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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. Solutions to Odd-Numbered Eercises The Cartesian Plane and the Distance Formula. (a) a. (a) a 0 b b c 0 0 c 7 09 (b) a b 9 c (b) a b c - (0, 0) c a (, ) b (, 0) (-, ) c a (7, ) b (7, ). (a) (b) d, ), ), ) ) ) ) 0 (c) Midpoint,, 7. (a) (b) d,,, 9 0 (c) Midpoint,, 9. (a) (b) d 0 (, ) (, 8) (, ) 8 7 (c) Midpoint,, 8

3 ) ) Chapter Functions, Graphs, and Limits. (a). d 0 7 (, ) d 0 0 (, ) 0, d 7 Since d d d, the triangle is a right triangle., 7) (b) d 8 d d (c) Midpoint, 0, 0, ) ) d, ) ). d d d d 0 0 d d 0 d 0 Since d d d, the points are collinear. (, ) d d (0, 0) (, ) d d (, ) ( 0, ) (, ) (,0) d d d 9. d 8. d d Since d d d, the points are not collinear., ) ) d , d d d ), ), ). d ±

4 Section. The Cartesian Plane and the Distance Formula. 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. 7. (a), (b),,, 7, 7, 0, 0, 0, 0, 9 0, 0, 9. (a),êê > feet (b) A square feet (a) 800 (b) 700 (c) 800 (d). (a) billion (b) billion (c) 0 billion (d) 8 billion 7. (a) Revenue midpoint 99 99,07 8,, 99,,80 Profit midpoint , 99, 87.9 (b) Actual revenue in 99:,7 million Actual profit in 99: million (c) Yes, the pattern seems linear. (d) Epenses for 99: $,90. million; 99: $,8 million; 99: $,0 million (e) Answers var.

5 Chapter Functions, Graphs, and Limits 9. (a) 0, 0 is translated to 0, 0,., is translated to,,., is translated to,,. (b). (a) Number of ear infections Medium clinic 70 0 Large clinic Small clinic 0 Number of doctors (b) The larger the clinic, the more patients a doctor can handle. Section. Graphs of Equations. (a) This is not a solution point since 0. (b) This is a solution point since 0. (c) This is a solution point since 0.. (a) This is not a solution point since (b) This is a solution point since 0. (c) This is not a solution point since The graph of is a straight line with -intercept at 0,. Thus, it matches (e). 7. The graph of is a parabola opening up with verte at,. Thus, it matches (c). 9. The graph of has a -intercept at 0, and has -intercepts at, 0 and, 0. Thus, it matches (a).. To find the -intercept, let 0 to obtain. The -intercept occurs at 0,. To find the -intercepts, let 0 to obtain Thus, the -intercept is 0,. To find the -intercept, let 0 to obtain,. 0 0 Thus, the -intercepts are, 0 and, 0.. Thus, the -intercept is, 0.. The -intercept occurs at 0, 0. To find the -intercepts, 7. The -intercept occurs at 0,. The -intercept occurs let 0 to obtain when the numerator equals zero and the denominator does not equal zero. Thus,, 0 is the onl -intercept , ±. Thus, the -intercepts are 0, 0,, 0, and, Let 0. Then 0 Þ 0. Let 0. Then 0 Þ 0. The -intercept and -intercept both occur at 0, 0.

6 Section. Graphs of Equations 7. The graph of is a straight line with intercepts at 0, and, The graph of is a straight line with intercepts at and 0,., (0, ) (, ( (-, 0) (0, -). 7. (0, ) (, 0 ) (0, ) (, 0) Intercepts:, 0 and 0, The graph is a parabola with verte at, 0 and intercepts at, 0 and 0, (0, 0) , Intercept: 0, (0, ) 0 ± ± ± 0 (, 0) ( 0, ) The graph is a parabola with verte at, 0 and intercepts at, 0, 0,, and 0,.

7 ) 8 Chapter Functions, Graphs, and Limits Since the point 0, 0 lies on the circle, the radius must be. the distance between 0, 0 and,. Radius Center midpoint 0, Radius distance from center to an endpoint Center:, Radius: 9 9 Center:, Radius: 0 8, ) ), ) 7 7. Center: Radius:, 9. 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,.. Solving for in the second equation ields 0 and substituting this into the first equation gives ,. The corresponding -values are and 0, so the points of intersection are, and, 0.

