1.3 LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION
|
|
- Jean Summers
- 6 years ago
- Views:
Transcription
1 . Limits at Infinit; End Behavior of a Function 89. LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION Up to now we have been concerned with its that describe the behavior of a function f) as approaches some real number a. In this section we will be concerned with the behavior of f) as increases or decreases without bound. LIMITS AT INFINITY AND HORIZONTAL ASYMPTOTES If the values of a variable increase without bound, then we write, and if the values of decrease without bound, then we write. The behavior of a function f) as increases without bound or decreases without bound is sometimes called the end behavior of the function. For eample, 0 and 0 ) are illustrated numericall in Table.. and geometricall in Figure... 0 / / values Table.. 0, , conclusion As the value of / increases toward zero. As + the value of / decreases toward zero. In general, we will use the following notation. Figure Horizontal asmptote f) L.. its at infinit an informal view) If the values of f) eventuall get as close as we like to a number L as increases without bound, then we write f) L or f) L as ) Similarl, if the values of f) eventuall get as close as we like to a number L as decreases without bound, then we write f) L or f) L as ) L f) L + Horizontal asmptote Figure.. illustrates the end behavior of a function f when f) L or f) L In the first case the graph of f eventuall comes as close as we like to the line L as increases without bound, and in the second case it eventuall comes as close as we like to the line L as decreases without bound. If either it holds, we call the line L a horizontal asmptote for the graph of f. Figure.. f) L Eample It follows from ) and ) that 0 is a horizontal asmptote for the graph of f) / in both the positive and negative directions. This is consistent with the graph of / shown in Figure...
2 90 Chapter / Limits and Continuit 6 Eample Figure.. is the graph of f) tan. As suggested b this graph, tan π and tan π 6) ^ so the line π/ is a horizontal asmptote for f in the positive direction and the line π/ is a horizontal asmptote in the negative direction. Figure.. e tan 6 Figure.. + Eample graph, Figure.. is the graph of f) + /). As suggested b this + ) e and + ) e 7 8) so the line e is a horizontal asmptote for f in both the positive and negative directions. LIMIT LAWS FOR LIMITS AT INFINITY It can be shown that the it laws in Theorem.. carr over without change to its at + and. Moreover, it follows b the same argument used in Section. that if n is a positive integer, then ) n f))n f))n ) n f) 9 0) provided the indicated it of f) eists. It also follows that constants can be moved through the it smbols for its at infinit: kf) k f) kf) k f) ) provided the indicated it of f) eists. Finall, if f) k is a constant function, then the values of f do not change as or as,so k k k k ) Eample a) It follows from ), ), 9), and 0) that if n is a positive integer, then ) n ) 0 and n n 0 n b) It follows from 7) and the etension of Theorem..e) to the case that + ) [ + ) ] / [ + ) ] / e / e INFINITE LIMITS AT INFINITY Limits at infinit, like its at a real number a, can fail to eist for various reasons. One such possibilit is that the values of f) increase or decrease without bound as or as. We will use the following notation to describe this situation.
3 . Limits at Infinit; End Behavior of a Function 9.. infinite its at infinit an informal view) If the values of f) increase without bound as or as, then we write f) + or f) + as appropriate; and if the values of f) decrease without bound as or as, then we write as appropriate. f) or f) LIMITS OF n AS ± Figure.. illustrates the end behavior of the polnomials n for n,,, and. These are special cases of the following general results: {, n,,,... n +, n,,,... n +, n,, 6,... 6) Figure.. Multipling n b a positive real number does not affect its ) and 6), but multipling b a negative real number reverses the sign. Eample +, 76, 76 LIMITS OF POLYNOMIALS AS ± There is a useful principle about polnomials which, epressed informall, states: The end behavior of a polnomial matches the end behavior of its highest degree term.
4 9 Chapter / Limits and Continuit More precisel, if c n 0, then c0 + c + +c n n) c n n 7) c0 + c + +c n n) c n n 8) We can motivate these results b factoring out the highest power of from the polnomial and eamining the it of the factored epression. Thus, c 0 + c + +c n n n c 0 + c ) n + +c n n As or, it follows from Eample a) that all of the terms with positive powers of in the denominator approach 0, so 7) and 8) are certainl plausible. Eample ) ) 8 LIMITS OF RATIONAL FUNCTIONS AS ± One technique for determining the end behavior of a rational function is to divide each term in the numerator and denominator b the highest power of that occurs in the denominator, after which the iting behavior can be determined using results we have alread established. Here are some eamples. Eample 7 + Find 6 8. Solution. Divide each term in the numerator and denominator b the highest power of that occurs in the denominator, namel,. We obtain ) 6 8 ) Divide each term b. Limit of a quotient is the quotient of the its Limit of a sum is the sum of the its. A constant can be moved through a it smbol; Formulas ) and ).
