CHAPTER P Preparation for Calculus

Transcription

1 CHAPTER P Preparation for Calculus Section P. Graphs and Models Section P. Linear Models and Rates of Change Section P. Functions and Their Graphs Section P. Fitting Models to Data Review Eercises Problem Solving

2 CHAPTER P Preparation for Calculus Section P. Graphs and Models.. 9 -intercept:, -intercepts:,,, -intercept:, -intercept:, Matches graph. Matches graph (d)... -intercepts:,,, -intercepts:,,,,, -intercept:, -intercept:, Matches graph (a). Matches graph (, 7) (, ) (, ) 8 8 (, ) 8 (, ) (, ) 8 (, ) (, ) (, ) (, )

3 ) Section P. Graphs and Models 9.. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) 8 (9, ) (, ) (, ) 8.. Undef. Undef. ), ) (, ) (, ), ) (, ) (, ) ), ) ), ) (, ), ), ) (, ) ) ). Xmin = - Xma = Xscl = Ymin = - Yma = Yscl =. Xmin = - Xma = Xscl = Ymin = - Yma = Yscl = Note that when. Note that when or.

4 Chapter P Preparation for Calculus 7. (., ) (,.7) (a),,.7,,.7 (a).,.,.7,., and,, 9.. -intercept: ;, -intercept: ;, -intercepts: -intercepts:, ;,,,, ±;,, ±,.. -intercept: ;, -intercept: ;, -intercepts: -intercept: ;,, ±;, ; ±,.. -intercept: None. cannot equal. -intercept: ;, -intercept: -intercepts: ;,, ;,,,. -intercept: ;, -intercept: ;,

5 Section P. Graphs and Models. -intercept: ;, 7. Smmetric with respect to the -ais since. -intercept: Note: ± ;, is an etraneous solution. 8. No smmetr with respect to either ais or the origin. 9. Smmetric with respect to the -ais since.. Smmetric with respect to the origin since. Smmetric with respect to the origin since... Smmetric with respect to the -ais since.. No smmetr with respect to either ais or the origin.. Smmetric with respect to the origin since.. Smmetric with respect to the origin since.. is smmetric with respect to the -ais since. 7. is smmetric with respect to the -ais since. 8. is smmetric with respect to the -ais 9. since. Intercepts:,,, Smmetr: none (, ),

6 Chapter P Preparation for Calculus... Intercepts:,,, Intercepts: 8,,, Intercepts:,,, Smmetr: none Smmetr: none Smmetr: none (, ) (, ) 8 (, ) (8, ) 8 (, ) (, )... Intercepts: Intercept:, Intercepts:,,,,, Smmetr: -ais,,, 9 Smmetr: -ais Smmetr: none (, ) (, ) (, ) 9 (, ) (, 9) 8 8 (, ) Intercepts:,,, Intercepts:,,, Intercepts:,,,,, Smmetr: none Smmetr: none Smmetr: origin ( ), (, ) (, ) (, ) (, ) (, ) (, ) 9.. Intercepts:,,, Smmetr: none Domain: (, ) (, ) 9 Intercepts:,,,,, Smmetr: -ais Domain:, (, ) (,) (, )

7 Section P. Graphs and Models 7.. Intercept:, Smmetr: origin. Intercepts:,,,,, Smmetr: -ais Intercepts: none Smmetr: origin (, ) (, ) (, ) (, ). Intercept:, Smmetr: -ais.. Intercepts:,,,,, Smmetr: -ais Intercepts:,,, Smmetr: none (, ) 8 (, ) 8 (, ) (, ) (, ) (, ) ± 9 Intercepts:,,,, 9, Smmetr: -ais (, ) ( 9, ) (, ) ± Intercepts:,,,,,,, Smmetr: origin and both aes Domain:, (, ) (, ) (, ) (, ) 9. Intercepts: Smmetr: -ais,,,,, ± (, ) (, ) (, ) 8. 8 Intercept: 8 ± 8, Smmetr: -ais 8 (, ). The corresponding -value is. Point of intersection:,

