Chapter One. Chapter One

Transcription

1 Chapter One Chapter One

2 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? (a) f() = (b) g() = (c) h() = (d) i() = (b) and (c). For g() =, the domain and range are all nonnegative numbers, and for h() =, the domain and range are all real numbers. It is worth considering the domain and range for all choices.. Which of the following functions have identical domains? (a) f() = (b) g() = / (c) h() = (d) i() = (e) j() = ln (a), (c), and (d) give functions which have the domain of all real numbers, while (b) and (e) give functions whose domain consists of all positive numbers. For alternate choices, replace (a) with / and (e) with ln.. Which of the following functions have identical ranges? (a) f() = (b) g() = / (c) h() = (d) i() = (e) j() = ln (a) and (d) give functions which have the identical range of nonnegative numbers. (c) and (e) also give functions which have the same range all real numbers. You could have students find a function with the same range as (b) : / for eample.

3 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons. Which of the graphs is that of = /? (b) (c) (d) (d). / is not defined at = 0. This is where ou could emphasize that the domain is a crucial part of the definition of a function. Follow-up Question. { Give another formula for the function = /., 0 Answer. = undefined, = The slope of the line connecting the points (, ) and (, 8) is (a) (b) (c) (d) (d). rise run = 8 = =. You might point out in finding slopes, the order of the points in the ratio ( )/( ) is immaterial.

4 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 6. Put the following linear functions in order of increasing slope. (a) = π + 9 (b) = + (c) = 0 (d) = (e) = (f) = 00 (c), (f), (e), (d), (b), (a). In order to put the lines in the correct order, consider the slope of each function. This question was used as an elimination question in a classroom session modeled after Who Wants to be a Millionaire?, replacing Millionaire b Mathematician. 7. List the lines in the figure below in the order of increasing slope. (The graphs are shown in identical windows.) (a) (b) (c) (d) (b), (c), (a), (d). You ma want to point out the difference between slope (numerical values) and steepness (absolute value of the slope).

5 8. Which of the following lines have the same slope? (a) = + (b) = 9 + (c) = + 6 (d) = 6 + CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 5 (a), (b), and (d). Solving for in the last three choices gives = +, = +, and = +. Thus (a), (b), and (d) have the same slope,. You could point out that (a) and (d) are the same line. 9. Which of the following tables could represent linear functions? (a) f() (b) g() 9 6 (c) h() (d) j() (b). This is the onl table with a constant difference,, for the same increase in. Therefore, (b) is the onl one representing a linear function. You could point out that (d) fails to be a linear function because the slope between consecutive points is not constant. This happens even though the function values increase b constant amounts because the values do not increase b constant amounts. 0. Which of the following lines have the same vertical intercept? (a) = + (b) = 9 + (c) = + 6 (d) = 6 + (a), (c), and (d). Solving for in the last three choices gives = +, = +, and = +. Thus (a), (c), and (d) have the same vertical intercept,. You could graph these four equations and then ask about the intercept.. Ever line has a vertical intercept. (a) True (b) False (b). The line = a, where a 0, does not have a vertical intercept. Note that ever non-vertical line has a vertical intercept.. Ever line has a horizontal intercept. (a) True (b) False (b). The line = a where a 0, does not have a horizontal intercept. Note that all lines with a nonzero slope have a horizontal intercept.

6 6 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Ever line has both a horizontal intercept and a vertical intercept. (a) True (b) False (b). Either a horizontal line or a vertical line, ecluding the horizontal and vertical aes, provides a countereample. Note that all lines of the form = m + b, where m 0, have both tpes of intercepts.. Ever non-horizontal line must have at most one horizontal intercept. (a) True (b) False (a). Lines of the form = m+b, where m 0, have b as the horizontal intercept, while vertical lines have the m form = a, where a is the horizontal intercept. Ask the students what this means geometricall. 5. The graph in Figure. is a representation of which of the following functions? (a) = (b) = + 6 (c) = + (d) = Figure. (b). The line has slope and -intercept 6. Other methods of reasoning could be used. For eample, the line shown has a negative slope, which eliminates choices (a) and (d). The -intercepts for choices (b) and (c) are 6 and, respectivel. From the graph the -intercept is 6.

7 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 7 6. The graph in Figure. is a representation of which of the following functions? (a) = (b) = +.5 (c) = + (d) = Figure. (c). The line has slope and -intercept. Note that the -intercept on the graph is positive, which eliminates choices (a) and (d). The lines given in choices (b) and (c) have the same slope, so the choice will depend on the -intercept, which appears to be, not Consider the function f() = +. Give an equation of a line that intersects the graph of this function (a) Twice (b) Once (c) Never (a) There are man answers here. An horizontal line of the form = a, where a >. (b) There are man answers here. One eample is =. (c) There are man answers here. An horizontal line of the form = a, where a <. If ou consider this question graphicall, then have our students draw non-horizontal lines that meet the requirements for (a) and (b). This could be a wa to introduce the idea of tangent lines. 8. Consider the function f() = sin. Give an equation of a line that intersects the graph of this function (a) Once (b) Never (c) An infinite number of times (a) There are man answers here. For eample an vertical line will do. (b) An horizontal line of the form = n where n >. (c) An horizontal line of the form = n where n. Follow-up Question. Draw a line that intersects the graph of this function (a) Twice (b) Three times (c) Four times You can also ask if it is possible for a line which intersects the curve at an intercept to intersect the curve an even number of times.