5 . Limits at Infinit; End Behavior of a Function 9 Eample 8 Find a) + b) Solution a). Divide each term in the numerator and denominator b the highest power of that occurs in the denominator, namel,. We obtain Divide each term b. ) ) Limit of a quotient is the quotient of the its. Limit of a difference is the difference of the its A constant can be moved through a it smbol; Formula ) and Eample. Solution b). Divide each term in the numerator and denominator b the highest power of that occurs in the denominator, namel,. We obtain + + 9) In this case we cannot argue that the it of the quotient is the quotient of the its because the it of the numerator does not eist. However, we have 0, ) +, Thus, the numerator on the right side of 9) approaches + and the denominator has a finite negative it. We conclude from this that the quotient approaches ; that is, + + A QUICK METHOD FOR FINDING LIMITS OF RATIONAL FUNCTIONS AS + OR Since the end behavior of a polnomial matches the end behavior of its highest degree term, one can reasonabl conclude: The end behavior of a rational function matches the end behavior of the quotient of the highest degree term in the numerator divided b the highest degree term in the denominator.
6 9 Chapter / Limits and Continuit Eample 9 Use the preceding observation to compute the its in Eamples 7 and 8. Solution LIMITS INVOLVING RADICALS 0 ) ) Eample 0 Find a) + b) In both parts it would be helpful to manipulate the function so that the powers of are transformed to powers of /. This can be achieved in both cases b dividing the numerator and denominator b and using the fact that. Solution a). As, the values of under consideration are positive, so we can replace b where helpful. We obtain ) 6 ) + 0) 6 0) ) ) + ) ) 6 ) TECHNOLOGY MASTERY It follows from Eample 0 that the function f) + 6 has an asmptote of in the positive direction and an asmptote of in the negative direction. Confirm this using a graphing utilit. Solution b). As, the values of under consideration are negative, so we can replace b where helpful. We obtain ) + + 6
7 6 + Eample a) Find. Limits at Infinit; End Behavior of a Function ) b) 6 + ) Solution. Graphs of the functions f) 6 +, and g) 6 + for 0, are shown in Figure..6. From the graphs we might conjecture that the requested its are 0 and, respectivel. To confirm this, we treat each function as a fraction with a denominator of and rationalize the numerator. ) ) 6 + ) a) 6 + ) ) ) for >0 ) , 0 Figure..6 b) We noted in Section. that the standard rules of algebra do not appl to the smbols + and. Part b) of Eample illustrates this. The terms 6 + and both approach + as, but their difference does not approach 0. There is no it as + or. sin 6 + ) for >0 END BEHAVIOR OF TRIGONOMETRIC, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS Consider the function f) sin that is graphed in Figure..7. For this function the its as and as fail to eist not because f) increases or decreases without bound, but rather because the values var between and without approaching some specific real number. In general, the trigonometric functions fail to have its as and as because of periodicit. There is no specific notation to denote this kind of behavior. In Section 0. we showed that the functions e and ln both increase without bound as Figures 0..8 and 0..9). Thus, in it notation we have Figure..7 ln + e + 0 ) For reference, we also list the following its, which are consistent with the graphs in Figure..8: e 0 ln ) 0 +
8 96 Chapter / Limits and Continuit e e e ln Figure..8 Figure..9 Finall, the following its can be deduced b noting that the graph of e is the reflection about the -ais of the graph of e Figure..9). e 0 e + ) QUICK CHECK EXERCISES. See page 00 for answers.). Find the its. a) ) b) ) ) c) ln d) e. Find the its that eist. a) b) c) + + sin + ). Given that f) and g) find the its that eist. a) [f) g)] b) c) d) f) g) f) + g) f) + g) 0 f)g). Consider the graphs of /, sin,ln, e, and e. Which of these graphs has a horizontal asmptote? EXERCISE SET. Graphing Utilit In these eercises, make reasonable assumptions about the end behavior of the indicated function.. For the function g graphed in the accompaning figure, find a) g) b) g). g). For the function φ graphed in the accompaning figure, find a) b) φ) φ). f) Figure E- Figure E-
9 . Limits at Infinit; End Behavior of a Function 97. For the function φ graphed in the accompaning figure, find a) φ) b) φ). f) b) Use Figure.. to find the eact value of the it in part a). 8. Complete the table and make a guess about the it indicated. f) / f) ,000 00,000,000,000 f) Figure E-. For the function G graphed in the accompaning figure, find a) G) b) G). G) Figure E-. Given that f), g), h) 0 find the its that eist. If the it does not eist, eplain wh. a) [f) + g)] b) [h) g) + ] c) e) g) [f)g)] d) [g)] + f) f ) g) h) + h) 6f) f) + g) 6. Given that f) 7 and g) 6 find the its that eist. If the it does not eist, eplain wh. a) c) e) g) [f) g)] b) [6f) + 7g)] [ + g)] d) [ g)] f)g) f ) [ f) + g) ] h) g) f) f) + )g) 7. a) Complete the table and make a guess about the it indicated. ) f) tan f) 0 + f) Find the its ) ) t t 7t s t t t s 7 s s ). ) e e.. + e + e. 7. ln e + e 6. e e ) ln + ) e + e e e ) + ) True False Determine whether the statement is true or false. Eplain our answer.. We have + ) + 0) + +.