8 8 Chapter P Preparation for Calculus.. 7 The corresponding -value is., The corresponding -values are (for ) and (for ). Points of intersection:,,, Point of intersection:,.. or The corresponding -values are and. Points of intersection:,,, or The corresponding -values are and. Points of intersection:,,, or The corresponding -values are and. Points of intersection:,,,,, or The corresponding -values are,, and. Points of intersection:,,,,, or The corresponding -values are and. Points of intersection:,,,,,,,,,, = + (, ) (, ) (, ) 8 = +

9 Section P. Graphs and Models ,,,,,,, = + (, ) (, ) (, ) = = + (, ) 7 = (, ) Points of intersection: Analticall,,,,,.,,,,.7 7. Points of intersection:, Analticall,,, 7 (, ) = (, ) 8 = - + or or. Hence,,,,. 7. (a) Using a graphing utilit, ou obtain.7t.8t.. 7. (a).t.t 7 For, t and 7. For, t and.. 7 acres. 7. C R.,.9..9,..8,8,,.8,8.,, Use the Quadratic Formula. units The other root, 99, does not satisf the equation R C. This problem can also be solved b using a graphing utilit and finding the intersection of the graphs of C and R.

10 Chapter P Preparation for Calculus 7.,77.7 If the diameter is doubled, the resistance is changed b approimatel a factor of. For instance,. and (other answers possible) 78. (other answers possible) 79. (i) k matches. Use, 7: 7 k k, thus,. (ii) k matches (d). Use, 9: 9 k k, thus,. (iii) k matches (a). Use, : k k, thus,. (iv) k matches. Use, : k k, thus,. 8. (a) If, is on the graph, then so is, b -ais smmetr. Since, is on the graph, then so is, b -ais smmetr. Hence, the graph is smmetric with respect to the origin. The converse is not true. For eample, has origin smmetr but is not smmetric with respect to either the -ais or the -ais. Assume that the graph has -ais and origin smmetr. If, is on the graph, so is, b -ais smmetr. Since, is on the graph, then so is,, b origin smmetr. Therefore, the graph is smmetric with respect to the -ais. The argument is similar for -ais and origin smmetr. 8. False; -ais smmetr means that if, is on the graph, then, is also on the graph. 8. True 8. True; the -intercepts are b ± b ac, a. 8. True; the -intercept is b a, Circle of radius and center, (, ) (, ) (, ) 8. Distance from the origin K Distance from, K, K K K K K K Note: This is the equation of a circle!

11 Section P. Linear Models and Rates of Change Section P. Linear Models and Rates of Change. m. m. m. m. m. m 7. m is undefined. 8. (, ) m = m = m = m = (, ) m = m = m = 9. m (, ) 7 (, ). m. m. m undefined (, ) (,) (, ) (, ) (, ) (, ). m (, ) (, ). m ( ) 7, 8 ( ),. Since the slope is, the line is horizontal and its equation is. Therefore, three additional points are,,,, and,.. Since the slope is undefined, the line is vertical and its equation is. Therefore, three additional points are,,,, and,. 7. The equation of this line is 7. Therefore, three additional points are and,.,,,, 8. The equation of this line is. Therefore, three additional points are and,.,,,,

12 Chapter P Preparation for Calculus 9. (a) Slope ft ft B the Pthagorean Theorem,. feet.. (a) m indicates that the revenues increase b in one da. m indicates that the revenues increase b in one da. m indicates that the revenues do not change from one da to the net.. (a). (a) r Population (in millions) Year ( 99) t 8 t The slopes of the line segments are: The population increased least rapidl between and. The slopes are: The rate changed most rapidl between and seconds. The change is. mphsec... Therefore, the slope is m and the -intercept is,. Therefore, the slope is m and the -intercept is,... The line is vertical. Therefore, the slope is undefined and there is no -intercept. The line is horizontal. Therefore, the slope is m and the -intercept is, (, ) (, )

13 Section P. Linear Models and Rates of Change (, ) (, ) (, ).. m. m (, ) 8 (, ) (, ) 8 8 (, ) 8 (,). m (, ) (, ). m (, ) (, ) m (, 8) 8 (, ) 8. m 9. m 8 Undefined 7 (, ) Vertical line (, ) (, 8) (, )

