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A Brief Introduction to Calculus1

ContentsIntroduction1. Functions and Graphs2. Linear Functions, Lines, and Linear Equations3. Limits4. Continuity5. Linear Approximation6. Introduction to the Derivative7. Product, Quotient, and Chain Rules8. Derivatives and Rates9. Increasing and Decreasing Functions10. Concavity11. Optimization12. Exponential and Logarithmic Functions13. Antiderivatives14. Integrals2

IntroductionThese notes are intended as a brief introduction to some of the mainideas and methods of calculus. They are very brief and are notintended as a mathematical exposition of the subject. They do notcontain recipes for solving problems. Hence, you will not be able tosolve homework problems by looking back through the notes andfinding similar examples.We feel that the only way one can really learn calculus (or any anothersubject) is to take basic ideas and apply those ideas to solve newproblems. Hence the learning process is accomplished primarily bysolving the problems.These notes were prepared with support from a National ScienceFoundation Grant.Copyright: Robert MolzonCopies of these notes may be made under the terms of the GeneralPublic License.3

1. Functions and GraphsA function is a rule that assigns one number to a given number. Ingeneral “number” will mean real number such as 1.25 or 6.498, 2 , orπ. The rule that defines the function can be described in severaldifferent ways. Perhaps the most common way of describing thefunction is an algebraic expression. For examplef t t 2is the function that assigns the square of number to any given number.The notation above is sometimes referred to as “functional notation”.We may compute the value the function assigns to a given number bysubstitution the given number into the algebraic expression and thenperforming the algebra. For example, with the above function f t t 2we evaluatef 2 2 2 4f 3 3 2 9f 4.1 4.1 2 16.81Functions may also be described by tables or lists. Suppose wemeasure some quantity such as weight or length at certain fixed timesand express the results in a table. For example, suppose ourmeasurements give the following results.Time Weight13.224.536.247.859.3We can then consider the table as a rule that assigns a number(weight) to a given time. Notice that this function is only defined for thevalues of time 1, 2, 3, 4, and 5. If we denote the function described bythe table using functional notation we could write W t to denote theweight at time t. In this case W t only makes sense for the t valuesabove.4

In some cases it may be necessary to combine the two approachesabove in order to adequately describe a function. For exampleh t t if t 0h t t if t 0describes the function h t by two algebraic expressions and theexpression that applies depends on the value of t. You may recognizethe above function as the “absolute value” function.Note that in the above examples, we have used the letter “t” to indicatethe number or value with which we start, and f t or h t as the numberassigned to this starting value. We could use other letters or symbolsinstead of “t” and you should get used to writing functions using avariety of symbols or letters. Basic texts in mathematics frequently use“x” to represent the starting value, but in applications (for example infinance or other business settings) more descriptive letters are usuallyused.Algebraic Operations with FunctionsWe can perform algebraic operations with functions just as we do withnumbers. Performing the algebraic operation results in a new function.For example iff t t and g t t 2 thenh t f t g t t t 2Another operation on functions is composition. For examplef t t 2 t 1 and g t t 3 thenf g t t 3 2 t 3 1Composition is also frequently written asf g t f g t It is very important to realize that the function is defined by the rule andnot by the symbol used to describe the rule. For example we couldhave presented the composition example above as follows.f t t 2 t 1 and g s s 3f g x x 3 2 x 3 1Note that if we use one symbol for a variable on the left side of anequation, we must use the same symbol on the right side. Thus theexpressionF t x 25

does not make sense.The values of the variable for which the function is defined is called thedomain of the function. If we writeF t 3t 4 for t 0then the domain of the function, F t , is t 0.When we write an algebraic expression to define a function withoutputting specific conditions on the values of the variable, then thedomain is assumed to be all values of the variable for which theexpression makes sense. For example the domain ofG x xis x 0. The domain of the functionG t 1tis t 0.GraphsGraphs give us a visual way of understanding functions. The functionrule is presented in a manner that allows us to see the properties of thefunction quickly if not precisely. The graph of a function (of onevariable) is a two dimensional object, and the graph is usuallydescribed algebraically as an equation. Here is an example.y f t 2 1t 4The symbol t is called the independent variable and the symbol y is6