8 8 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 9. Which of the following functions is not increasing? (a) The elevation of a river as a function of distance from its mouth (b) The length of a single strand of hair as a function of time (c) The height of a person from age 0 to age 80 (d) The height of a redwood tree (c). In general, people stop growing when the are oung adults and, before the are 80, the begin to lose height. This question epands a student s idea of a function. You could ask students to suppl some more functions that are increasing. 0. Considering as a function of, which of the following lines represent decreasing functions? (a) + = (b) = (c) = 6 (d) + = 6 (a) and (d). Lines with negative slopes are decreasing functions. (a) has a slope of and (d) has a slope of, and thus are decreasing functions. These equations could also be graphed to show the slopes geometricall.. Which of the graphs does not represent as a function of? (a) (b) a (c) (d) (b). For > a there are two function values corresponding to the same value of. It ma be worth noting that we are assuming that the entire graph is shown that the function does not behave differentl outside the given window.

9 . All linear functions are eamples of direct proportionalit. (a) True (b) False CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 9 (b). An linear function whose graph does not pass through the origin is not an eample of direct proportionalit. Students should tr to find eamples as well as countereamples antime a definition is introduced.. Which of the following graphs represent as directl proportional to? (a) (b) Figure. Figure. (c) (d) Figure.5 Figure.6 (a) and (c). If is directl proportional to, then = k, where k is a constant. (a) and (c) are graphs of such equations. Note that graphs representing as directl proportional to are lines through the origin. Students should recognize the graphical properties of being directl proportional to. Notice that (d) could be a representation of being directl proportional to some even power of.

10 0 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Which of the graphs represents the position of an object that is slowing down? (a) distance (b) distance (c) distance t (d) distance t t t (b). If the object is slowing down, then the changes in distance over various time intervals of the same length will decrease as time increases. You could have our students describe how the object is traveling in the other choices.

11 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 5. Which of the graphs represents the position of an object that is speeding up and then slowing down? (a) distance (b) distance (c) distance t (d) distance t t t (b). The graph has a positive slope everwhere, which is increasing for t near zero and decreasing for larger times. You could have students describe how the object is traveling in the other choices.

12 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 6. There are three poles spaced 0 meters apart as shown in Figure.7. Joe walks from pole C to pole B, stands there for a short time, then runs to pole C, stands there for a short time and then jogs to pole A. Which of the following graphs describes Joe s distance from pole A? A B C Figure.7 (a) distance (b) distance (c) distance time (d) distance time time time (d). The graph describes the distance from pole A, so it cannot start at the origin. This eliminates choices (a) and (c). Joe will spend more time walking between poles than running. This eliminates (b). You can also reason using slopes ( slope = speed) as follows: In the first interval he is walking, then he is going three times as fast (running), and finall he is jogging (half the speed of running). Follow-up Question. Give a scenario that describes the remaining graphs. ConcepTests and Answers and Comments for Section.. The graph of a function is either concave up or concave down. (a) True (b) False (b). A function can change concavit or can be a straight line and have no concavit. Show the different possibilities.

13 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons For Problems 5, use the graphs (I) (IV). (I) (II) (III) (IV). Which graph shows a function that is increasing and concave down? (III) Follow-up Question. Is this still true if the graph is shifted up one unit?. Which graph shows a function that is increasing and concave up? (I) Follow-up Question. Is this still true if the graph is reflected across the -ais?. Which graph shows a function that is decreasing and concave down? (IV) Follow-up Question. Is this still true if the graph is reflected across the -ais? 5. Which graph shows a function that is decreasing and concave up? (II) Follow-up Question. Is this still true if the vertical distance from the origin is doubled at ever point on the graph?

14 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 6. Which of the graphs is that of =? (a) (b) (c) (d) (c) Have students eplain wh each of the other choices is not appropriate.

15 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 5 7. Which of the graphs is that of =? (a) (b) (c) 5 (d) (c) Have students eplain wh each of the other choices is not appropriate. 8. Ever eponential function has a vertical intercept. (a) True (b) False (a). Eponential functions have the form P 0 a t, where a > 0, but a. The vertical intercept is P 0. Show this graphicall include positive and negative values of P 0 and values of a which are between 0 and as well as values of a which are greater than. 9. Ever eponential function has a horizontal intercept. (a) True (b) False (b). For eample, = has no horizontal intercept. Eponential functions are of the form = P 0 a t with 0 < a < or a >. Point out that an eponential function does not have a horizontal intercept unless P 0 = 0.

16 6 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 0. Which of the following tables could represent an eponential function? (a) f() (b) g() (c) h() (d) k() /6 9 0 /8 5 / 6 / / 8 6 (a). In (a) each term in this table is found b multipling the previous term b. In fact the entries in this table have the form (/6). Answer (b) is incorrect since each term in this table is found b multipling the previous term b /, but eponential functions need ab, where b > 0. Although in (c) the ratio of adjacent terms is constant, the values do not change b a constant amount. Answer (d) is incorrect since the ratio between adjacent terms is not constant. Have our students solve for a and b in h() = ab in choice (c) using the first two values and then check what happens to the last two values.. Let f() = ab f( + h), b > 0. Then = f() (a) b h (b) h (c) b +h b (d) a (a), since f( + h) = ab +h = ab b h = f()b h. This fact, in a different format, is used to find the derivative of b. This introduces the algebraic manipulations required for the definition of the derivative later in the tet.. How man asmptotes does the function f() = 0 have? (a) Zero (b) One (c) Two (d) Three (b). The onl asmptote is a horizontal one, namel =. Show that f() ma be thought of as 0 reflected across the -ais and then shifted up unit.. Estimate the half-life for the eponential deca shown in Figure t Figure.8 Depending on how much ou enlarge the graph, some variation in answers should be allowed here.