8 Section. Graphs of Equations 9. B equating the -values for the two equations, we have 0 0 0, ±. The corresponding -values are 0,, and, so the points of intersection are 0, 0,,, and,. 7. B equating the -values for the two equations, we have 0 0 0, ±. The corresponding -values are, 0, and 0, so the points of intersection are, 0, 0,, and, (a) C (b) B equating R and C, we have (c) R 9. R C units. Profit Revenue Cost units.. 0.8, ,000 0,000 R C., ,000 units , ,000,000,000 R C 0, units 0 00, (a) The greater the value of c, the steeper the line. Farm work force (percent of total work force) Year (990 «0) (b) The model predicts.9% for the ear 000 t 00. (c) The model is not valid. 9. If C and R represent the cost and revenue for a business, the break-even point is that value of for which C R. For eample, if C 00,000 0 and R 0, then the break-even point is 0,000 units. 7. Intercept: 0, Intercepts:.780, 0,.8, 0, 0,

9 0 Chapter Functions, Graphs, and Limits Section. Lines in the Plane and Slope. The slope is m since the line rises one unit verticall. The slope is m 0 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 0. Thus, the line is horizontal., (, ) (, ) 9. The points are plotted in the accompaning graph. The slope is undefined since m (undefined slope) 8 8. Thus, the line is vertical.. The points are plotted in the accompaning graph and the slope is m 0. Thus, the line is horizontal. (, ) (, ) 8 ) ) 8, ) 8, ). The equation of this horizontal line is. Therefore, three additional points are 0,,,, and,.. The equation of this line is 8. Therefore, three additional points are 0, 8,,, and,. 7. The equation of the line is 7 0. Therefore, three additional points are 0, 0,,, and,. 9. The equation of this vertical line is 8. Therefore, three additional points are 8, 0, 8,, and 8,.. 0. Therefore, the slope is m and the -intercept is 0,. 7 7 Therefore, the slope is m 7 and the -intercept is 0,.. Since the line is vertical, the slope is undefined and there is no -intercept.

10 Section. Lines in the Plane and Slope 7. The slope of the line is m 0 Using the point-slope form, we have 0.. 0, ) ), ) ) 9. The slope of the line is m 0 0. (, ) Using the point-slope form, we have 0. (0, 0). The slope of the line is undefined, so the line is vertical and its equation is 0. (,). The slope of the line is m 0, so the line is horizontal and its equation is 0. (, ) (, ) (, ). Using the slope-intercept form, we have 7. Using the slope-intercept form, we have (0, ) - (0, 0) Using the slope-intercept form, we have Using the slope-intercept form, we have 0. (-, 7) (0, ) Using the slope-intercept form, we have ( ) 0, -

11 Chapter Functions, Graphs, and Limits. The slope of the line joining, and, 0 is 0. The slope of the line joining, 0 and, is 0. Since the slopes are different, the points are not collinear. d 0. d d 9 Since d d d, the points are collinear. 7. Since the line is vertical, it has an undefined slope and its equation is Given line: 7,Ê m 0 (a) Parallel; m (b) Perpendicular; m + = (-, ) -. Given line: 7,Ê m (a) Parallel; m (b) Perpendicular; m + = 7 0 (-, ) Given line: is horizontal and m 0 (a) Parallel: - 0 or the -ais (-, 0) (b) Perpendicular: or 0 = - -. Given line: (a) Parallel; m (b) Perpendicular; m (, ) = - -

12 Section. Lines in the Plane and Slope 7. is a horizontal line has slope and -intercept 0,.. (a) 78 t 0 0.t The slope m. tells ou that the population increases. thousand per ear. (b) If t, thousand. (c) If t,. 0. thousand. (d) 99: 07. thousand 99: 00.9 thousand The estimates are good. (e) The model could possibl be used to predict the population in 000 t 0 if the population continues to grow at the same linear rate. 7. Using F 9 ou have C C, 9 F. (a) If F 0., then C (b) If F 8, then C Since 0 < 9.8, gallium is solid.

13 Chapter Functions, Graphs, and Limits 9. (a) W (union plan) W.. (b) (corporation plan) W. Þ.87,. (c) The point of intersection indicates the number of units.87 a worker needs to produce for the two plans to be equivalent. If a worker produces more than units, the corporation plan pas better Use the points 0, 8,000 and, 7, (a) The slope is Hence, 80. 8,000 7,000 8,000 t p 80 8,000 0,000t 0,000t 8,000,Ê Ê0 ² t ² (b) If (c) If p. p, p 9, units. 9 9 units. 7. (a) Y 79 Y t 79 (b) If t,ê Y 79 billion. (c) If t,ê Y 79 billion. (d) 99:. billion 99: 9. billion 0 79 t ,00 00 ² 00,000 0, ² 7,00 ².77 ² units or.7 units if fractional units are allowed C 8,7 0 ² 00, C 7,00 89 ² 00,000 0 ² 8, 89 ²,00 ² ² 7.8 ² 70 units ² 7 units 0,000 0,

