Calculus 6 edition swokowski pdf free download

I n mathematics we use the phrase continuous. Intuitively, we regard a continuous function as a function whose graph has no breaks , holes , or vertical asymptotes. To illustrate , the graph of each function in Figure 2. Note that in i of the figure, f c is not defined. The graph of a function f is not one of these types if f satisfies the three conditions listed in the next definition.

D efinition 2. Intuitively we know that condition iii implies that as x gets closer to c, the function value f x gets closer to f c. More precisely , we can make f x as close to f c as desired by choosing x sufficiently close to c. Certain types of discontinuities are given special names. The discontin- uities in i and ii of Figure 2. The discontinuity in iii of the figure is a jump discontinuity , so nam ed because of the appearance of the graph.

If f x approaches CfJ or - CD as x approaches c from either side, as, for example, in iv of the figure, we say that f has an infinite discontinuity at c. In the following illustration, we reconsider some specific functions that were discussed in Sections 2. The next th eo rem states that polynomial functions and rational func- tions quotients of polynomial functions arc continuous at every number in their domain s.

Hence I is continuous at every real number. Since x and -. It remains to be shown that f is continuous at O. Since the right-hand and left-hand limits are equal. Hence f is continuous at O. By factoring we ob tain.

Setting each factor equal to zero, we see that the discontinuities of I arc at O. If a function I is continuous at every number in an open interval a, h , we say that f is continuous on the interval a, b. Simi larl y. The next definition covers the case of a closed interval. The function f is continuous on [a, b] if it is continuous on 0. If a function I has either a right-hand or a left -h a nd limit of the type indicated in Definition 2. So lving for. Hencel is continuous at c by Definition 2.

All that remains is to check - 3. Thus,f is continuous from the right at - 3 and from the left a t 3. By Def- inition 2. Strictly speaking. However, it is lIot customary to usc the phrase dis- continllolls ilt c if c is in an open interval throughout which f is undefined.

We may also define continuity on other types of intervals. For exam- ple, a function I is con tin uous on [a, b or [a, x if it is continuous at every numbcr greater than a in the interval and if, in addition. I is con- tinuous from the right at a.

For intervals of the form a,hJ o r -x ,hJ , we require continuity at every number less than h in the interval and also cont inuit y from the left at h. Using facts stated in Theorem 2. Consequentl y,. Pa rts ii - iv are proved III similar fa shion. If, in addi tio n , g c 0 for every c in t he in- terval, then f i g is co ntinu o us o n the interva l.

T hese res ul ts m ay be ex - tended to mo re th a n two fun cti o ns; that is, sums , d iffere nces , products , or quotients in vo lving a ny number of cont in uo us fun ctions are cont in uo us provided ze ro de no min a tors do no t occu r. Hence , by Theo rem 2. A proof of the next res ult on the lim it of a co m posite fun ct io n fog is given in Appendix II. The principal use of Theorem 2. To illustrate, let us use Theorem 2. Conclusion of theorem 2. Applying Theorem 2. Part i of the next theorem follows from Theorem 2.

Part ii is a restatement of i using the composite function! Since both I and yare continuous func- 0. A proof of the following property of continuous functions may be found in more advanced texts on calculu s. Intermediate value theorem 2. The intermediate va lue theorem states that IIS. If the graph of the continuous function f is regarded as extending in an unbroken manner from the point a J a to the point hJ h , as illustrated in Figure 2. Thus, if the point a, I a on the graph of a continuous function lies below the x-axis and the point h, I h lies above the x-axis, or vice versa, then the graph crosses the x-axis at some point c.

Since I I and f 2 have opposite signs. Example 6 illustrates a scheme for locating real Leros of polynomials. By using a method of sueeessire approximatioll. Another useful consequence of the intermediate value theorem is the following. The interval referred to may be either closed.