17 . Estimate the doubling time for the eponential growth shown in Figure.9. CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons Figure.9 t 7 Depending on how much ou enlarge the graph, some variation in answers should be allowed here. 5. List at least five properties of eponential functions of the form () = ab for a > 0 and either 0 < b < or b >. (a) Domain: all real numbers (b) Range: > 0 (c) Horizontal asmptote: = 0 (d) (0) = a (e) Alwas concave up (f) Alwas increasing for b >, and alwas decreasing for 0 < b < (g) No vertical asmptote You can have our students list more properties. Discuss what happens if a < 0.

18 8 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 6. Which of the following graphs is that of = ab if b >? (a) 5 (b) 5 (c) (d) (b). If b >, we need a graph that is increasing, concave up, and does not go through the origin. Here it is worth having the students point out specificall wh the other choices do not work. Follow-up Questions. What is the value of a? How does the graph change if a < 0? 7. During 988, Nicaragua s inflation rate averaged.% a da. Which formula represents the preceding statement? Assume t is measured in das. (a) I = I 0 e 0.0t (b) I = I 0 (.0) t (c) I = I 0 (.0)t (d) I = I 0 (.) t (b) Follow-up Question. What happens if the statement is changed to During 988, Nicaragua s inflation rate grew continuousl at a rate of.% each da.?

19 ConcepTests and Answers and Comments for Section. CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 9 For Problems, the graph in Figure.0 is that of = f(). Use the graphs (I) (IV) for the answers. (I) Figure.0 (II) (III) (IV). Which could be a graph of cf()? (III) and (IV). In (III) the graph could be /f() and in (IV) the graph could be f(). You could ask students to verbalize the relationship between f() and cf() for c >, 0 < c <, and c <.. Which could be a graph of f() k? (I) could be f(). You could ask students to verbalize the relationship between f() and f() k for k > 0 and k < 0.. Which could be a graph of f( h)? (II) could be a graph of f( + ). You could ask students to verbalize the relationship between f() and f( h) for h > 0 and h < 0.

20 0 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Which of the following functions is the sum of the functions in Figures. and.? Figure. Figure. (a) (b) (c) (d) (a). On the interval < < 0, the sum should be negative and concave up. You could follow up b asking for the graph of the difference of the functions in Figures. and. (both differences).

21 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 5. Given the graph of = sin in Figure., determine which of the graphs are those of sin() and sin()? = sin 6 6 Figure. (I) (II) (III) (IV) (a) (I) = sin() and (II) = sin() (b) (I) = sin() and (III) = sin() (c) (II) = sin() and (III) = sin() (d) (II) = sin() and (IV) = sin() (e) (III) = sin() and (IV) = sin()

22 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons (b). Replacing b means the first positive -intercept will be at = π/, so (I) is that of = sin(). Similarl the first positive zero of = sin() is at = π/, so (III) is that of = sin(). Have students also determine the equation for the graphs labeled (II) and (IV). 6. Which of the graphs is that of = + e? (a) (b) (c) (d) (b) The graph has a -intercept of /. You could have students tell of properties evident in the graphs of the other choices that conflict with those of = /( + e ).

23 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons For Problems 7, let f and g have values given in the table. f() g() f(g()) = g() = 0, so f(g()) = f(0) =. You can also consider f(g()) for = and =. 8. f(g(0)) = g(0) =, so f(g(0)) = f() =. You can also consider f(g()) for = and =. 9. f(g( )) = g( ) =, so f(g( )) = f() =. You can also consider f(g()) for = and =. 0. If f(g()) =, then = f( ) =, and g() =, so =. You can also consider f(g()) = a for a =,,.. If f(g()) = 0, then = f( ) = 0, and g( ) =, so =. You can also consider f(g()) = a for a =,,.. If g(f()) =, then = g(0) =, and f( ) = 0, so =. You can also consider g(f()) = a for a =, 0,.. If g(f()) =, then = g() =, and f() =, so =. You can also consider g(f()) = a for a =, 0,.

24 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons For Problems 8, let the graphs of f and g be as shown in Figure.. Estimate the values of the following composite functions to the nearest integer. 8 6 f() g() 6 8 Figure.. g(f(0)) f(0) 0, so g(f(0)) g(0). You ma want to point out that since f(0) = 0, this composition is similar to composing a function with the identit function. 5. g(f(8)) f(8) 8, so g(f(8)) g(8) 5. You ma want to point out that since f(8) = 8, this composition is similar to composing a function with the identit function. 6. g(f()) f(), so g(f()) g(). When ou are computing g(f(a)) from the graphs of g and f, it is not alwas necessar to compute f(a). For eample, when the horizontal and vertical scales are the same, ou can measure the height of f(a) with a straightedge. This distance placed on the -ais is the new value from which to measure the height of g. The result will be g(f(a)). 7. f(g()) g(), so f(g()) f(). See the Comment for Problem f(g( )) g( ) 0, so f(g( )) f(0) 0. See the Comment for Problem 6.