10 98 Chapter / Limits and Continuit. If L is a horizontal asmptote for the curve f), then f) L and f) L. If L is a horizontal asmptote for the curve f), then it is possible for the graph of f to intersect the line L infinitel man times.. If a rational function p)/q)has a horizontal asmptote, then the degree of p) must equal the degree of q). FOCUS ON CONCEPTS. Assume that a particle is accelerated b a constant force. The two curves v nt) and v et) in the accompaning figure provide velocit versus time curves for the particle as predicted b classical phsics and b the special theor of relativit, respectivel. The parameter c represents the speed of light. Using the language of its, describe the differences in the long-term predictions of the two theories. Velocit c v v nt) Classical) Time v et) Relativit) t Figure E- 6. Let T ft) denote the temperature of a baked potato t minutes after it has been removed from a hot oven. The accompaning figure shows the temperature versus time curve for the potato, where r is the temperature of the room. a) What is the phsical significance of t 0 + ft)? b) What is the phsical significance of t + ft)? Temperature ºF) 7. Let Find a) 00 r T T ft ) Time min) +, < 0 f) + +, 0 t Figure E-6 f) b) f). 8. Let Find a) t + t t + 6, t <,000,000 gt) 6t 00, t >,000,000 t gt) b) t + gt). 9. Discuss the its of p) ) n as and for positive integer values of n. 0. In each part, find eamples of polnomials p) and q) that satisf the stated condition and such that p) + and q) + as. a) c) p) p) b) q) q) 0 p) + d) [p) q)] q). a) Do an of the trigonometric functions sin, cos, tan, cot, sec, and csc have horizontal asmptotes? b) Do an of the trigonometric functions have vertical asmptotes? Where?. Find c 0 + c + +c n n d 0 + d + +d m m where c n 0 and d m 0. [Hint: Your answer will depend on whether m<n, m n, orm>n.] FOCUS ON CONCEPTS These eercises develop some versions of the substitution principle, a useful tool for the evaluation of its.. a) Eplain wh we can evaluate e b making the substitution t and writing e t + et + b) Suppose g) + as. Given an function f), eplain wh we can evaluate f [g)] b substituting t g) and writing f [g)] ft) t + Here, equalit is interpreted to mean that either both its eist and are equal or that both its fail to eist.) c) Wh does the result in part b) remain valid if is replaced everwhere b one of, c, c, or c +?. a) Eplain wh we can evaluate e b making the substitution t and writing e t et 0 cont.)
11 . Limits at Infinit; End Behavior of a Function 99 b) Suppose g) as. Given an function f), eplain wh we can evaluate f [g)] b substituting t g) and writing f [g)] ft) t Here, equalit is interpreted to mean that either both its eist and are equal or that both its fail to eist.) c) Wh does the result in part b) remain valid if is replaced everwhere b one of, c, c, or c +? 6 Evaluate the it using an appropriate substitution.. 0 e/ e/ 7. 0 ecsc ecsc ln 9. [Hint: t ln ] ln 60. [ln ) ln + )] [Hint: t ] 6. ) [Hint: t ] 6. + ) [Hint: t /] 6. Let f) b, where 0 <b. Use the substitution principle to verif the asmptotic behavior of f that is illustrated in Figure 0... [Hint: f) b e ln b ) e ln b) ] 6. Prove that 0 + ) / e b completing parts a) and b). a) Use Equation 7) and the substitution t / to prove that ) / e. b) Use Equation 8) and the substitution t / to prove that 0 + ) / e. 6. Suppose that the speed v in ft/s) of a skdiver t seconds after leaping from a plane is given b the equation v 90 e 0.68t ). a) Graph v versus t. b) B evaluating an appropriate it, show that the graph of v versus t has a horizontal asmptote v c for an appropriate constant c. c) What is the phsical significance of the constant c in part b)? 66. The population p of the United States in millions) in ear t can be modeled b the function pt) +.e 0.0t 990) a) Based on this model, what was the U.S. population in 990? b) Plot p versus t for the 00-ear period from 90 to 0. c) B evaluating an appropriate it, show that the graph of p versus t has a horizontal asmptote p c for an appropriate constant c. d) What is the significance of the constant c in part c) for the population predicted b this model? 67. a) Compute the approimate) values of the terms in the sequence.0 0,.00 00, , , , What number do these terms appear to be approaching? b) Use Equation 7) to verif our answer in part a). c) Let a 9 denote a positive integer. What number is approached more and more closel b the terms in the following sequence?.0 a0a,.00 a00a,.000 a000a,.0000 a0000a, a00000a, a000000a... The powers are positive integers that begin and end with the digit a and have 0 s in the remaining positions). 68. Let f) + ). a) Prove the identit f ) f ) b) Use Equation 7) and the identit from part a) to prove Equation 8) The notion of an asmptote can be etended to include curves as well as lines. Specificall, we sa that curves f) and g) are asmptotic as + provided [f) g)] 0 and are asmptotic as provided [f) g)] 0 In these eercises, determine a simpler function g) such that f) is asmptotic to g) as or. Use a graphing utilit to generate the graphs of f) and g) and identif all vertical asmptotes. 69. f) [Hint: Divide into.] 70. f) + 7. f) f) + 7. f) sin + 7. Writing In some models for learning a skill e.g., juggling), it is assumed that the skill level for an individual increases with practice but cannot become arbitraril high. How do concepts of this section appl to such a model?