I x has the same sign throughout the interval. If this conclusion were. By our preceding remarks , this, in turn. Thus, the conclusion must be true. In Chapter 4 we sha ll apply Theorem 2. Classify 3 y the discontinuities of J as removable, jump, or infinite. Actually, 9 varies with latitude. If is the interval. We begin this chapter by considering two applied problems.

The first is to find the slope of the tangent line at a point on the graph of a function, and the second is to define the velocity of an object moving along a line. Our discussion provides insight into the power and generality of mathematics. Specifically, in Section 3. This allows us, in later sections, to apply the derivative concept to any quantity that can be represented by a function.

Since quantities of this type occur in almost every field of knowledge, applications of derivatives are numerous and varied, but each concerns a rate of change. Thus, returning to the two problems that started it all, the slope of a tangent line may be used to ,describe the rate at which a graph rises or falls , and' velocity is the rate at which distance changes with respect to time.

Our main objective in this chapter is to introduce derivatives and develop rules that can be used to find them without employing limits. We shall consider some applications here and many more in subsequent chapters. In geometry the tangent line I at a point P on a circle may be interpreted as the line that intersects the circle only at P, as illustrated in Figure 3. We cannot extend this interpretation to the graph of a function f, since a line may "touch" the graph of f at some isolated point P and then inter- sect it again at another point, as illustrated in Figure 3.

Our plan is to define the slope of the tangent line at P, for if the slope is known, we can find an equation for I by using the point-slope form 1. This line is called a secant line for the graph. We shall use the following notation:. If Q is close to P, it appears that I11pQ is an approximation to moo Moreover, we would expect this approximation to improve if we take Q x closer to P. If Q approaches P from the right, we have the situation illustrated in Figure 3. In Figure 3.

We could also let Q approach P in other ways , such as by taking points on the graph that are alternately to the left and to the right of P.

If I11pQ has a limiting value- that is, if rnpQ gets closer to some number as Q approaches P - then that number is the slope l11a of the tangent line I. Let us rephrase this discussion in terms of the function f. Referring to Figure 3. This motivates the following definition for the slope. It is often desirable to usc an alternative form for " obtained by changing from the variable x to a variable h as follows. DefinitIOn 3.

If the limit in Definition 3. Applying Definition 3. Using the point-slope form 1. The limit in Dellnition 3. One of the most familiar is the determination of the speed , or velocity, of a mov- ing object.

Let us consider the case of rectilinear motion, in which the object travels along a line. To find the at'!! Solving for I' gives us the following definition. To illustrate, if an automobile leaves city A at I P. The average velocity gives no information whatsoever about the veloc- ity at any instant. For example, at 2: 30 P. If we wish to determine the rate at which the automobile is traveling at 2: 30 P.

For example , suppose at P. Substituting thesc numbers in DefIni- tion 3. This result is still not an accurate indication of the velocity at P. Evidently, we obtain a better approximation by usi ng the average velocity during a smaller time inter va l. I t appears that the best procedure would be to take smaller and smaller time intervals near P.

This leads us into a limiting process similar to that discussed for tangent lines. I n order to make our discussion more precise, let us represent the position of an object moving rectilinearly by a point P on a coordinate FIGURE 3.

We sometimes refer to the motion of the point P on I, or the motion Time Position of P of an object whose position is specified by P. We shall assume that we o know the position of P at every instant in a given interval of time. To define the velocity of P at time a, we first determine the average velocity in a small time interval near a.

This number may be o positive, negative, or zero. By Definition 3. Thus , we defille the velocity as the limit , as It approaches 0, of ['av, as in the following definition.

Definition 3. The velocity Va of P at time a is. The limit in Definition 3. If s t is in miles and t in hours, then velocity is in miles per hour. Other units of measurement may, of course. We sha ll return to the velocity concept in Chapter 4, where we will show that if the velocity is positive in a given time interval, then the point is moving in the positive direction on f.