25 9. If f() = (a) e ( +) (b) e + (c) e + (d) e + + and g() = e then f(g()) = (b) Substituting g() into f() gives f(g()) = CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 5 (e ) + = e +. Students have trouble simplifing ( e ). Net ou could have them compute g(f()). 0. For which values of m, n, and b is f(g()) = g(f()) if f() = + n and g() = m + b? (a) m =, n and b could be an number (b) n =, m and b could be an number (c) n = 0, m and b could be an number (d) m =, n and b could be an number, or n = 0, m and b could be an number. (d). After the composition we have: m + b + n = m + mn + b. Thus mn = n which implies m = or n = 0. This provides an opportunit to point out the difference between subtracting the same quantit from both sides (alwas allowable) and canceling the same term from both sides (sometimes letting ou lose information like n = 0).

26 6 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Given the graphs of the functions g and f in Figures.5 and.6, which of the following is a graph of f(g())? f() g() Figure.5 Figure.6 (a) (b) (c) (d) (a). Because f() =, we have f(g()) = g(). Follow-up Question. Which graph represents g(f())?

27 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 7 For Problems, consider the four graphs. (I) (II) (III) (IV). Which of these graphs could represent even functions? (III) could be the graph of an even function. You could ask students to give geometric and analtic definitions here.. Which of these graphs could represent odd functions? (II) and (IV) could be graphs of odd functions. You could ask students to give geometric and analtic definitions here.

28 8 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Which of the following could be graphs of functions that have inverses? (a) (b) 0.5 (c) (d) (b) and (c). Because these graphs pass the horizontal line test, the could have inverses. You might redefine the other two functions b limiting their domains so the will have inverses. 5. If P = f(t) = + t, find f (P) If P = + t, then P = t and t = P. Thus f (P) = P. You could also use f(t) = + 8t, which uses a bit more algebra.

29 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 9 6. Which of the following graphs represents the inverse of the function graphed in Figure.7? Figure.7 (a) (b) (c) (d) (d). The graph of the inverse is a reflection of the function across the line =. You could ask the students wh each of the other choices fails to be the inverse.

30 0 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons ConcepTests and Answers and Comments for Section.. Which is a graph of = ln? (a) (b) (c) (d) (c). = ln is an increasing function passing through the point (, 0). You could discuss which properties each of the remaining graphs possess that made the student conclude that it was not an appropriate choice. Follow-up Question. Find possible formulas for the remaining graphs. Answer. (a) =, (b) = e, and (d) = ln.

31 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons. The graph in Figure.8 could be that of (a) = ln + (b) = ln ( (c) = ln + ) ( (d) = ln ) Figure.8 (c). Note that (a) is the graph of ln shifted up, (b) is shifted down, (d) is shifted to the right. You could also distinguish the four graphs b their horizontal intercepts.. Which of the following functions have vertical asmptotes of =? ( (a) = ln ) (b) = ln( ) (c) = ln( + ) (d) = ln (b). Note that (a) and (d) have vertical asmptotes at = 0, while (c) has one at =, and (b) has one at =, as desired. Follow-up Question. Do an of these functions have horizontal asmptotes? If so, what are the? Answer. No, the range of these functions is all real numbers.. Without caculating the following quantities, use the properties of logarithms to decide which of them is largest. (a) ln(0) ln() (b) ln (c) ln + ln ln (d) ln (b). ln(0) ln() = ln(5), ln() = ln(6), ln() + ln() = ln(), and ln() ln() = ln( ) = ln ln ln = = ln(e ). Since e < 9 and ln is an increasing function, ln(6) is the largest number. Point out that using the rules of logarithms enables us to compare eact values. If the comparison were made using a computer or calculator, we would likel be comparing approimate values.

32 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 5. The graph of a logarithmic function has a horizontal asmptote. (a) True (b) False (b). The range of logarithmic functions consists of all real numbers. You could also ask about vertical asmptotes. ( ) M N 6. ln = M + N (a) ln M (b) ln N (c) ln N (d) ln(m N) ln(m + N) (d) ( ) M N Follow-up Question. What is the value of ln? M N Answer. ln(m + N). 7. If log 0 ( a) = n, then = (a) 0 a+n (b) a + 0 n (c) n + 0 a (d) n + a 0 (b). Compose each side with the eponential function 0 since it is the inverse function of log 0. You could ask the same question with the natural logarithm rather than the logarithm base Which of the following functions are increasing and concave up? (a) (b) (c) ln (d) ln (b). Note that (a) and (d) are decreasing, and (c) is concave down. You could also ask about asmptotes (horizontal and vertical) and intercepts for all four functions. 9. Which of the following functions are decreasing and concave up? (a) ln( + ) (b) (c) (d) ln( ) (a) and (c). Note that (b) is increasing and (d) is concave down. You could also ask about asmptotes (horizontal and vertical) and intercepts for all four functions.

33 0. Which of the following do not have a horizontal asmptote? (a) = ln (b) = (c) = 5 (d) = / (a) and (d). The range of ln and / is all real numbers. Follow-up Question. Which of the above functions does not have an asmptotes? Answer. (d). The domain and range of = / is all real numbers.. Give a formula for the inverse of the following function: (a) f (P) = 6 e P (b) f (P) = (c) f (P) = ( ) ln6 P ln P ln6 (d) f (P) = ( ) P ln 6 (d). If P = 6e t, then P = 6 et and ln ( P 6 Students ma find that f (P) = CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons P = f(t) = 6e t ) = t. This gives t = ln( P 6). logarithms. For an alternate question, ou could also use f(t) = e t.. Give a formula for the inverse of the following function: (a) f (P) = e6p (b) f (P) = ep/6 (c) f (P) = ( ) P ln 6 ( ) (d) f ln6 (P) = P (ln P ln6). This would be an ecellent time to review the properties of P = f(t) = 6 ln(t) (b). If P = 6 ln(t), then P 6 = ln(t) and ep/6 = t. This gives t = ep/6. You could also use f(t) = 6 + ln(t ).. Solve for if 8 = e. (a) = ln 8 + ln + ln (b) = ln ln8 + ln (c) = ln 8 + ln ln (d) = ln ln8 ln (c). If 8 = e, then 8 = e and ln ( 8 ) =, so = ln 8 + ln ln. This is a good place to point out the man was answers can be epressed using logarithms.

