CHAPTER 1 Functions, Graphs, and Limits

Transcription

1 CHAPTER Functions, Graphs, and Limits Section. The Cartesian Plane and the Distance Formula Section. Graphs of Equations Section. Lines in the Plane and Slope Section. Functions Section. Limits Section. Continuit Review Eercises Practice Test

2 CHAPTER Functions, Graphs, and Limits Section. The Cartesian Plane and the Distance Formula Solutions to Odd-Numbered Eercises. (a) a b c a b 9 c. (a) a b c 7 9 a b 9 9 c (, ) c a (, ) b (, ) (, ) c a (7, ) 8 b (7, ). (a) a 7. (a) b c a b c (, ) (, ) (, ) (, ) b c (, ) a (, ) d (c) Midpoint,, 9. (a) ( ), ( ), d 9 ( ), (c) Midpoint,,

3 Chapter Functions, Graphs, and Limits. (a) (, ). (a) (, ) (, ) d 7 (, 8) (c) Midpoint,, 8 8 (, ) d 8 +, (c) Midpoint,,. d 7 7. d d 7 Since d d d, the triangle is a right triangle. (, 7) d d (, ) d (, ) d d d d (, ) d (, ) d d (, ) d (, ) 9. d. d d Since d d d, the points are collinear. d 8 d d Since d d d, the points are not collinear. (, ) d (, ) d d d d (, ) (, ) d (, ) (, ). d 7 7 8,. d ±

4 Section. The Cartesian Plane and the Distance Formula 7 7. Midpoint, The point one-fourth of the wa between, and, is the midpoint of the line segment from, to which is The point three-fourths of the wa between, and, is the midpoint of the line segment from to,, which is Thus,,,,,,,,,., and,,., are the three points that divide the line segment joining, and, into four equal parts. 9. (a),,,,,,, 7, 7. (a),, 9,, d, d > d 8 d 8.7 feet A square feet. 7 Answers will var. The number of subscribers appears to be increasing linearl.. (a), 8,9 (c) 8, (d), 7. (a) $8 thousand $ thousand (c) $ thousand (d) $9 thousand

5 8 Chapter Functions, Graphs, and Limits 9. (a) Revenue midpoint 999 7,89,, Revenue estimate for : $,7 million Profit midpoint , Profit estimate for : $89.7 million Actual revenue: $, million Actual profit: $88. million (c) Yes, the increase in revenue is approimatel $7 million per ear. Yes, the increase in profit is approimatel $ million per ear. (d) 999 Epenses: 7,89. $7,.9 million Epenses:, 88. $,77. million Epenses:, 7. $,7.7 million (e) Answers will var.,,7, (a), is translated to,,., is translated to,,., is translated to,,.. (a) Number of ear infections 7 Medium clinic Large clinic Small clinic Number of doctors The larger the clinic, the more patients a doctor can handle. Section. Graphs of Equations. (a) This is not a solution point since. This is a solution point since. (c) This is a solution point since.. (a) This is a solution point since. This is not a solution point since (c) This is not a solution point since (a) This is not a solution point since 8. This is a solution point since. (c) This is a solution point since. 7. The graph of is a straight line with -intercept at,. Thus, it matches (e). 9. The graph of is a parabola opening up with verte at,. Thus, it matches (c).

6 Section. Graphs of Equations 9. The graph of has a -intercept at, and has -intercepts at, and,. Thus, it matches (a).. To find the -intercept, let to obtain. The -intercept occurs at,. To find the -intercepts, let to obtain. Thus, the -intercept is,. To find the -intercept, let to obtain,. Thus, the -intercepts are, and,.. Thus, the -intercept is,. 7. The -intercept occurs at,. To find the -intercepts, let to obtain 9, ±. Thus, the -intercepts are,,,, and,. 9. The -intercept occurs at,. The -intercept occurs. Let. Then. when the numerator equals zero and the denominator does not equal zero. Thus,, is the onl -intercept. Let. Then. The -intercept and -intercept both occur at,.. is a line with intercepts, and,. 7. ( ), (, ) (, ) (, ) (, ) Intercepts:,, ±,

