Think Ahaiit It In Exercises 67-7(1. create a function whose yrapli lias tlie indicated

In Kxertises 1-4. match the fjraph of/ in the left column with that of lis (leii\ alive in tlie rij;lil column.

In Kxertises 1-4. match the fjraph of/ in the left column with that of lis (leii\ alive in tlie rij;lil column.

In Kxertises 1-4. match the fjraph of/ in the left column with that of lis (leii\ alive in tlie rij;lil column.

In Kxertises 1-4. match the fjraph of/ in the left column with that of lis (leii\ alive in tlie rij;lil column.

Graphical Keasoniiif; The graph of / is given in the figure. (a) Hor which \alucs ol .v is fix) zero? Positive? Negative? (b) l-or which \ahics ol a is /"(a) zero? Positive'.' Negative? (c) On what interval is /' an increasing function? (d) For which value of a is /'(a) niiniinum? For this \'aluc of \. how docs the rate ol change ot / compare with the laic ol change ol / loi oihci \alues of a' FAplain,

Graphical Reasoning Identify the leal numbers v,,. v,. v,. .v,. and Aj in llie figure sucli that each of the following is true. (a) /'(a) = (h) /"(a) = (c) /'(a) does not exist. (d) / has a relatixe nia.vinuini. (e) / has a point of inneetion,

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. > = .V- A-"+3

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. ^' - x- + 1

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = 1 A - 2

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. > x^-9

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. -)v 11. \ =

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. /( 1 '^-

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. 1,.(a) = A 15. ,/(a) = 17. v = -^' V- + 1

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. fix) = X + '

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.,/(a) = 17. v = -^' V- + 1 V- + 1

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.Ji.x) ="

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.v = -^' V- + 1 V- + 1 .V 6a + 12

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.. 5a + 5

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = xj4 - .

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.,?(.v) = A./9"

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. h{x) = X JV

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.V = aV16 - A

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.3a-"

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = 3(a- 1)-/' - (a- D

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.3a- -I- 3

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.y = -{(a' - 3a -I- 2)

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.y = 2 - A - A

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results./(a) = {(a - \? + 2

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results./(a) = 3a' - 9a -I- 1

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results./(a) = (a -I- IKa - 2)(a

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = ix-' + 4a"

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.V = 3a-' - 6a-

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. fix) - A-* - 4a' -I- KiA

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results./(a) = A-* - 8a' + 18a- 16a +

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.y = A^ - 5a

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.y = (a - 1 )'

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.^ - |2a - 3|

In F^xercises 7-38. analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. y = |.v- 6a + 5

In Exercises 39-46, sketch a graph of the function over the indi- cated interval. Use a graphing utility to verify your graph. y = sin A - 40. \' = cos A j^ sin 3x < A < 277

In Exercises 39-46, sketch a graph of the function over the indi- cated interval. Use a graphing utility to verify your graph. \' = cos A < A < 277

In Exercises 39-46, sketch a graph of the function over the indi- cated interval. Use a graphing utility to verify your graph. y = 2a - tan V "2 ^ ' ^ 2

In Exercises 39-46, sketch a graph of the function over the indi- cated interval. Use a graphing utility to verify your graph. \ = 2(a - 2) + cot A < A < 7T

In Exercises 39-46, sketch a graph of the function over the indi- cated interval. Use a graphing utility to verify your graph. V = 2(csc A + sec a) < A < ^

In Exercises 39-46, sketch a graph of the function over the indi- cated interval. Use a graphing utility to verify your graph. \ = sec!S - 3 < A < 3

In Exercises 39-46, sketch a graph of the function over the indi- cated interval. Use a graphing utility to verify your graph.g(x) = A tan A 377 377 y < A < y

In Exercises 39-46, sketch a graph of the function over the indi- cated interval. Use a graphing utility to verify your graph.g(.x) = A cot A 2 77 < A- < 2 77

^ In Exercises 47-5(1. use a computer algebra system to analyze and graph the function. Identify any relative extrenia. points of intlection, and asymptotes. fix) = 49. /(a) = 2Q A- +

^ In Exercises 47-5(1. use a computer algebra system to analyze and graph the function. Identify any relative extrenia. points of intlection, and asymptotes. /(a) = 5 - 4 4a Vx- + 15 +

^ In Exercises 47-5(1. use a computer algebra system to analyze and graph the function. Identify any relative extrenia. points of intlection, and asymptotes. /(a) = 2Q A- + 1 A v'a~ + 7

^ In Exercises 47-5(1. use a computer algebra system to analyze and graph the function. Identify any relative extrenia. points of intlection, and asymptotes. fix 1 A - 4 4a Vx- + 15

In Exercises 51 and 52. the graphs of/./', and / "are shown on the same set of coordinate axes. Which is which? To print an enlarged copy of the graph, go to the website www.mathgraphs.ciiin.

In Exercises 51 and 52. the graphs of/./', and / "are shown on the same set of coordinate axes. Which is which? To print an enlarged copy of the graph, go to the website www.mathgraphs.ciiin.

In Exercises 53-56, use the graph of/' to sketch a graph of / and the graph of / ". To print an enlarged copy of the graph, go to the website www.mathgraplis.com.

In Exercises 53-56, use the graph of/' to sketch a graph of / and the graph of / ". To print an enlarged copy of the graph, go to the website www.mathgraplis.com.

In Exercises 53-56, use the graph of/' to sketch a graph of / and the graph of / ". To print an enlarged copy of the graph, go to the website www.mathgraplis.com.

In Exercises 53-56, use the graph of/' to sketch a graph of / and the graph of / ". To print an enlarged copy of the graph, go to the website www.mathgraplis.com.