34 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Solve for if = e +. (a) = ln ln ln( ) (b) = ln (c) = ln ln (d) = ln( e) ln (d). If = e +, e = ln( e) and ln( e) = ln. This gives =. ln You could ask what errors could have been made in obtaining the other choices. 5. For what value of is + = 6? If + = 6, then ( + ) = 6 =. Division gives =, so = and =. Students ma tr to take logarithms of both sides of the original equation. ConcepTests and Answers and Comments for Section.5. The amplitude and period of the graph of the periodic function in Figure.9 are (a) Amplitude:. Period:. (b) Amplitude:. Period:. (c) Amplitude:. Period: /. (d) Amplitude:. Period:. (e) Amplitude:. Period: /. 6 5 Figure.9 (d). The maimum value of the function is 5, and the minimum value is, so the amplitude is 5 ( ) =. The function repeats itself after units, so the period is. Point out that the function is oscillating about the line =. Have the students find a formula for the function shown in Figure.9.

35 . The amplitude and period of the graph of the periodic function in Figure.0 are (a) Amplitude:. Period:. (b) Amplitude:. Period:. (c) Amplitude:. Period: /. (d) Amplitude:. Period:. (e) Amplitude:. Period: /. CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons Figure.0 (c). The maimum value of the function is 5 and the minimum is, so the amplitude is 5 =. The function repeats itself after / unit, so the period is /. It is easiest to find the period using the etreme values of the function.. Which of the following could describe the graph in Figure.. ( (a) = sin + π ) ( (b) = sin + π ) (c) = cos() ( ( (d) = cos (e) = sin() (f) = sin ) ) 6 6 Figure. (a) and (d). Note that (b), (c), and (e) have period π, with the rest having period π. Answer (f) has (0) = 0. (a) and (d) could describe the graph. The fact that the same graph ma have more than one analtic representation could be emphasized here.

36 6 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Figure. shows the graph of which of the following functions? (a) = cos( + π/6) (b) = cos( π/6) (c) = sin( π/6) (d) = sin( + π/6) (e) None of these π/ π π/ π Figure. (b) You could ask what the graphs of the other choices look like.

37 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 7 5. Which of the graphs are those of sin(t) and sin(t)? (a) (I) = sin(t) and (II) = sin(t) (b) (I) = sin(t) and (III) = sin(t) (c) (II) = sin(t) and (III) = sin(t) (d) (III) = sin(t) and (IV) = sin(t) (I) (II) π/ π/ π/ π/ t π/ π/ π/ π/ t (III) (IV) π/ π/ π/ π/ t π/ π/ π/ π/ t (b). The period is the time needed for the function to eecute one complete ccle. For sin(t), this will be π and for sin(t), this will be π/. You could ask what equations describe the graphs of (II) and (IV). This question is the same as Question 5 in Section..

38 8 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 6. Consider a point on the unit circle (shown in Figure.) starting at an angle of zero and rotating counterclockwise at a constant rate. Which of the graphs represents the horizontal component of this point as a function of time. Figure. (a) (b) π/ π π/ π t π/ π π/ π t (c) (d) π/ π π/ π t π/ π π/ π t (b) Use this question to make the connection between the unit circle and the graphs of trigonometric functions stronger. Follow-up Question. Which of the graphs represents that of the vertical component of this point as a function of time? Answer. (c)

39 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 9 7. Which of the following is the approimate value for the sine and cosine of angles A and B in Figure.? A B Figure. (a) sin A 0.5, cos A 0.85, sin B 0.7, cos B 0.7 (b) sin A 0.85, cos A 0.5, sin B 0.7, cos B 0.7 (c) sin A 0.5, cos A 0.85, sin B 0.7, cos B 0.7 (d) sin A 0.85, cos A 0.5, sin B 0.7, cos B 0.7 (b) You could use this question to make the connection between the unit circle and the trigonometric functions stronger. You could also identif other points on the circle and ask for values of sine or cosine. 8. Figure.5 shows the graph of which of the following functions? π/ π π/ π Figure.5 (a) = sin( π/) (b) = cos( π/) (c) = sin( + π/) (d) = cos( + π/) (e) None of these (c) You could ask what the graphs of the other choices look like.

40 0 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 9. Which of the following are properties of both = arcsin and = (a) (0) = 0 (b) Increasing everwhere (c) Concave down for < 0 (d) Domain: [, ] (e) Range: (, )? (a), (b), and (c). The domain of = is (, ) and the range of arcsin is [ π/, π/]. You ma want to have the students sketch graphs of both functions. Follow-up Question. What is the concavit for each function when > 0? Answer. Both functions are concave up when > Which of the following have the same domain? (a) arcsin (b) arctan (c) ln (d) e (e) (a) and (e), also (b) and (d). Note that the domain of arcsin and is [, ], the domain of arctan and e is (, ). Students could be thinking of the range of sin for (a) and the range of tan for (b). Follow-up Question. Does changing (c) to ln affect the answer? Answer. No, because the domain of ln is all real numbers ecept = 0.. Which of the following have the same range? (a) arcsin (b) arctan (c) ln (d) e (e) [ (a) and (b). The range of (a) and (b) is π, π ]. Follow-up Question. Does changing (c) to ln affect the answer? Answer. No, the range of ln is [0, ) while the range of e is (0, ).. If = arcsin, then cos = (a) (b) (c) (d) (e) (a). If = arcsin, then π < < π giving cos = sin =. You could also solve this problem b drawing a right triangle with hpotenuse of and labeling the side opposite angle as.