If the velocity is negative, the point is moving in the negative direction. Although these facts have not been proved, we shall use them in the following example. The negative sign indicates that the motion of the sandbag is in the nega- tive direction downward on f.

There are many other applications that require limits similar to those in 3. In some , the independent variable is time t, as in the definition of velocity. For example, over a period of time, a chemist may be interested in the rate at which a certain substance dissolves in water; an electrical engineer may wish to know the rate of change of current in part of an electrical circuit ; a biologist may be concerned with the rate at which the bacteria in a culture increase or decrease.

We can also consider rates of change with respect to quantities other than time. If the pressure is changing, a typical problem is to find the rate at which the volume is changing per unit change in pressure. This rate is known as the instantaneous rate of change of L' IVith respect to p. We define rates of change of a variable.

We shall use the phrase rate of change interchangeably with instantaneous rate of change. If, in Definition 3. To interpret ii of Defil1ltlOn 3. Tn Figure 3. A physical application of Definition 3. If R is increasing, find the instantaneous rate of change of I with respect to R at la any resistance R Ib a resistance of 20 ohms.

The negative sign indicates that the current is decreasing. Determine the tangent line to the graph of the equation at the point whether a creature will be hit if the pla yer shoots when with x-coordinate a.

Find the athlete's velocity. Aft er 1 seco nd s. If, for a certain e Find an approximate equation of the tangent line gas. The position function 5 of an object moving on a coor- a Use the graph to estimate the slope of the tangent dinate line is given by line at P IA ,f IA.

This limit is the basis for one of the fun- damental concepts of calculus, the derivatire , defined next. The symbol I' in Definition 3. It is important to note that in determining f' x we regard x as an arbitrary real number and consider the limit as h approaches zero. The statement f' x exists means that the limit in Definition 3.

If f' x exists , we say that f is differentiable at x , or that f has a deriva- tive at x. If the limit does not exist, then f is not differentiable at x.

Occasionally we will find it convenient to use the following alternative form of Definition 3. Alternative definition of derivative 3. This was the first formula used to define mil on page The following applications are restatements of Definitions 3. These interpretations of the derivative are very impor- tant and will be used in many examples and exercises throughout the text. Applications of the derivative 3. As a special case of 3. A function f is differentiable on an open interval a , b if f' x exists for every x in a , h.

We shall also consider functions that are differentiable on an infinite interval a, 0 , - x;, a , or - x. For closed intervals we use the following convention, which is analogous to the definition of continuity on a closed interval given in 2. The one-sided limits in Definition 3. For the left-hand derivative. If f is defined on a closed interval [a, b] and is undefined elsewhere, then the right-hand and left-hand derivatives define the slopes of the tangent lines at the points P a.

For the slope of the tangent line at P, we take the limiting value of the slope of the secant line through P and Q as Q approaches P frol11 the right.

For the tangent line at R. Differentiability on an interval of the form [a , b. The domain of the derivative f' consists of all numbers at which f is differentiable and also possible endpoints of the domain of f. The functions whose graphs are sketched in Figure 3. Since the slopes of 11 and 12 are unequal , rea does not exist.

The graph of I has a corner at Pea , I a if I is continuous at a and if the right-hand and left-hand derivatives at a exist and are unequal or if one of those derivatives exists at a and I1' x I As indicated in the next definition , a rertical tan!

DefinitlOf1 3. If P is an endpoint of the domain of f, we can state a similar definition using a right-hand or left-hand derivative. Some typical vcrtical tangent lines are illustrated in Figure 3. As indicated in the next definition , the point P in Figure 3. Hence the domain of f' is IR. The corresponding value of ' is - 4.

Hence the tangent line is horizontal at the point Q 2 , - 4. The graph of I a parabola and the tangent lines at P and Q are sketched in Figure 3. Note that the vertex of the parabola is the point Q 2, - 4. Note that the domain of I -l consists of all nonnegative numbers. Using Definition 3. The last limit shows that the graph of I has a vertical tangent line the y-ax is at the point 0,0. Using the limits in Definition 3.