7 Chapter Functions, Graphs, and Limits (, ) (, ) (, ) (, ) Intercepts:,,, Intercepts:, and,. has intercepts, and, (, ) (, ) (, ) (, ) Intercepts:,,,. 7. (, ), (, ) (, ).. The graph is a parabola with verte at, and intercepts at,,,, and,. ± ± ±

8 ) ) Section. Graphs of Equations. Since the point, lies on the circle, the radius must be. the distance between, and,. Radius Center midpoint, Radius distance from center to an endpoint Center:, Radius: 9 9 Center:, Radius: 8, ), ) 7. Center: Radius:,. 7 Center: Radius: 9, 7.,..,.. Solving for in the equation ields, and solving for in the equation ields. Then setting these two -values equal to each other, we have. The corresponding -value is, so the point of intersection is,. 7. Solving for in the second equation ields and substituting this into the first equation gives 7 8,. The corresponding -values are and, so the points of intersection are, and,.

9 Chapter Functions, Graphs, and Limits 9. B equating the -values for the two equations, we have, ±. The corresponding -values are,, and, so the points of intersection are,,,, and,.. B equating the -values for the two equations, we have, ±. The corresponding -values are,, and, so the points of intersection are,,,, and,.. (a) C.8,. R C R 9...8, C R.7,.8, 9., 7. units,,.7, units (c) P R C 9..8,, 7.,. Hence, units ,,,, R C 9., 9 units p 8 p units, in thousands, 7. (a) Model:.77t.99t t correponds to 99 t Model Eact The model is a good fit. Answers will var. For, t and $,7 million.

10 Section. Lines in the Plane and Slope t 7. Model:.t t 7 corresponds to 997 (a) Year Salar Answers will var. (c) For, t and $ Yes, this prediction seems reasonable Intercept:, The greater the value of c, the steeper the line. 79. Intercepts:.78,,.8,,, Intercepts:,,.7 Section. Lines in the Plane and Slope. The slope is m since the line rises one unit verticall. The slope is m since the line is horizontal. for each unit of horizontal change from left to right.. The points are plotted in the accompaning graph and the slope is m. 7. The points are plotted in the accompaning graph and the slope is m. Thus, the line is horizontal. (, ) (, ) (, ) (, )

11 Chapter Functions, Graphs, and Limits 9. The points are plotted in the accompaning graph. The slope is undefined since m (undefined slope) 8 8. Thus, the line is vertical. ( 8, ) ( 8, ). The points are plotted in the accompaning graph and the slope is m. (, ) (, ) 8. The points are plotted in the accompaning graph and the slope is m The points are plotted in the accompaning graph and the slope is m 8. ( ) 8, ( ), (, ) ( ), 7. The equation of this horizontal line is. Therefore, three additional points are,,,, and,. 9. The equation of this line is 8. Therefore, three additional points are and,.,, 9,,. The equation of the line is 7. Therefore, three additional points are,,,, and,.. The equation of this vertical line is 8. Therefore,. three additional points are 8,, 8,, and 8,. Therefore, the slope is m and the -intercept is, Therefore, the slope is m 7 and the -intercept is,. Therefore, the slope is and the -intercept is,.. Since the line is vertical, the slope is undefined and there is no -intercept.. Therefore, the slope is and the -intercept is,.

12 Section. Lines in the Plane and Slope. The slope of the line is 7. The slope of the line is m. m. Using the point-slope form, we have Using the point-slope form, we have. (, ). (, ) (, ) (, ) 9. The slope of the line is undefined, so the line is vertical and its equation is. (, ) (, ). The slope of the line is m, so the line is horizontal and its equation is. (, ) (, ). The slope of the line is m Using the point-slope form, we have. The slope of the line is m 8. Using the point-slope form, we have ( ), ( ), 8 (, ) 8 8. (, 8 ) 8 7. Using the slope-intercept form, we have (, ).