called the dependent variable. Points in the two dimensional drawingare indicated by a pair of numbers, a, b , with a as the value of theindependent variable, t, and b the value of the dependent variable, y.The horizontal axis is in this case the t axis and the vertical axis is the yaxis. Note that one may use y as a horizontal axis.In this example the domain of the function is the set of all t 2. Wesee immediately from the graph the behavior of the function for largepositive or negative values of t as well as the behavior for values of tnear 2 and 2. Hence for getting an overall idea of the behavior of thefunction, the graph is often more useful than the algebraic expression.Of course, for computations we need the algebraic expression.When we graph a function that is defined by different algebraicexpressions for different values of the (independent) variable, we mustbe careful to indicate how the function is defined at the break. Forexample iff x 1 if x 2 andf x 1 if x 2we can place a solid dot at the point 2, 1 to indicate that the value ofthe function at 2 is 1.Often an open dot or circle is used at the point 2, 1 to indicate thatthe value at 2 is not 1.If the graph of a function crosses an axis, for example the t axis, thenthe point of crossing is called a t intercept.7

2. Linear Functions, Lines, and Linear EquationsLines are the simplest geometric object after a point, and linearfunctions are the simplest algebraic functions. The approximation ofcomplicated functions by linear functions is one of the basic tools inmathematics and its applications. Hence, an absolutely firmunderstanding of linear functions is essential to an understanding ofmore complicated functions.A linear function of the variable x is a function of the formL x a bxwhere a and b are constants. For example,L x 3 5xis a linear function. Note that the functionL x A B x c is also a linear function, although it is written in a slightly different form.For example,L x 8 5 x 1 can be rewritten asL x 8 5 x 1 8 5x 5 8 5 5x 3 5xThis particular form of a linear function, L x A B x c , will beuseful later.Lines and Linear EquationsThe graph of a linear function is a line. If L x is a linear function andwe introduce the dependent variable y, theny L x is a linear equation. Note the terminology we use here. Be sure youunderstand the difference between a linear function, line, and linearequation, since they represent three different mathematical objects. If8

L x a bx then the equation of the line ( the graph of the functionL x ) can be written asy L x y a bxIf L x A B x c then the equation of the line representing thegraph of L x isy A B x c For example if L x 8 5 x 1 we havey 8 5 x 1 If we graph the linear function L x 8 5 x 1 we getWe can obtain this graph by plotting a couple of points and thendrawing the line through the two points. For example, a table of valuesfor L x isx L x 18213If we plot the two points 1, 8 and 2, 13 and then draw the linethrough the two points, we obtain the graph of L x .Notice that when xincreases by one unit (from 1 to 2 ) the value of L x (or y ) increasesby 5 units. The number 5 appeared as the coefficient of x 1 in theexpression for L x . We would have seen exactly the same thing if wehad used different values for x. For example,9

xy5 286 33The increase in L x or y for a unit increase in x always appears as thecoefficient of x in the expression for L x . The value is called the slopeof the line y L x . In applications this value is also called themarginal value of L. If, for example, L x represents the cost ofproducing x items and L x 8 5 x 1 then the marginal cost is 5.It represents the cost of producing one additional unit. The slope of aline has a nice geometric interpretation. It measures the steepness ofthe graph of the linear function. The larger the value of the slope, thesteeper the line. If the slope is negative, then the linear functiondecreases as the value of the independent variable increases, and thegraph “ heads downhill”.The above example gives us two ways to determine the expression fora linear function or equivalently, to find the equation of a line. Firstsuppose we are given two points on the line - perhaps by a table. Wewill use the values above as an example since we already know whatwe should get as an expression for L x . So, for example, suppose thetwo points are 1, 8 and 2, 13 . The value of L x for x 1 must be 8,so we writeL x 8 m x 1 where m represents some constant that we must determine. (Think ofm as the marginal value). How do we determine m? When x 2 ,L x 13 so we substitute these values into the expression for L x .We getL 2 138 m 2 1 13Now we can solve this last equation for m. In two steps we get8 m 2 1 138 m 13m 5Hence the expression for L x is L x 8 5 x 1 which is asexpected.10