12 00 Chapter / Limits and Continuit 7. Writing In some population models it is assumed that a given ecological sstem possesses a carring capacit L. Populations greater than the carring capacit tend to decline toward L, while populations less than the carring capacit tend to increase toward L. Eplain wh these assumptions are reasonable, and discuss how the concepts of this section appl to such a model. QUICK CHECK ANSWERS.. a) + b) c) d) 0. a) b) does not eist c) e. a) 9 b). /, e, and e each has a horizontal asmptote. c) does not eist d). LIMITS DISCUSSED MORE RIGOROUSLY) In the previous sections of this chapter we focused on the discover of values of its, either b sampling selected -values or b appling it theorems that were stated without proof. Our main goal in this section is to define the notion of a it precisel, thereb making it possible to establish its with certaint and to prove theorems about them. This will also provide us with a deeper understanding of some of the more subtle properties of functions. MOTIVATION FOR THE DEFINITION OF A TWO-SIDED LIMIT The statement a f) L can be interpreted informall to mean that we can make the value of f) as close as we like to the real number L b making the value of sufficientl close to a. It is our goal to make the informal phrases as close as we like to L and sufficientl close to a mathematicall precise. To do this, consider the function f graphed in Figure..a for which f) L as a. For visual simplicit we have drawn the graph of f to be increasing on an open interval containing a, and we have intentionall placed a hole in the graph at a to emphasize that f need not be defined at a to have a it there. f) L f ) L + e L f) L + e f) L f) f) a L e L e 0 a 0 a Figure.. a) b) c) Net, let us choose an positive number ɛ and ask how close must be to a in order for the values of f) to be within ɛ units of L. We can answer this geometricall b drawing horizontal lines from the points L + ɛ and L ɛ on the -ais until the meet the curve f), and then drawing vertical lines from those points on the curve to the -ais Figure..b). As indicated in the figure, let 0 and be the points where those vertical lines intersect the -ais.
1.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS
.6 Continuit of Trigonometric, Eponential, and Inverse Functions.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS In this section we will discuss the continuit properties of trigonometric
More informationInfinite Limits. Let f be the function given by. f x 3 x 2.
0_005.qd //0 :07 PM Page 8 SECTION.5 Infinite Limits 8, as Section.5, as + f() = f increases and decreases without bound as approaches. Figure.9 Infinite Limits Determine infinite its from the left and
More information2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:
SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert
More information2.5. Infinite Limits and Vertical Asymptotes. Infinite Limits
. Infinite Limits and Vertical Asmptotes. Infinite Limits and Vertical Asmptotes In this section we etend the concept of it to infinite its, which are not its as before, but rather an entirel new use of
More informationx c x c This suggests the following definition.