Suppose /'(;) < for all t in the inter\al (2.8). Explain ; why/(3) >/(5).

Suppose /(O) = 3 and 2 < f'(x) < 4 for all a in the interval i [-5. 5], Determine the areatest and least possible values of ! /(2).

In Exercises 59 and 60, use a graphing utility to graph the function. I'se the graph to determine whether it is possible for the graph of a function to cross its horizontal asymptote. Do \ou think it is possible for the graph of a function to cross its vertical asymptote? \\ hy or why not? /(a) = 4(a - 1 ri- 4v + .'

In Exercises 59 and 60, use a graphing utility to graph the function. I'se the graph to determine whether it is possible for the graph of a function to cross its horizontal asymptote. Do \ou think it is possible for the graph of a function to cross its vertical asymptote? \\ hy or why not? ,d.v) 3a-' - 5.V + 3 A-" + 1

Writing In Exercises 61 and 62, use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function may indicate that there should be one. hix) = 3

Writing In Exercises 61 and 62, use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function may indicate that there should be one.gix) .X-. + -VI

Writing In Exercises 63 and 64, use a graphing utility to graph the function and determine the slant asymptote of the graph. Zoom out repeatedly and describe how the graph on the display appears to change. W h\ does this occur? fix) 3a I

Writing In Exercises 63 and 64, use a graphing utility to graph the function and determine the slant asymptote of the graph. Zoom out repeatedly and describe how the graph on the display appears to change. W h\ does this occur? i'(A) = Tv- \5

Graphical ReuMiiiiiig Consider the liuict /(vl y.v- + 1 < A < 4. (a) Use a computer algebra system to graph the function and use the graph It) visually approximate the critical numbers. (b) Use a computer algebra system to find /' and approximate the critical numbers. Are the results the same as the visual approximation in part (a) ' Explain.

66. Grapliical Kensonin^ Consider llic lunclion fix) = tan(siii 77-v). (a) Use a graphing iililil\ to graph llie liincUon. (b) IdenlilX an_\ s_\ninielr\ ol Ihe graph. (e) Is tlie Innelion [leiiodie' It so. what is the period? (dl Identit) an\ extrenia on I I. II (e) Use a graphing iitilit\ to determine the eonca\ity ol the graph on (0. I ).

Think Ahaiit It In Exercises 67-7(1. create a function whose yrapli lias tlie indicated characteristics. (There is more than one correct answer. I Vertical asymptote: v = 5 Horizontal asymptote: 1 =

Think Ahaiit It In Exercises 67-7(1. create a function whose yrapli lias tlie indicated characteristics. (There is more than one correct answer. I Vertical asymptote: v = 3 Hori/oiital asy mptote: None

Think Ahaiit It In Exercises 67-7(1. create a function whose yrapli lias tlie indicated characteristics. (There is more than one correct answer. IVertical asymptote: -V = 5 .Slant asymptote: y = .\v + 2

Think Ahaiit It In Exercises 67-7(1. create a function whose yrapli lias tlie indicated characteristics. (There is more than one correct answer. I Vertical asymptote: v = Slant as\ niptote: y = .v

(inipluail Rciisdiunti Consider the fiinelion (.V - /ilia) Determine the el'teet on the graph ol / iT /) ^ and a is \aried. Consider eases where 11 is positi\e and a is negative. (h) Deternime the elTeet on the graph ol / \\ n * and /) is varied

Consider the Innelion /(.v) = \Ul\)- - iax). a * 0. la) Deteririine the changes |if any) in the intercepts. cMiema. and concaxity of the graph of / when a is varied. lb) In the same \ ievving w indow. use a graphing utility to graph the function for four different values of d.

Investigation Consider the function f(x) = 3x x + 1 for nonnegative integer values of it (a) [Discuss the relationship between the value of n and the symmetry ol the graph. (b) For which values of ;i will the .v-a.xis be the horizontal asymptote' (c) For which \alue ol will \- = 3 be the horizontal asymptote'.' (d) What is the asyinptote of the graph when u = 5? (e) Use a graphing utility to graph / for the indicated values of ;( in the table. Use the graph to determine the number of extrema A7 and the nuniber of inflection points N of the graph.

Iincstis^iitidii Let /^l v. v,,! be an arbitrary pomi on the graph ot / sueh that fix,,) i= 0, as indicated in the figure. Verity each ol the followini;. (a) The .v-intereept ol' the tangent line is .v,. ,/'(.v)- (h) The \-inlereept of the tangent line is (0./(.v,,) - .v,, /'(.v,,)). (e) The .v-inteicept ol Ihe normal line is (.v,, + /( V|,)/'(.V|,). 0). Id) The i-inteivept of the normal line is I 0. y,, + ' " 1. /(-v) le) \BC\ = If) \PC\ = n.v,,) (g) \AB\ = |/(.v)/'(.v)| ./(v ,)x/l + [./'(A )? ./Xv, ) (h) |.4P| = |/(.v)|.I +[/'(.v)]-

Modeliiif; Data The data in the table show the number N of hactei"ia m a cultuie at lime 1. where 1 is measured in da\s. A model for this data is gi\en by ., 24.670 - 3.S.l.i3/ + 13.250/- N = ^ . 1 < f < 8 100 - y)t + lt(a) Use a graphing utility to plot the data and graph the model. lb) Use the model to estimate Ihe number of bacteria when t = 10. (e) Approximate the day when the nuniber of bacteria was greatest. (dl Use a computer algebra system to determine the time when the rate of increase in the number of bacteria was greatest. (e) Find lim /V(/).