Thus, not every continuous Junction is differentiable. In contrast, the next theorem states that every differentiable function is continuous.

Theorem 3. Thus, by Definition 2. By using one-sided limits , we can extend Theorem 3. Y The process of finding a derivative by means of Definition 3. The graph of J is the line that has slope and y-intercept b see Fig- 0. As indicated in the figure, the tangent line I at any point P co- incides with the graph of J and hence has slope Thus, from i of 3. We can also prove this fact directly from Defi- x nition 3. This gives us the following rule. Derivative of a linear function 3.

The following result is the special case of 3. Derivative of a constant function 3. Thc prcceding result is also graphically evident, because the graph of a constant function is a horizontal line and hence has slope O. Some special cases of 3. Many algebraic expressions contain a variable x raised to some power II. The next result , appropriately called the power rule, provides a simple formula for finding the derivative if 11 is an integer. Some special cases of the power rule are listed in the next illu stration.

We can extend the power rule to rational exponents. In particular, in Appendix II we show that for every positive integer 11 , 1 then. By using rule 3. In Chapter 7 we will prove that the power rulc holds for every real number Some special cases of the power rule for rational exponents are given in the next illustration. By using the same type of proof that was used for the powcr rule 3.

We shall conclude this section by introducing addition a l notations for derivatives. All of these notations are used in mathematics and applications, and you should become familiar with the different forms. For example, we can now write d. If we use a different independent variable, say I, then we write.

Each of the sy mbols Dx and di dx is called a differential operator. Sta nding alone , D, or di dx has no practical significance; howeve r, when either symbo l has an expression to its right, it denotes a derivati ve.

We say that Dx or dldx operates on the expression, and we call D , y or dy dx the derivative of y with respect to x. We shall justify the notation dyldx in Section 3. The next illustratio n contains some examples of the use of 3.

Note that in 3. If f' has a deriva- tive, it is denoted by f" read J double prime and is called the second derivative of f.

As we have indicated, the operator symbol D; is used for second deriva- tives. The third derivative f'" of J is the derivative of the second derivative. In general, if n is a positive integer, then P" denotes the nth derivative of J and is found by starting with J and differentiating, successively, n times.

The integer n is called the order of the deri vative P" x. Notations for higher derivatives 3. Sketch the graph of I' and determine where f is not 31 y differentiable. In these situations. Find the rate at [ - I, I J and estimate where f is not differentiable.

This section contains some general rules that simp lify the task of finding derivatives. The first three parts of the following theorem were proved in Section 3. Parts v and vi of Theorem 3. Ivl The derivative of a sum is the sum of the derivatives.

Ivll The derivative of a difference is the difference of the derivatives. These results can be extended to sums or differences of any number of functions. Since a polynomial is a sum of terms of the form ex", where c is a real number and n is a nonnegative integer, we may use results on sum s and differences to obtain the derivative , as illustrated in the next example. Formulas for derivatives of products o r quotients are more compli- cated than those for sums and differences.

In particular, the derivative of a product generally is not equal to the product of the derivatives. The derivative of any product f x g x may be expressed in terms of derivatives of f x and g x as in the following rule. Product rule 3.

Since f is differentiable at x, it is continuous at x see Theorem 3. Finally, applying the definition of derivative to f x and g x , we obtain.

The product rule may be phrased as follows: The deril'ative of a prod- uct equals the first factor times the derivative of the second factor , plus the second times the derivative of the first. Ibl The tangent line to the graph of f is ho ri zo ntal if its slope is zero.

We shall next obtain a formula for the derivative of a quotient. Note that the derivative of a quotient generally is not equal to the quotient of the derivatives. Quotient rule 3. Subtracting and adding g x f x in the numerator of the last quotient, we obtain. The quotient rule may be stated as follows : The derivative of a quotient is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator , divided by the square of the denominator.

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