13 Chapter Functions, Graphs, and Limits 9. Since the slope is undefined, the line is vertical and its equation is.. Since the slope is, the line is horizontal and its equation is 7. 8 (, ) (, 7). Using the point-slope form, we have.. Using the slope-intercept form, we have ( ), (, ) 7. The slope of the line joining, and, is. The slope of the line joining, and, is. Since the slopes are different, the points are not collinear. d. d 9. d 9 Since d d d, the points are not collinear. 9. Since the line is vertical, it has an undefined slope and its equation is.. Slope and -intercept at :.. Given line: 7, m (a) Parallel; m Perpendicular; m (, ) + = 7 9 9

14 Section. Lines in the Plane and Slope 7. Given line: 7, m (a) Parallel; m 7 8 Perpendicular; m ( ) 7, 8 + = Given line: is horizontal and m (a) Parallel: or the -ais (, ) Perpendicular: or + = 9. Given line: is vertical (a) Parallel: m is undefined, Perpendicular: m, (, ) = 7. is a horizontal line has slope and -intercept,.

15 8 Chapter Functions, Graphs, and Limits 79. (a) 997, 8,, t 7 9.t. The slope m 9. tells ou that the population is increasing 9. thousand per ear. For 999, t 9 and 98.8 thousand (,98,8). (c) For, t and 7. thousand (,7,). (d) 999: 97 thousand : thousand The estimates were close to the eact values. (e) The model could possible be used to predict the population in if the population continues to grow at the same linear rate.,,, F or C F.8C 9 C C 9 F ) 8. C. 8. (a) (c),,, C C 8 A or A 8 C C 9 F 8 A F 8 7A C 8 A A A F 7A A A 7 (d) C ; F (e) A 8. F 7 A 9 or A 7 F (a) The equipment depreciates $ per ear, so the value is t, where t. (c) When t, the value is $.. (d) The value is $ when t.7 ears (a) The slope is Hence, (a) Y 97 If (c) If p 8 p. p, p 9, 9 9 units. units. Y 7t 878 t corresponds to 99 For 999, t 9 and Y $78 billion. (c) For, t and Y $9 billion. (d) 999: $778. billion : $899. billion t 7 7

16 Section. Functions 9 9.,, 9. 7,.77 units or.7 units if fractional units are allowed., C 8,7,, 8, units C 7, 89, , units, C,, 8,. units,. C,.,.,, units,, Section. Functions. ± is not a function of since there are two values of for some.. isa function of since there is onl one value of for each.. is a function of since there is onl one value of for each. 7. ± is not a function of since there are two values of for some.

17 Chapter Functions, Graphs, and Limits 9. Domain:, Range:.,. Domain:,, Range:,. Domain:, Range:, 7. Domain:,, 7. Domain:, 9. Domain:, Range:,, Range:, Range:,. f. (a) f f 9 (c) f (d) f g (a) (c) (d) g g g g g. f f f, 7. g g,

18 Section. Functions 9. f f,. is not a function of.. isa function of.. (a) f g f g f (c) g (d) f g f (e) g f g 7. (a) f g f g f (c) g (d) f g f (e) g f g 9. (a) (c) (d) f g f g f g, f g f. (e) g f g,. f, g (a) f g f f g f g g (c) g f g (d) f g f (e) f g f (f) g f g,. The data fits the function g with c.. The data fits the function (d) r with c. 7. f g f 9. g f g f f g f g f g9 9 9, g 9 f g 9

19 Chapter Functions, Graphs, and Limits. f. f f f f f f f. f 9, f, f f 9, f f f = f 9. f 7 is one-to-one f. f is not one-to-one since f f. f f is not one-to-one since f f.