14 Section. Functions Section. Functions. (a) f 0 0 (b) f 9 (c) f (d) f. (a) g (b) (c) g (d) g g g. f f,ê Ê 0 7. f f,ê Ê 0 9. ±. 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. ± is not a function of since there are two values of for some. 7. Domain:, 9. Domain:, 0 0, Range: 0, Range:,. Domain:,, Range:,,

15 Chapter Functions, Graphs, and Limits. Domain:, Range:,. Domain:, Range:, 7. is not a function of. 9. isa function of.. (a) f g (b) f g f (c) g (d) f g f (e) g f g. (a) f g,ê Ê ² (b) f g,ê Ê ² f (c) g,êê < (d) f g f,ê Ê ² (e) g f is not defined since the domain of g is, and the range of f is,. The range of f is not in the domain of g.. f g f 7. g f g f f g f g f g9 9 9,Ê Ê ³ 0 9 f g g 9 9. f. f f f f f f f. f 9,Ê Ê0 ² ² 9 9 f f 9 9 f 9,ÊÊ0 ² ²

16 Section. Functions 7. f 7 f,ê Ê ³ 0 7. is one-to-one. 7 f f f f. f is not one-to-one since f f. f f is not one-to-one since f 0 f f,ê Êg (a) (c) f g f f 0 0 g f 0 g0 (b) g f g g 0 (e) f g f (f) g f g,ê Ê ³ 0 (d) f g f. The data fits the function (b) g with c. 7. The data fits the function (d) r with c. 9. (a) (b) (c) (d) (e) (f)

17 8 Chapter Functions, Graphs, and Limits. (a) Shifted three units to the left: (b) 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:. (a) (b) C C (c) C <.9 < < < since > Must sell 809 units before the average cost per unit falls below the selling price. Cost Cost on land Cost underwater (a) If 0 ² ² 00, then p 90. If 00 < ² 00, (b) then p If > 00, then p 7. Thus, p 90, 9 0.0, 7, 0 ² ² < ² 00 > 00. P p , P , 7 0, 0, 0.0,, 0 ² ² < ² 00 > 00 0 ² ² < ² 00 > r 8 0.0n 80,Ê Ên ³ (a) Revenue R rn 8 0.0n 80n (b) n R f 9 Zeros: 0, ± The function is not one-to-one. (c) The revenue begins to decrease for n > gt t t 7. f Zero: t The function is one-to-one. 9 9 Zeros: ± The function is not one-to-one Answers will var.

18 Section. Limits 9 Section. Limits f f undefined f undefined f undefined 0 9. (a) f (b) f. (a) g (b) 0 0 g (-, ) (-, ) (0, ) (0, ) (a) f g c c f c g. (a) f (b) 9 f g c f c g c 9 7 f f (c) c g c g 9 c (b) f (c) f (, ) - - -

19 0 Chapter Functions, Graphs, and Limits 7. (a) (b) f 0 9. (a) f 0 (c) f 0 (, ) f (b) f (c) does not eist. - (, ) - (, -) does not eist t t t 0 0 Therefore, t 0 t t t t t 0 does not eist.. f. f Therefore, t 0 t t t t t t t t tt t t t t t t 0 t tt t t t 0 t t t t 0 t

20 Section. Limits f Because.7 f ,000. undefined decreases without bound as tends to from the left, the it does not eist f ,000 undefined Because f decreases without bound as tends to from the left, the it does not eist.. C,000p,ÊÊ0 ² p < p (a) If p 0, C, (b) If C 00,000,000p then 00 p, 00 p p 00 p p 80 Þ 80%. $,000. (c) C p 00 The cost function increases without bound as p approaches 00%.. (a) f undefined (b) (c) Domain:, 0 0, Range:, e e, - 0 -

21 Chapter Functions, Graphs, and Limits 7. f f 0 does not eist (a) 00 (b) For 0., A.. For, A 8.9. (c) e $9. Continuous compounding 0 00 Section. Continuit. The polnomial f is continuous on. The rational function f is not continuous on the entire real line. the entire real line. It is continuous at all ±.. f is continuous on,. 7. f is continuous on,,. 9. f is continuous on,.. f is continuous on,,.. f is continuous on,.. f is continuous on 9 0,,,. 7. f is continuous on,,. 9.. f Since f. f, f is continuous on the entire real line. f f f is continuous on,. Since f does not eist, f is continuous on,,.