110 Chapter 1 / Limits and Continuit 1.5 CONTINUITY A thrown baseball cannot vanish at some point and reappear someplace else to continue its motion. Thus, we perceive the path of the ball as an unbroken
More informationFIRST- AND SECOND-ORDER IVPS The problem given in (1) is also called an nth-order initial-value problem. For example, Solve: Solve:
.2 INITIAL-VALUE PROBLEMS 3.2 INITIAL-VALUE PROBLEMS REVIEW MATERIAL Normal form of a DE Solution of a DE Famil of solutions INTRODUCTION We are often interested in problems in which we seek a solution
More informationProperties of Limits
33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate
More informationFinding Limits Graphically and Numerically. An Introduction to Limits
8 CHAPTER Limits and Their Properties Section Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach Learn different was that a it can fail to eist Stud and use
More informationFinding Limits Graphically and Numerically. An Introduction to Limits
60_00.qd //0 :05 PM Page 8 8 CHAPTER Limits and Their Properties Section. Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach. Learn different was that a it can
More informationLimits and Continuous Functions. 2.2 Introduction to Limits. We first interpret limits loosely. We write. lim f(x) = L
2 Limits and Continuous Functions 2.2 Introduction to Limits We first interpret limits loosel. We write lim f() = L and sa the limit of f() as approaches c, equals L if we can make the values of f() arbitraril
More information1. d = 1. or Use Only in Pilot Program F Review Exercises 131
or Use Onl in Pilot Program F 0 0 Review Eercises. Limit proof Suppose f is defined for all values of near a, ecept possibl at a. Assume for an integer N 7 0, there is another integer M 7 0 such that f
More informationLimits. or Use Only in Pilot Program F The Idea of Limits 2.2 Definitions of Limits 2.3 Techniques for Computing.
Limits or Use Onl in Pilot Program F 03 04. he Idea of Limits. Definitions of Limits.3 echniques for Computing Limits.4 Infinite Limits.5 Limits at Infinit.6 Continuit.7 Precise Definitions of Limits Biologists
More informationDerivatives of Multivariable Functions
Chapter 0 Derivatives of Multivariable Functions 0. Limits Motivating Questions In this section, we strive to understand the ideas generated b the following important questions: What do we mean b the limit
More informationFitting Integrands to Basic Rules. x x 2 9 dx. Solution a. Use the Arctangent Rule and let u x and a dx arctan x 3 C. 2 du u.
58 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8 Basic Integration Rules Review proceres for fitting an integrand to one of the basic integration rules Fitting Integrands
More informationLimits and Their Properties
Limits and Their Properties The it of a function is the primar concept that distinguishes calculus from algebra and analtic geometr. The notion of a it is fundamental to the stud of calculus. Thus, it
More information4.3 Exercises. local maximum or minimum. The second derivative is. e 1 x 2x 1. f x x 2 e 1 x 1 x 2 e 1 x 2x x 4
SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 297 local maimum or minimum. The second derivative is f 2 e 2 e 2 4 e 2 4 Since e and 4, we have f when and when 2 f. So the curve is concave downward
More informationIn everyday speech, a continuous. Limits and Continuity. Critical Thinking Exercises
062 Chapter Introduction to Calculus Critical Thinking Eercises Make Sense? In Eercises 74 77, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 74. I evaluated
More information39. (a) Use trigonometric substitution to verify that. 40. The parabola y 2x divides the disk into two
35. Prove the formula A r for the area of a sector of a circle with radius r and central angle. [Hint: Assume 0 and place the center of the circle at the origin so it has the equation. Then is the sum
More informationLimits and Their Properties
Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus
More informationExact Differential Equations. The general solution of the equation is f x, y C. If f has continuous second partials, then M y 2 f
APPENDIX C Additional Topics in Differential Equations APPENDIX C. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Chapter 6, ou studied applications
More information7.7. Inverse Trigonometric Functions. Defining the Inverses
7.7 Inverse Trigonometric Functions 57 7.7 Inverse Trigonometric Functions Inverse trigonometric functions arise when we want to calculate angles from side measurements in triangles. The also provide useful
More information2.2 SEPARABLE VARIABLES
44 CHAPTER FIRST-ORDER DIFFERENTIAL EQUATIONS 6 Consider the autonomous DE 6 Use our ideas from Problem 5 to find intervals on the -ais for which solution curves are concave up and intervals for which
More informationIn this note we will evaluate the limits of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0. f(x)
L Hôpital s Rule In this note we will evaluate the its of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0 f() Suppose a f() = 0 and a g() = 0. Then a g() the indeterminate
More informationSection 1.2: A Catalog of Functions
Section 1.: A Catalog of Functions As we discussed in the last section, in the sciences, we often tr to find an equation which models some given phenomenon in the real world - for eample, temperature as
More informationRepresentation of Functions by Power Series. Geometric Power Series
60_0909.qd //0 :09 PM Page 669 SECTION 9.9 Representation of Functions b Power Series 669 The Granger Collection Section 9.9 JOSEPH FOURIER (768 80) Some of the earl work in representing functions b power
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More informationSEPARABLE EQUATIONS 2.2
46 CHAPTER FIRST-ORDER DIFFERENTIAL EQUATIONS 4. Chemical Reactions When certain kinds of chemicals are combined, the rate at which the new compound is formed is modeled b the autonomous differential equation
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationCHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 0 Section. Rolle s Theorem and the Mean Value Theorem. 07 Section. Increasing and Decreasing Functions and the First
More informationx f x
MATC 00 Class Notes - Sec.. Limits Idea: Look at the behavior of f as gets closer and closer to a specific number. Let f. We want to know the behavior of f when is close to a specific number, sa. Look
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationCHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval................... 0 Section. Rolle s Theorem and the Mean Value Theorem...... 0 Section. Increasing and Decreasing Functions and
More information3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23
Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical
More informationx = 1 n (x 1 + x 2 + +x n )
3 PARTIAL DERIVATIVES Science Photo Librar Three-dimensional surfaces have high points and low points that are analogous to the peaks and valles of a mountain range. In this chapter we will use derivatives
More informationFitting Integrands to Basic Rules
6_8.qd // : PM Page 8 8 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8. Basic Integration Rules Review proceres for fitting an integrand to one of the basic integration
More informationb. Create a graph that gives a more complete representation of f.