20 Section. Functions. (a) (c) (d) (e) (f) 7. (a) Shifted three units to the left: Shifted three units upward: (c) Shifted three units to the right, si units upward, and reflected: (d) Shifted si units to the left, three units downward, and reflected: 9. Total sales R R R 8 8 8t.8t.78t 7 7.t.8t, t,,,,,, 7. (a)..7 p.7p..7 p.p.7 p p 7 p.7 7. units 7. C 7 7 t t Ct Ct 8t 7 This is the weekl cost per t hours of production.

21 Chapter Functions, Graphs, and Limits 7. (a) If, then p 9. If <, 77. r 8.n 8, n 8 then p If (a) Revenue R rn 8.n 8n >, then p 7. Thus, p 9, n 9 9., < R , >. P p 9, P 9., 7,,.,, < > < > (c) The revenue begins to decrease for n >. 79. f 9 8. gt t t 9 9 Zeros: 9, f is not one-to-one. 9 Zero: t The function is one-to-one. 8. f 8. g Zeros: ± The function is not one-to-one. Domain: Zeros: ± g is not one-to-one. 87. Answers will var. Section. Limits f

22 Section. Limits f... undefined f undefined f undefined. 9. (a) f f. (a) g g (, ) (, ) (, ) (, ). (a) f g c c 9 f g c f c g c 9 7 f f (c) c g c g 9 c f c g. f c (a) f c f 8 c (c) c f 7. (a) f f (c) f (, ) 9. (a) f f (c) f (, )

23 Chapter Functions, Graphs, and Limits. (a) f f (c) f does not eist. (, ). (, ) does not eist.. 9. t t t t t t t t t t t Therefore, t does not eist Therefore, f f...

24 Section. Limits 7 7. t t t t t t t t t tt t t t t t t t tt t t t t t t t t f Because.7 f...,. undefined decreases without bound as tends to from the left, the it does not eist f, undefined Because f decreases without bound as tends to from the left, the it does not eist.. f f does not eist

25 8 Chapter Functions, Graphs, and Limits 7. (a) f undefined (c) Domain:,, Range:, e e, 9. C,p, p < 7. (a) p (a) If p, C, $,. If C,,p then p, p p p p 8 8%. (c) C p The cost function increases without bound as p approaches %. For., A.. For, A 8.9. (c). e $9. Continuous compounding Section. Continuit. The polnomial f is continuous on the entire real line.. The rational function f is not continuous on the entire real line. It is continuous at all ±.. The rational function f is continuous on the entire real line. 7. f is not continuous on the entire real line. f is not defined at,. 9. g is not continuous on the entire real line. g is not defined at ±.. f is continuous on, and,.. f is continuous on, and,.. f is continuous on,. 7. f f is not defined at ±. Continuous on the intervals,,,, and,. 9. f is continuous on,.. f is continuous on,,, and,. 9

26 Section. Continuit 9. f is continuous on all intervals of the form. f c, c, where c is an integer. That is, f is continuous on...,,,,,,,.... f f is not continuous at all points c, where c is an integer. f is continuous on,. 7. f and f. Hence 9. f is continuous on, and,. Since f does not eist, f is continuous on, and,... c, c c is an integer. c, c c is an integer. h f g f, > Thus, h is continuous on its entire domain,. Since does not eist, f is continuous on all c intervals c, c.. Since f is a polnomial, it is continuous on,. 7. Since f has a nonremovable discontinuit at on the closed interval,., 9. f, Removable discontinuit at Continuous on, and, 8. f Removable discontinuit at Continuous on, and,,. f,, < Nonremovable discontinuit at Continuous on,,,

27 Chapter Functions, Graphs, and Limits. f 8 f a a Therefore, 8 a and a. 7. h 9. f,, > h is not continuous at and. f is not continuous at. 7.. f is continuous on,. f is not continuous at all integers c.. f is continuous on all intervals of the form c where c is an integer., c, 7. f appears to be continuous on,. But it is not continuous at (removable discontinuit). 9. A 7. t, t (a) A,,,, 9, 8, 7, 8 t The graph has nonremovable discontinuities at t,,,,,... (ever months). For t 7, A 7. 7 $, c , 9.8., >, not an integer >, an integer c is not continuous at,,,...