14 Chapter P Preparation for Calculus. m. m 7. m (, ) (, ) 7 (, ) (, ) 7 ( ) 7, 8 ( ),.. m b a (, ) b a b b b a a b (, b) ( a, ).. 7. a a a a a a 8. a 9. a a a. a a

15 Section P. Linear Models and Rates of Change... (, ) (a) The lines do not appear perpendicular. The lines appear perpendicular. The lines are perpendicular because their slopes and are negative reciprocals of each other. You must use a square setting in order for perpendicular lines to appear perpendicular. Answers depend on calculator used. 8. (a) The lines do not appear perpendicular. The lines appear perpendicular. The lines are perpendicular because their slopes and are negative reciprocals of each other. You must use a square setting in order for perpendicular lines to appear perpendicular. Answers depend on calculator used (a) m (a) 7 m

16 Chapter P Preparation for Calculus (a) m 7 8 (a) m The given line is vertical.. (a) (a). The slope is. V when t.. The slope is.. V when t. V t t V.t.t 8 7. The slope is. V, when t. 8. The slope is. V, when t. V t, t 8, V t, t 7, 9. (, ) 7., (, ) (, ) (, ) 9 9 You can use the graphing utilit to determine that the points of intersection are, and,. Analticall,,,. The slope of the line joining, and, is m. Hence, an equation of the line is. You can use the graphing utilit to determine that the points of intersection are, and,. Analticall,,,. The slope of the line joining, and, is m. Hence, an equation of the line is.

17 Section P. Linear Models and Rates of Change 7 7. m 7. m m m m m The points are not collinear. m m The points are not collinear. 7. Equations of perpendicular bisectors: c c a b c a b c Setting the right-hand sides of the two equations equal and solving for ields. Letting in either equation gives the point of intersection:, a b c c. a b ( b, c) b a This point lies on the third perpendicular bisector,. ( ) (, ) b a, c a b c ( a, ) ( a, ) 7. Equations of medians: c b Solving simultaneousl, the point of intersection is b, c. c a a b c a a b 7. Equations of altitudes: a b a c b a b a c Solving simultaneousl, the point of intersection is b, a b c. ( ) b a, c ( a, ) ( b, c) ( ) a + b, c (, ) ( a, ) ( a, ) ( b, c) ( a, ) 7. The slope of the line segment from to b, a b b is:, c c m a b c c b b a b c c b The slope of the line segment from to, a b c b is:, c c a b c bc m a b c c c b m m Therefore, the points are collinear. a b c c c b a b c bc

18 8 Chapter P Preparation for Calculus 77. Find the equation of the line through the points, 78. and,. m 8 9 F 9 C C. If 7, C.7 $9.8. or F 9C For F 7, C.. F 9 C C 9F 79. (a) W.7. W. 9. Using a graphing utilit, the point of intersection is Analticall,, 7. (, 7) Both jobs pa $7 per hour if units are produced. For someone who can produce more than units per hour, the second offer would pa more. For a worker who produces less than units per hour, the first offer pas more. 8. (a) Depreciation per ear: 87 $ where $ ears 8. (a) Two points are, 8 and 7,. The slope is 8. (a) quiz score, test score m 8 7. p 8 p 7 8 or p If 7, (d) The slope shows the average increase in eam score for each unit increase in quiz score. (e) The points would shift verticall upward units. The new regression line would have a -intercept greater than before: If p, units. If p 9, 9 9 units.

19 Section P. Linear Models and Rates of Change 9 8. The tangent line is perpendicular to the line joining the point, and the center,. 8. The tangent line is perpendicular to the line joining the point, and the center of the circle,,. (, ) (, ) 8 (, ) (, ) Slope of the line joining, and, is. The equation of the tangent line is 9. 9 Slope of the line joining, and, is. Tangent line: 8. d 8. d d 88. d A point on the line is,. The distance from the point, to is d. 9. A point on the line is,. The distance from the point, to is d If A, then B C is the horizontal line CB. The distance to, is d C B B C B A B C. A B If B, then A C is the vertical line CA. The distance to, is d C A A C A A B C. A B (Note that A and B cannot both be zero.) CONTINUED