From the above calculation, it should be clear that a linear function iscompletely determined by its value at two points (two x values).Now suppose we are given a point and the slope of the linear function.How do we determine the expression for the linear function? It is eveneasier than in the above case (given two points). For example supposethe point is 6, 33 and the slope is 5. Then the expression for L x isL x 33 m x 6 since we must get 33 when x 6. But we are given the value of theslope, m 5. Hence the expression for L x isL x 33 5 x 6 So we really did not need any calculation at all! You should now seethe advantage of the formL x A B x c for a linear equation.Note that we could rewrite L x asL x 33 5 x 6 33 5 x 1 5 33 5 x 1 5 5 33 5 x 1 25 8 5 x 1 If we multiplied out the 5 and the x 1 we would obtainL x 8 5x 5 3 5xThis is the form we had at the very beginning.The above method of finding the expression for a linear function maybe slightly different than what you are familiar with from algebra.However it has several advantages and you should try to understandthis method. First of all, given a point and slope, the method allows youto write down immediately the expression for the linear function.Second, you are much less likely to get mixed up if symbols other thanx and y are used as the independent and dependent variablesrespectively. For example, suppose you know the relationship between11

cost, c, and interest rate, r, is linear. You are given a cost rate table.cr300 7400 8Suppose you want to find c as a linear function of r. Writec L r 300 m r 7 Now find m.400 300 m 8 7 300 m100 mHencec L r 300 100 r 7 Of course you can work this problem using the other methods forfinding the equation of a line, but it is easy to get things backwards(usually the mistake that is made is getting the reciprocal of the slope).We should finally mention one particular line that is not the graph of afunction. Recall again that a line is a geometric object. Suppose wehave drawn a vertical line in the x, y plane. Then there is no linearfunction L x such that the vertical line is the graph of the function L x ,since a function L x may take on at most one value for a given x,However, it is possible to write an equation that represents the line. Forexample, consider the vertical line through the point 2, 5 . Since theline is vertical, the x value of any point on the line must be 5, There isno restriction on the y value of a point on the line. Hence the equationof the line isx 2Finally, we summarize three different forms for the equation of a line.Each has an advantage in some setting.y a m x b , the equation of the line through a, b with slope my c mx, the equation of the line through 0, c with slope mAx By C 0, the equation of a line that can be used to describe vertical lines12

3. LimitsThe concept of limit is one idea that allows calculus to solve problemsthat are impossible to solve with algebra alone. The combination of theideas of limit and linear approximation together provide most of theresults of calculus. We are not going to give a precise mathematicaldefinition of limit here, since we believe it is important that you firstdevelop a feel for the idea through computations and examples beforeattempting to understand a precise definition. Here are a couple ofexamples of problems that require the use of limit.Suppose you deposit 1000 in an account that pays 10% interestcompounded one time per year. After one year the account contains 1100. Now suppose the interest is compounded two times per year.The amount in the account after one year is now 1102. 50. If theinterest is compounded four times per year the amount after one yearis 1103. 80. Here is a table that gives the resulting amount in theaccount after one year as a function of the number of times per yearthat the interest is compounded.Times Compounded Amount (in dollars)1110021102.5041103.80121104.70(You may not be familiar with compound interest, but your bank andevery bank uses this method to compute interest. Don’t worry aboutthe details of this now, but just keep in mind that it is a simple algebraiccomputation that you can do on your calculator.) We see that as thenumber of times we compound the interest increases, the amount inthe account at the end of the year increases. An obvious question isthis; does this amount get larger and larger without bound. Forexample, if we compound enough times per year might we end up with 2000 in the account at the end of the year? The answer is no. In factno matter how many times we compound per year we would never endup with more than 1, 105. 20 (approximately). To obtain this result wecould try increasing the number of rows in our table above, andcompute the amount in the account at the end of the year. If you do13