28 Review Eercises for Chapter. (a) Ct.,..t,..t, C is not continuous at t,,,.... C9..9 $. < t t >, t is not an integer. t >, t is an integer. C t. (a) Nonremovable discontinuities at t,,, 8,... N when t,,, 8,..., so the inventor is replenished ever two months. 7. There are nonremovable discontinuities at t,,,,, and. N Rabbit population 8 Time (in months) t Review Eercises for Chapter. Matches (a). Matches. Distance 7. 9 Distance Midpoint 9, 7,. Midpoint, 8 8,. P R C. The tallest bars represent revenues. The middle bars represent costs. The bars on the left of each group represent profit, because P R C.. The translated vertices are,, 7,,, 8, and, 8,. 7. Bar graph for data: Mtilus 7 Gammarus 78 Littorina Nassarius Arbacia Ma 8

29 Chapter Functions, Graphs, and Limits , -intercept,,, -intercepts r 7 r 9 r 7 r Center:, Radius:, 8 (, ) Points of intersection:,,, 8 7. Points of intersection:,,,

30 Review Eercises for Chapter 9. (a) C. C R R , or 8 units... Slope: (horizontal line) Slope: -intercept:, -intercept:, Slope: -intercept:, 7. Slope Slope 7.. (a) 7 8., 7, 7, ; slope (c) The line through, and, has slope m (a) p p 7 If p., (c) If p.,. 7.. (d) Slope of perpendicular is.

31 Chapter Functions, Graphs, and Limits 7. Yes 9. No. f (a) f 7 f 7 (c) f. f. f 7. f Domain:, Domain:, Domain:, Range:, Range:, Range:, 9. (a) (c) f g f g f g 7. f has an inverse b the horizontal line test. (d) (e) f g f f g f (f) g f g 7. f does not have an inverse b the horizontal line test t 79. t t t 8. t t does not eist. 89. t t t t t t t t t t t t t t t t t

32 Review Eercises for Chapter f The statement is false since The statement is false since is undefined. 99. The statement is false since f.. f is continuous on the intervals. f is continuous on the intervals, and,., and,.. f is continuous on all intervals of the form c, c, where c is an integer. 7. f is continuous on the intervals, and,. 9. f f a 8 a 8 Thus, a 8 and a.. (a) 7. (a) Graphing utilit graph of A not continuous at n, n a positive integer. Graph as shown below. If, A $... t Debt Model t Debt Model t Debt Model (c) For 8, t 8 and D $,7. billion

33 Chapter Functions, Graphs, and Limits Practice Test for Chapter. Find the distance between, 7 and,.. Find the midpoint of the line segment joining, and,.. Determine whether the points,,,, and, are collinear.. Find so that the distance between, and, is 7.. Sketch the graph of.. Sketch the graph of. 7. Sketch the graph of. 8. Write the equation of the circle in standard form and sketch its graph Find the points of intersection of the graphs of and.. Find the general equation of the line passing through the points 7, and,.. Find the general equation of the line passing through the point, with a slope of m.. Find the general equation of the line passing through the point, 8 with undefined slope.. Find the general equation of the line passing through the point, and perpendicular to the line given b 7.. Given f, find the following. (a) f f (c) f (d) f. Find the domain and range of f.. Given f and g, find the following. (a) f g g f 7. Given f, find f. 8. Find. 9. Find.

34 Practice Test for Chapter 7. Find.. Find.. Find where f, f,, >.. Find the discontinuities of f 8 Which are removable?.. Find the discontinuities of f Which are removable?.. Sketch the graph of f. Graphing Calculator Required. Solve the equation for and graph the resulting two equations on the same set of coordinate aes. 7. Use a graphing calculator to graph f 9 and find f. Is the graph displaed correctl at?