22 Section. Continuit 7.. c, c c is an integer. c, c c is an integer. h f g f,êê > Thus, h is continuous on its entire domain,. Since does not eist, f is continuous on all c intervals c, c. 9. Since f is a polnomial, it is. Since f has a nonremovable discontinuit at on the closed interval,, continuous on,... f,ê Ê. Removable discontinuit at Continuous on,, f Removable discontinuit at 0 Continuous on, 0 0,,Ê Ê f,, Nonremovable discontinuit at 0 Continuous on, 0, 0, < 0 ³ 0 9. f 8 f a a Therefore, 8 a and a.. h h is not continuous at and.. f is continuous on,.. f is continuous on all intervals of the form c where c is an integer., c,

23 Chapter Functions, Graphs, and Limits 7. f appears to be continuous on,. But it is not continuous at 0 (removable discontinuit). 9. A t,êêt ³ 0 (a) A,000,000,000 0,000 9,000 8,000 7, t The graph has nonremovable discontinuities at t,,,,,... (ever months). (b) For t 7, A $, (a) Ct.0,.0 0.t,.0 0.t, C is not continuous at t,,,.... C 0 < t ² t > t, t is not an integer. t >, t is an integer.. Nonremovable discontinuities at t,,, 8,... N 0 when t,,, 8,..., so the inventor is replenished ever two months t (b) C $.. There are nonremovable discontinuities at t,,,,, and. N 0 8 t Review Eercises for Chapter. Population of Teas matches (a).. Number of U.S. Business Failures matches (b).. Distance Distance Midpoint 9, 7,. Midpoint 0, 8 8,

24 Review Eercises for Chapter. The taller bars in the back represent revenues. The middle bars represent costs. The smaller bars in front represent profits, since P R C.. The translated vertices are,, 7,,, 8, and, 8, intercept: 0 Þ Þ 0, 7. -intercept: 0 Þ Þ, r r r 9 r r Elevation: Center:, Radius:, ) ) 8. Þ. Þ Point of intersection:, 0 0 Points of intersection: 0, 0,,,,. (a) C (b) R C R Slope: -intercept: 0, 0 shirts

25 Chapter Functions, Graphs, and Limits Slope: 0 (horizontal line) -intercept: 0, Slope: -intercept: 0,. Slope Slope (a) 7 8 (b) Þ 7 ;ÊÊslope (c) The line through 0, 0 and, has slope (d) Þ. Þ Slope of perpendicular is.., 70, 7, 700 m Yes (a) 70 0p 0p 070 (b) If p.0,ê (c) If p.00,ê No 7. f (a) (b) f 7 f 7 (c) f 0 9. f. f Domain:, Domain:, Range: 0, Range:,

26 Review Eercises for Chapter 7. f Domain:, Range:,. (a) f g (b) f g (c) f g (d) f g (e) f g f (f) g f g 7. f has an inverse b the horizontal line test. 9. f does not have an inverse b the horizontal line test. f t 7. 0 t t 77. t t 0 t t t 0 t t t 0 t does not eist does not eist f

27 8 Chapter Functions, Graphs, and Limits 89. The statement is false since The statement is false since is undefined The statement is false since f f is continuous on the intervals,,. 97. f is continuous on the intervals,,. 99. f is continuous on all intervals of the form c, c, where c is an integer. 0. f is continuous on the intervals, 0 0,. 0. f f a 8 a 8 Thus, a 8 and a. 0. (a) (b) t Debt Model (c) In 999, t 9 and D 800. billion. 07. Yellow sweet maize: White flint maize: Intercepts: 0,,, 0 Intercepts: 0, 0,., 0 0 Line: 0 0 Line:

28 Practice Test for Chapter 9 Practice Test for Chapter. Find the distance between, 7 and,.. Find the midpoint of the line segment joining 0, and,.. Determine whether the points 0,,,, and, are collinear.. Find so that the distance between 0, 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 0, and perpendicular to the line given b 7.. Given f, find the following. (a) f (b) f (c) f (d) f. Find the domain and range of f.. Given f and g, find the following. (a) f g (b) g f 7. Given f, find f. 8. Find. 9. Find.

29 0 Chapter Functions, Graphs, and Limits 0. Find.. Find. 0. Find where f, f,, ² >.. 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 Use a graphing calculator to graph f 9 and find f. Is the graph displaed correctl at?

ODD03.PDF

ODD03.PDF 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