have the origin as their lower left hand corner, and organize your results in a table like this: and so there base is on the x-axis. x f(a) f(a + x) y y=x 0:1 . . . . . . . . . . . . 35. Group Problem. 0:01 . . . . . . . . . . . . LetA(x) be the area of a square with sidex, and let 0:001 . . . . . . . . . . . . L(x) be the perimeter of the square (sum of the lengths 6 10 . . . . . . . . . . . . of all its sides). Using the familiar formulas for A(x) and 12 0 10 . . . . . . . . . . . . 1 L(x) show that A (x) = L(x). 2 Look carefully at the ratios y=x. Do they look like Give a geometric interpretation that explains why 1 they are converging to some number? Compare the values A L(x)x for small x. 2 y of with the true value you got in the beginning of x 36. Let A(r) be the area enclosed by a circle of radius this problem. r, and let L(r) be the length of the circle. Show that 30. Simplify the algebraic expressions you get when you 0 A (r) =L(r). (Use the familiar formulas from geometry compute y and y=x for the following functions for the area and perimeter of a circle.) 2 (a) y =x 2x + 1 37. Let V (r) be the volume enclosed by a sphere of ra- 1 dius r, and let S(r) be the its surface area. Show that (b) y = 0 4 3 x V (r) = S(r). (Use the formulas V (r) = r and x 3 (c) y = 2 2 S(r) = 4r .) 19