41 . If = arcsin, then tan = (a) (d) (b) (e) (d). If = arcsin, then π < < π sin giving tan = CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons (c) sin sin =. cos = You could also solve this problem b drawing a right triangle with hpotenuse of and labeling the side opposite angle as.. If = arctan, then cos = + (a) (b) (d) + (e) + (c) + (c). If = arctan, then π < < π. Because tan + = cos, cos = tan + = +. You could also solve this problem b drawing a right triangle with the side opposite angle labeled as and the side adjacent to angle labeled as. 5. If = arctan, then sin = + (a) (b) (d) + (e) + (c) + (b). If = arctan, then π < < π and sin = tan cos. Because tan + = cos, cos = and sin = +. tan + You could also solve this problem b drawing a right triangle with the side opposite angle labeled as and the side adjacent to angle labeled as. ConcepTests and Answers and Comments for Section.6. Graph =, =, =, = 5. Make at least three observations about the graphs. (a) = and = have the same general shape that of a U. (b) = and = 5 have the same general shape that of a seat. (c) For >, as the power increases, the function grows faster. (d) When 0 < <, we have > > > 5. (e) For > 0, all functions are increasing and concave up. (f) All functions intersect at (, ) and (0, 0). (g) For < 0, the functions = and = are decreasing and concave up. (h) For < 0, the functions = and = 5 are increasing and concave down. This question could also be used as an eplorator activit.

42 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Graph =, =, =, =. Make at least three observations about the graphs. (a) = and = have the same general shape and the are alwas positive. (b) = and = have the same general shape. (c) As the power decreases, the function approaches 0 faster as increases. (d) For > 0, the are all concave up. (e) The intersect at (,). (f) Each has a vertical asmptote at = 0 and a horizontal asmptote at = 0. (g) For < 0, the functions = and = are increasing and concave up. (h) For < 0, the functions = and = 5 are decreasing and concave down. This question could also be used as an eplorator activit.. Graph = /, = /, = /, = /5. What do ou observe about the growth of these functions? The smaller the power of the eponent, the slower the function grows for >. Your students ma observe other properties. For Problems 8, as which function dominates, (a) or (b)? (That is, which function is larger in the long run?). (a) 0. (b) 0 0 (a). Power functions with the power greater than one and with a positive coefficient grow faster than linear functions. You could ask about the behavior as as well. 5. (a) 0.5 (b) 5,000 (a). Note that 0.5 is an increasing function whereas 5,000 is a decreasing function. One reason for such a question is to note that global behavior ma not be determined b local behavior. 6. (a) 0.9 (b) ln (b). Note that 0.9 has a horizontal asmptote whereas the range of ln is all real numbers. Students should realize that the graph the calculator displas can be misleading. 7. (a) (b) (b). Eponential growth functions grow faster than power functions. You could ask about the behavior as as well. 8. (a) 0( ) (b) 7,000 (a). Eponential growth functions grow faster than power functions, no matter how large the coefficient. One reason for such a question is to note that global behavior ma not be determined b local behavior.

43 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 9. List the following functions in order from smallest to largest as (that is, as increases without bound). (a) f() = 5 (b) g() = 0 (c) h() = 0.9 (d) k() = 5 (e) l() = π (a), (c), (d), (e), (b). Notice that f() and h() are decreasing functions, with f() being negative. Power functions grow slower than eponential growth functions, so k() is net. Now order the remaining eponential functions, where functions with larger bases grow faster. This question was used as an elimination question in a classroom session modeled after Who Wants to be a Millionaire?, replacing Millionaire b Mathematician. 0. The equation = is represented b which graph? (a) (b) (c) (d) (b). The graph will have a -intercept of 6 and not be that of an even function. You ma want to point out the various tools students can use to solve this problem, i.e. intercepts, even/odd, identifing the zeros, etc. You could also have students identif a propert in each of the other choices that is inconsistent with the graph of =

44 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. The graph in Figure.6 is a representation of which function? (a) = + (b) = ( )( + ) (c) = ( 6)( ) (d) = ( )( + ) Figure.6 (b). The graph is a parabola with -intercepts of and. You could ask students to describe the graphs of the other equations.

45 . The equation = is represented b which graph? CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 5 (a) (b) (c) (d) 6 (d). The graph will be a parabola with -intercepts of and. Have students identif a propert in each of the other choices that is inconsistent with the graph of = +5+6.

46 6 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Which of the graphs could represent a graph of = a + b + c + d + e? Here a, b, c, d and e are real numbers, and a 0. (a) (b) (c) 0 (d) (a), (b), (c), and (d) Follow-up Question. What propert could ou observe in a graph to know it could not be that of a fourth degree polnomial? Answer. The graph of the function has opposite end behavior ( as ), or the graph turns more than three times, are a few properties students might observe.