or Use Onl in Pilot Program F 96 Chapter Limits 6 7. Steep secant lines a. Given the graph of f in the following figures, find the slope of the secant line that passes through, and h, f h in terms of h,
More information1.5. Analyzing Graphs of Functions. The Graph of a Function. What you should learn. Why you should learn it. 54 Chapter 1 Functions and Their Graphs
0_005.qd /7/05 8: AM Page 5 5 Chapter Functions and Their Graphs.5 Analzing Graphs of Functions What ou should learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals
More information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More informationMATH 1010E University Mathematics Lecture Notes (week 8) Martin Li
MATH 1010E University Mathematics Lecture Notes (week 8) Martin Li 1 L Hospital s Rule Another useful application of mean value theorems is L Hospital s Rule. It helps us to evaluate its of indeterminate
More informationChapter 4 Analytic Trigonometry
Analtic Trigonometr Chapter Analtic Trigonometr Inverse Trigonometric Functions The trigonometric functions act as an operator on the variable (angle, resulting in an output value Suppose this process
More informationAnalytic Trigonometry
CHAPTER 5 Analtic Trigonometr 5. Fundamental Identities 5. Proving Trigonometric Identities 5.3 Sum and Difference Identities 5.4 Multiple-Angle Identities 5.5 The Law of Sines 5.6 The Law of Cosines It
More informationSection 1.5 Formal definitions of limits
Section.5 Formal definitions of limits (3/908) Overview: The definitions of the various tpes of limits in previous sections involve phrases such as arbitraril close, sufficientl close, arbitraril large,
More information2.1 Rates of Change and Limits AP Calculus
. Rates of Change and Limits AP Calculus. RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important
More informationOrdinary Differential Equations
58229_CH0_00_03.indd Page 6/6/6 2:48 PM F-007 /202/JB0027/work/indd & Bartlett Learning LLC, an Ascend Learning Compan.. PART Ordinar Differential Equations. Introduction to Differential Equations 2. First-Order
More informationINTRODUCTION TO DIFFERENTIAL EQUATIONS
INTRODUCTION TO DIFFERENTIAL EQUATIONS. Definitions and Terminolog. Initial-Value Problems.3 Differential Equations as Mathematical Models CHAPTER IN REVIEW The words differential and equations certainl
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More information2.1 Limits, Rates of Change, and Tangent Lines. Preliminary Questions 1. Average velocity is defined as a ratio of which two quantities?
LIMITS. Limits, Rates of Change, and Tangent Lines Preinar Questions. Average velocit is defined as a ratio of which two quantities? Average velocit is defined as the ratio of distance traveled to time
More informationWith topics from Algebra and Pre-Calculus to
With topics from Algebra and Pre-Calculus to get you ready to the AP! (Key contains solved problems) Note: The purpose of this packet is to give you a review of basic skills. You are asked not to use the
More informationAPPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
More informationGrade 12 Pre-Calculus Mathematics Achievement Test. Booklet 2
Grade 2 Pre-Calculus Mathematics Achievement Test Booklet 2 June 207 Manitoba Education and Training Cataloguing in Publication Data Grade 2 pre-calculus mathematics achievement test. Booklet 2. June 207
More information3.7 InveRSe FUnCTIOnS
CHAPTER functions learning ObjeCTIveS In this section, ou will: Verif inverse functions. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one.
More informationKEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1
Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation
More informationTrigonometric Functions
Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle
More information11.4 Polar Coordinates
11. Polar Coordinates 917 11. Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of assigning ordered pairs of numbers to points in the plane.
More informationPACKET Unit 4 Honors ICM Functions and Limits 1
PACKET Unit 4 Honors ICM Functions and Limits 1 Day 1 Homework For each of the rational functions find: a. domain b. -intercept(s) c. y-intercept Graph #8 and #10 with at least 5 EXACT points. 1. f 6.