this with a calculator you will quickly come to the limiting value of 1,105.20. In fact if you compounded 1000 times per year and then2000 times per year the difference in the amount in the account at theend of the year would be so small that it could not be indicated with thenumber of digits available on the calculator.Here is a second example of a problem that requires the concept oflimit to solve. Find the area of a circle of radius two (2 . You mightobject and say that every high school student knows the answer: π2 2or just 4π. This is of course correct, but one might ask - what is thedecimal value of π. The answer to this question requires the concept oflimit.Finally, here is a third example of the use of limit. In this case theresulting limit is rather obvious, but the example still might help youunderstand the concept. Suppose I start with 1000 in an account (the account does not pay interest). I spend 1/2 of the money on 1January. The next day I spend 1/2 of the remaining amount ( 500).The day after that, I spend 1/2 of the amount remaining. I continue thisprocess day after day. What is the limiting amount in my account? Theanswer is clearly 0. It should also be clear that this will never happen!Even after 10 years I would have some minuscule amount remaining inthe account (assuming we can subdivide currency as small as we like).With these three examples, we are ready to consider an example usingfunctional notation. Suppose F x 1/x. Since this function is notdefined (does not make sense) if x 0, we consider only positivevalues of x. consider the following question. What is the limiting valueof F x as x gets larger and larger. (You should see the connection withthe previous example). If we substitute larger and larger values for xinto the expression for F x it becomes clear that the limiting value is 0.We want to introduce some notation that will summarize the questionand the answer. First we need notation for the concept of “x gets largerand larger”. You are probably already familiar with the notation that willbe used, but it can create a great deal of confusion. So be aware thatthe concept of “infinity” is a very difficult one to define and understand.The notation we use for “x gets larger and larger” isx and we say this as ”x tends to infinity”. Since we are considering larger14

and larger positive values of x we can be more precise and say ”xtends to positive infinity”. Now using this notation and the idea of limitthat we have intuitively introduced above we have the notationlimF x lim 1/x 0x x How do we compute limits? In the first example above we suggestedcomputing the limit of the amount in the account at the end of the yearby doing computations using more and more compounding periods.Frequently this may be necessary to evaluate the limit. It can obviouslybe quite time consuming, and we would like to have easier methods. Inmany cases algebraic simplification can be used to put a functionexpression in a form so the limiting value becomes obvious. Forexample, supposeG x 1 x2x xand we want to computelimG x x If we rewriteG x 1 x2 1 x 1x 1 x xx x(which we can do if the value of x is positive) then we seelimG x lim 1/x 0x x With the example F x 1/x we could also consider what happens tothe values of F x for positive values of x that get smaller and smaller;in other words as these positive values get closer and closer to 0. Thenotation for this islim 1/x x 0A couple of computations should convince you that the values becomelarger and larger positive numbers. Note that if you did the samecomputations with negative values of x very close to zero, you wouldget more and more negative values of the function. To distinguishthese various possibilities we can augment our notation in the followingway. The meaning of the following should be clear from our discussion.lim 1/x x 0lim 1/x x 0 15

It will help to understand this notation by graphing the function 1/x.The above notation x 0 can be extended in an obvious way. If wewrite x 3 we mean that “x tends to 3 from the right”. Similarly, x 3 means that “x tends to 3 from the left”. This will become importantwhen we consider continuity in the next section.It can happen that the limit of a certain function may not exist, that is itmay not make sense to talk about the limit. For example suppose afunction oscillates back and forth between 1 and 1. An example isthe function sin x whose graph looks likeIn this case we writelimsin x does not exist.x Note thatlim 1/x does not exit (be sure you can explain why not)x 016

Limits of functions satisfy certain nice properties that makecomputations easier than one might guess. If f x , g x , and h x arefunctions, a is a number, and the limitslimf x , limg x , and limh x x ax ax aall

Contents Introduction 1.FunctionsandGraphs 2.LinearFunctions,Lines,andLinearEquations 3.Limits 4.Continuity 5.LinearApproximation 6.IntroductiontotheDerivative

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