47 . Which of the graphs (a) (d) could be that of a function with a double zero? CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 7 (a) 5 (b) 5 (c) 5 (d) 5 The graphs in (b) and (c) touch the -ais, but do not cross there. So, the have double zeros. Students should remember that a double zero means the function will bounce off the -ais. 5. Let f() = and g() =, then f() = g(). (a) True (b) False + (b). The domain of f() is not equal to the domain of g(). Students should realize that some algebraic manipulations can onl be applied to functions if the domain is stated. Follow-up Question. How can ou change the statement to make it true? Answer. Keep f() defined as in the problem, and remove = from the domain of g(). Alternativel, replace f() = g() with f() = g() if.

48 8 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 6. Without using our calculator, which of the following is a graph for =? (a) 0 (b) (c) 0 (d) (c). The equation indicates -intercepts at ± and a vertical asmptote at =. (c) is the onl graph where this happens. Students should analze the long-term behavior of the function as well as the short-term behavior (i.e. asmptotes). You could also ask the students to analze the similarities and differences of the functions (i.e. zeros, intercepts, etc.).

49 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 9 7. Which of the graphs represents =? (a) (b) (c) (d) (b). The graph goes through the origin and is positive for >. Have students identif a propert in each of the other choices that is inconsistent with the graph of =.

50 50 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 8. Which of the graphs represents = +? (a) (b) (c) (d) (d). The graph goes through the origin and has vertical asmptotes at = and. It is also positive for 0 < <, and for large, the function is negative. You could have the students verbalize wh the ecluded choices (a), (b), and (c).

51 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 5 9. Which of the graphs represents = +? (a) (b) (c) (d) (c). The graph goes through the origin and has vertical asmptotes at = and. It is negative for 0 < <, and the function is positive for large. Alternativel, the function has a horizontal asmptote =. You could have the students verbalize wh the ecluded choices (a), (b), and (d).

52 5 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 0. Which of the following functions represents the higher of the two functions in Figure.7 divided b the lower of the two functions? 6 5 (a) Figure.7 (b) (c) (d) (d). The ratio of the higher of the two functions to the lower equals at = 0, =, and =. You might want to point out that the division is undefined when =. Follow-up Question. What does the graph of the reciprocal of this ratio look like? Answer. = { if undefined if =.

53 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 5. Which of the following functions represents the higher of the two functions in Figure.8 divided b the lower of the two functions? Figure.8 (a) (b) (c) (d) (b). Notice this ratio starts at and approaches as approaches. The division of the two functions is not defined when =. Follow-up Question. Could the graph of the reciprocal of this ratio be one of these choices? Answer. No, the graph of the reciprocal passes through the point (0, /).

54 5 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Which of the following functions represents the higher of the two functions in Figure.9 divided b the lower of the two functions? Figure.9 (a) (b) (c) (d) (b). Because as increases the lower graph is approaching zero faster than the upper graph, their ratio will be increasing. The division of the two functions is not defined when =. Follow-up Question. Could the graph of the reciprocal of this ratio be one of these choices? Answer. Yes, (d), because the lower function could be a quadratic, and a quadratic function divided b a linear function is linear where it is defined.

55 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 55. Which of the following functions represents the ratio of the function starting at the point (0, 8) to the function starting at the point (0, ) as shown in Figure.0. (a) Figure.0 (b) (c) (d) (d). The ratio is alwas positive, decreasing, and less than after the curves cross. Alternativel, the ratio has the value when. Follow-up Question. Could the graph of the reciprocal of this ratio be one of the choices? Answer. No, none of these graphs passes through the point (0, /).

56 56 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Ever eponential function has a horizontal asmptote. (a) True (b) False (a). The horizontal ais is a horizontal asmptote for all eponential functions. You ma need to remind students that an eponential function has the form = ab where a 0 and b > 0 but b. 5. Ever eponential function has a vertical asmptote. (a) True (b) False (b). Eponential functions do not have vertical asmptotes. The domain for an eponential function is all real numbers. You ma point out that analzing the domain of a function is another wa to decide if vertical asmptotes eist. ConcepTests and Answers and Comments for Section.7 For Problems 6, decide whether the function is continuous on the given interval.. f() = on [0, ] f() is not continuous because f() is not defined. Students should realize that changing the domain affects the answer. You ma want to ask for the definition of continuit on an interval.. f() = on [, 0] f() is continuous. You ma want to ask for the definition of continuit on an interval. [. f() = e tan θ on π, π ] f() is not continuous since tan θ is a not defined for = π, π. Follow-up Question. Is f() = e tan θ continuous on ( π/, π/)?. f() = on (, ) + f() is not continuous since f( ) is not defined. Ask the same question, but change the interval so that = is not in the interval. {, 5. f() = + Answer f() is continuous., = You might spend more time discussing piecewise defined functions. 6. The cost of a first class stamp over the past 00 ears. This function is not continuous. Stamp prices increase b a discrete amount.

57 Have our students describe other functions that are not continuous. 7. The range of which of the following functions can take on all values between and. (a) = arcsin() (b) = arctan (c) = + cos (d) = ( ) CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 57 (e) = 9 [ (b). The range of = arctan is [ π, π], whereas that of arcsin() is π, π ], so the function in (a) cannot have the value of. The functions in (c) and (d) are alwas positive, while (e) can never equal 0. Follow-up Question. Which functions have the same range? Answer. The range of (b) and (c) is all real numbers. ConcepTests and Answers and Comments for Section.8. Possible criteria for a limit: As ou get closer and closer to the limit point the function gets closer and closer to the limit value. Which of the following is an eample that meets the criteria but does not have the stated limit? (a) As increases to, f() = gets closer and closer to, so the limit at = of f() is. (b) As increases to 00, f() = / gets closer and closer to 0, so the limit as goes to 00 of f() is 0. (c) As increases to, f() = ( + ) gets closer and closer to 6, so the limit as goes to of f() is 6. (d) None of these show a problem with this criteria for a limit. (b) meets this criteria, but this limit is /00, not 0. If ou do not want to talk about one-sided limits, then ou ma not want to use this question.