More informationFunctions. Introduction
Functions,00 P,000 00 0 70 7 80 8 0 000 00 00 Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of work b MeasuringWorth)
More informationCalculus 1 (AP, Honors, Academic) Summer Assignment 2018
Calculus (AP, Honors, Academic) Summer Assignment 08 The summer assignments for Calculus will reinforce some necessary Algebra and Precalculus skills. In order to be successful in Calculus, you must have
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs
More information3 Polynomial and Rational Functions
3 Polnomial and Rational Functions 3.1 Quadratic Functions and Models 3.2 Polnomial Functions and Their Graphs 3.3 Dividing Polnomials 3.4 Real Zeros of Polnomials 3.5 Comple Zeros and the Fundamental
More informationPolynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
More informationAdditional Topics in Differential Equations
0537_cop6.qd 0/8/08 :6 PM Page 3 6 Additional Topics in Differential Equations In Chapter 6, ou studied differential equations. In this chapter, ou will learn additional techniques for solving differential
More information10.5 Graphs of the Trigonometric Functions
790 Foundations of Trigonometr 0.5 Graphs of the Trigonometric Functions In this section, we return to our discussion of the circular (trigonometric functions as functions of real numbers and pick up where
More information6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
More information8.1 Exponents and Roots
Section 8. Eponents and Roots 75 8. Eponents and Roots Before defining the net famil of functions, the eponential functions, we will need to discuss eponent notation in detail. As we shall see, eponents
More informationy sin n x dx cos x sin n 1 x n 1 y sin n 2 x cos 2 x dx y sin n xdx cos x sin n 1 x n 1 y sin n 2 x dx n 1 y sin n x dx
SECTION 7. INTEGRATION BY PARTS 57 EXAPLE 6 Prove the reduction formula N Equation 7 is called a reduction formula because the eponent n has been reduced to n and n. 7 sin n n cos sinn n n sin n where
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More informationIntroduction to Differential Equations
Introduction to Differential Equations. Definitions and Terminolog.2 Initial-Value Problems.3 Differential Equations as Mathematical Models Chapter in Review The words differential and equations certainl
More informationDerivatives of Multivariable Functions
Chapter 10 Derivatives of Multivariable Functions 10.1 Limits Motivating Questions What do we mean b the limit of a function f of two variables at a point (a, b)? What techniques can we use to show that
More information16.5. Maclaurin and Taylor Series. Introduction. Prerequisites. Learning Outcomes
Maclaurin and Talor Series 6.5 Introduction In this Section we eamine how functions ma be epressed in terms of power series. This is an etremel useful wa of epressing a function since (as we shall see)
More informationName Please print your name as it appears on the class roster.
Berkele Cit College Practice Problems Math 1 Precalculus - Final Eam Preparation Name Please print our name as it appears on the class roster. SHORT ANSWER. Write the word or phrase that best completes
More informationChapter One. Chapter One
Chapter One Chapter One CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons ConcepTests and Answers and Comments for Section.. Which of the following functions has its domain identical with its range?
More information4.317 d 4 y. 4 dx d 2 y dy. 20. dt d 2 x. 21. y 3y 3y y y 6y 12y 8y y (4) y y y (4) 2y y 0. d 4 y 26.
38 CHAPTER 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS sstems are also able, b means of their dsolve commands, to provide eplicit solutions of homogeneous linear constant-coefficient differential equations.
More information67. (a) Use a computer algebra system to find the partial fraction CAS. 68. (a) Find the partial fraction decomposition of the function CAS
SECTION 7.5 STRATEGY FOR INTEGRATION 483 6. 2 sin 2 2 cos CAS 67. (a) Use a computer algebra sstem to find the partial fraction decomposition of the function 62 63 Find the area of the region under the
More information2.2. Calculating Limits Using the Limit Laws. 84 Chapter 2: Limits and Continuity. The Limit Laws. THEOREM 1 Limit Laws
84 Chapter : Limits and Continuit. HISTORICAL ESSAY* Limits Calculating Limits Using the Limit Laws In Section. we used graphs and calculators to guess the values of its. This section presents theorems
More informationJones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION. Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION
FIRST-ORDER DIFFERENTIAL EQUATIONS 2 Chapter Contents 2. Solution Curves without a Solution 2.. Direction Fields 2..2 Jones Autonomous & Bartlett First-Order Learning, DEs LLC 2.2 Separable Equations 2.3
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet Day All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. The only way to guarantee the eistence of a it is to algebraically prove it.
More informationL 8.6 L 10.6 L 13.3 L Psud = u 3-4u 2 + 5u; [1, 2]
. Rates of Change and Tangents to Curves 6 p Q (5, ) (, ) (5, ) (, 65) Slope of PQ p/ t (flies/ da) - 5 5 - - 5 - - 5 5-65 - 5 - L 8.6 L.6 L. L 6. Number of flies 5 B(5, 5) Q(5, ) 5 5 P(, 5) 5 5 A(, )
More informationLimits 4: Continuity
Limits 4: Continuit 55 Limits 4: Continuit Model : Continuit I. II. III. IV. z V. VI. z a VII. VIII. IX. Construct Your Understanding Questions (to do in class). Which is the correct value of f (a) in
More informationCHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. Section. Rolle s Theorem and the Mean Value Theorem. 7 Section. Increasing and Decreasing Functions and the First Derivative
More informationQuick Review 4.1 (For help, go to Sections 1.2, 2.1, 3.5, and 3.6.)