58 58 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons. Which of the graphs has an ǫ and δ which would satisf the definition of a limit at = c? (a) 5 δ ǫ (b) δ ǫ (c) c (d) c 6 δ ǫ δ ǫ c c 6 (d) You might want to make sure that students realize that the δ and ǫ shown should be placed on both sides of c and f(c), respectivel. You can give students a specific = f() to enter in their calculator, along with a range of f(c) ǫ < < f(c) + ǫ and ask them to find the -window so the graph of f eits the side (rather than the top). Then ask them to eplain how that could give them a value δ which works in the limit definition.. Give an eample of a function showing wh each of the following could not be the onl requirement in the definition of a limit. (a) lim f() = L if f(a) = L a (b) lim f() = L if f() becomes closer to L as becomes closer to a a { /( a), a (a) Consider f() = L, = a (b) Consider f() = ( a) and L = You could have students describe these eamples graphicall instead of giving equations.

59 CHAPTER ONE Hughes Hallett et al c 009, John Wile & Sons 59. Which of the following statements is true of the limit of the function sin(/) as goes to zero? (a) If ou give me a number, ǫ, no matter how small, I can alwas get close enough to = 0 so that sin(/) stas within that number of 0. Therefore the limit of sin(/) as goes to 0 is 0. (b) If ou give me a number, ǫ, no matter how small, I can alwas get close enough to = 0 so that sin(/) stas within that number of. Therefore the limit of sin(/) as goes to 0 is. (c) If ou give me a number, ǫ, no matter how small, there is no point on the -ais close enough to 0 such that sin(/) is confined to that number. Therefore the limit of sin(/) does not eist. (c). For an interval containing 0, the value of sin(/) takes on the value of 0 an infinite number of times (at = /(nπ), where n is an integer) and sin(/) takes the value of an infinite number of times (at = /(mπ) where m is a member of the sequence, 5,9,,...). Follow-up Question. What happens to the graph of sin(/) as approaches 0? Answer. The graph of sin(/) approaches 0 as approaches Possible criteria for continuit of a function at a point: If the limit of the function eists at a point, the function is continuous at that point. Which of the following eamples fits the above criteria but is not continuous at = 0? (a) f() =. The limit of f(), as goes to 0, is 0 so this function is continuous at = 0. (b) f() = /. The limit of f(), as goes to 0, is 0 therefore f() is continuous at = 0. (c) f() = /. The limit of f(), as goes to 0, is therefore f() is continuous at = 0. (d) None of these show a problem with this criteria. (b). f(0) is not defined, so f cannot be continuous at = 0. You could ask students wh the limit does not eist in choice (c).

60 60 CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons 6. Definition of continuit of a function at a point: If the limit of the function eists at a point and is equal to the function evaluated at that point, then the function is continuous at that point. Which of the following is not continuous at = c? (a) (I) onl (b) (II) onl (c) (III) onl (d) (I) and (II) (e) (I) and (III) (f) (II) and (III) (I) c (III) (II) c 5 5 c (c) You could have students give reasons wh the other choices are continuous functions.

61 Chapter Two Chapter Two

62 6 CHAPTER TWO Hughes Hallett et al c 005, John Wile & Sons ConcepTests and Answers and Comments for Section. For Problems, assume our car has a broken speedometer.. In order to find m average velocit of a trip from Tucson to Phoeni, I need (a) The total distance of the trip (b) The highwa mile markers (c) The time spent traveling (d) How man stops I made during the trip (e) A friend with a stop watch (f) A working odometer (g) None of the above (a) and (c) The choices are intentionall vague. This is meant to provide discussion. Your students ma select more than one item.. In order to find m velocit at the instant I hit a speed trap, I need (a) The total distance of the trip (b) The highwa mile markers (c) The time spent traveling (d) How man stops I made during the trip (e) A friend with a stop watch (f) A working odometer (g) none of the above (e) and (f). After I pass the speed trap I can watch m odometer as it increases b 0. miles while m friend (simultaneousl) records the time it took to travel 0. miles. Using (e) and (f) as the odometer increases b 0. is a good estimate of the velocit at an instant. It ma be beneficial to point out that if the odometer measured in hundredths of a mile, then ou could compute an even better estimate of the instantaneous velocit.. Regarding the speed trap in Problem, when should our friend first start the stopwatch? (a) When the driver of an oncoming vehicle warns ou of the speed trap ahead b flashing his/her bright headlights (b) When ou spot the cop (c) Either scenario (d) Neither scenario (c). You can use an estimation of the average velocit before (or after) ou hit the speed trap to estimate our actual velocit. The focus of this discussion should be on how h can be either positive or negative in order to estimate the derivative.

63 CHAPTER TWO Hughes Hallett et al c 009, John Wile & Sons 6. Which graph represents an object that is slowing down where t is time and D is distance. Assume the units on each ais is the same for all graphs. (a) D (b) D (c) D t (d) D t t t (c) Students have a good idea of what the words slowing down mean, but tr to make the connection between the words and slope of the tangent line at different points.