Section 4. Etreme Values of Functions 93 EXPLORATION Finding Etreme Values Let f,.. Determine graphicall the etreme values of f and where the occur. Find f at these values of.. Graph f and f or NDER f,,
More informationR3.6 Solving Linear Inequalities. 3) Solve: 2(x 4) - 3 > 3x ) Solve: 3(x 2) > 7-4x. R8.7 Rational Exponents
Level D Review Packet - MMT This packet briefly reviews the topics covered on the Level D Math Skills Assessment. If you need additional study resources and/or assistance with any of the topics below,
More informationSEE and DISCUSS the pictures on pages in your text. Key picture:
Math 6 Notes 1.1 A PREVIEW OF CALCULUS There are main problems in calculus: 1. Finding a tangent line to a curve though a point on the curve.. Finding the area under a curve on some interval. SEE and DISCUSS
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationPreCalculus Final Exam Review Revised Spring 2014
PreCalculus Final Eam Review Revised Spring 0. f() is a function that generates the ordered pairs (0,0), (,) and (,-). a. If f () is an odd function, what are the coordinates of two other points found
More informationMcKinney High School AP Calculus Summer Packet
McKinne High School AP Calculus Summer Packet (for students entering AP Calculus AB or AP Calculus BC) Name:. This packet is to be handed in to our Calculus teacher the first week of school.. ALL work
More information17 Exponential Functions
Eponential Functions Concepts: Eponential Functions Graphing Eponential Functions Eponential Growth and Eponential Deca The Irrational Number e and Continuousl Compounded Interest (Section. &.A). Sketch
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More information11.4. Differentiating ProductsandQuotients. Introduction. Prerequisites. Learning Outcomes
Differentiating ProductsandQuotients 11.4 Introduction We have seen, in the first three Sections, how standard functions like n, e a, sin a, cos a, ln a may be differentiated. In this Section we see how
More informationModule 3, Section 4 Analytic Geometry II
Principles of Mathematics 11 Section, Introduction 01 Introduction, Section Analtic Geometr II As the lesson titles show, this section etends what ou have learned about Analtic Geometr to several related
More informationSet 3: Limits of functions:
Set 3: Limits of functions: A. The intuitive approach (.): 1. Watch the video at: https://www.khanacademy.org/math/differential-calculus/it-basics-dc/formal-definition-of-its-dc/v/itintuition-review. 3.
More informationChapter 3: Three faces of the derivative. Overview
Overview We alread saw an algebraic wa of thinking about a derivative. Geometric: zooming into a line Analtic: continuit and rational functions Computational: approimations with computers 3. The geometric
More informationIndeterminate Forms and L Hospital s Rule
APPLICATIONS OF DIFFERENTIATION Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at certain points. INDETERMINATE FORM TYPE
More informationDifferential Calculus I - - : Fundamentals
Differential Calculus I - - : Fundamentals Assessment statements 6. Informal ideas of limits and convergence. Limit notation. Definition of derivative from first principles f9() 5 lim f ( h) f () h ( h
More informationJones & Bartlett Learning, LLC, an Ascend Learning Company. NOT FOR SALE OR DISTRIBUTION
stefanel/shutterstock, Inc. CHAPTER 2 First-Order Differential Equations We begin our stu of differential equations with first-order equations. In this chapter we illustrate the three different was differential
More informationCoordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general
A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate
More informationExponential, Logistic, and Logarithmic Functions
CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.1 Eponential and Logistic Functions 3.2 Eponential and Logistic Modeling 3.3 Logarithmic Functions and Their Graphs 3.4 Properties of Logarithmic
More informationPolynomial and Rational Functions
Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan;
More informationReview Topics for MATH 1400 Elements of Calculus Table of Contents
Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical
More informationRolle s Theorem and the Mean Value Theorem. Rolle s Theorem
0_00qd //0 0:50 AM Page 7 7 CHAPTER Applications o Dierentiation Section ROLLE S THEOREM French mathematician Michel Rolle irst published the theorem that bears his name in 9 Beore this time, however,
More informationSANDY CREEK HIGH SCHOOL
SANDY CREEK HIGH SCHOOL SUMMER REVIEW PACKET For students entering A.P. CALCULUS BC I epect everyone to check the Google classroom site and your school emails at least once every two weeks. You will also
More information