20 Chapter P Preparation for Calculus 9. CONTINUED The slope of the line A B C is AB. The equation of the line through, perpendicular to A B C is: B A A A B B B A B A The point of intersection of these two lines is: A B C A AB AC B A B A B AB B AB A B AC B AB A B C AB B BC AC B AB A B B A B A AB A AB A A B BC AB A () () (B adding equations () and ()) () () (B adding equations () and ()) BC AB A A B AC B AB, BC AB A point of intersection A B A B The distance between, and this point gives us the distance between, and the line A B C. d AC B AB A B BC AB A A B AC AB A AC B A A B A B C A B A B A B C A B A B BC AB B A B BC A B A B 9. m m d A B C A B m m 9 9 m m The distance is when m. In this case, the line contains the point,. (, ) 8

21 Section P. Linear Models and Rates of Change 9. For simplicit, let the vertices of the rhombus be,, a,, b, c, and a b, c, as shown in the figure. The slopes of the diagonals are then ( b, c) ( a + b, c) m and m c c a b b a. Since the sides of the rhombus are equal, and we have a b c, (, ) ( a, ) m m c a b c b a c b a c c. Therefore, the diagonals are perpendicular. 9. For simplicit, let the vertices of the quadrilateral be,, a,, b, c, and d, e, as shown in the figure. The midpoints of the sides are a,, a b, c, b d, c e and, d, e. ( ) d, e ( d, e) ( ) b + d, c + e ( b, c) ( ) a + b, c The slope of the opposite sides are equal: (, ) ( ) a, ( a, ) c a b a c e e b d d c b e a d c a b c e b d e a d Therefore, the figure is a paralleogram. 9. Consider the figure below in which the four points are collinear. Since the triangles are similar, the result immediatel follows. 9. If m m, then m m. Let L be a line with slope m that is perpendicular to L. Then m m. Hence, m m L and L are parallel. Therefore, L and L are also perpendicular. (, ) (, ) ( *, * ) ( *, * ) 97. True. a b c a b c b m a b 98. False; if m is positive, then m m is negative. b a c b a c a m b a m m

22 Chapter P Preparation for Calculus Section P. Functions and Their Graphs. (a) Domain of f:. (a) Domain of Range of f: Domain of g: Range of g: f g f g for (d) f for (e) g for, and Range of g: f g f g for and (d) f for, (e) f: Range of f: Domain of g: g for. (a) f. (a) f f 9 f b b f 9 f, undefined (d) f (d) f. (a) g g g (d) gt t t t. (a) (d) g g 9 8 gc c c c c gt t t t t t 8t t 7. (a) f cos cos f f cos cos cos cos 8. (a) f sin f sin f sin 9. f f,. f f,. f f,. f f,

23 Section P. Functions and Their Graphs. h. Domain:, Range:, g. Domain:, Range:, t f t sec t k t k Domain: all t k, k an integer Range:,,. ht cot t 7. Domain: all t k, k an integer Range:, f Domain:,, Range:,, 8. g Domain:,, Range:,, 9. f. and and Domain: f. Domain: or Domain:,, g cos cos cos Domain: all n, n an integer. h sin. f. g sin Domain: all n, n, n integer sin Domain: all Domain: all ±. f, <,. f,, > (a) f (a) f f f f f (d) f t t t (d) f s s s 8s (Note: t for all t) (Note: s > for all s) Domain:, Domain:, Range:,, Range:,

24 Chapter P Preparation for Calculus 7. f (a), <, f 8. f,, > (a) f f f f f (d) f b b b (d) f Domain:, Domain:, Range:,, Range:, 9. f. g. h Domain:, Domain:,, Domain:, Range:, Range:,, Range:, 8. f. f 9. f Domain:, Domain:, Domain:, Range:, Range:, Range:,,.8 8 -intercept:, -intercept:, (, ( (, ). gt sin t. Domain:, Range:, t h cos Domain:, Range:, π π θ

25 Section P. Functions and Their Graphs 7. The student travels during the first mimin 8. minutes. The student is stationar for the following minutes. Finall, the student travels mimin during the final minutes. d t t t t 9. ±. is not a function of. Some vertical lines intersect the graph twice. is a function of. Vertical lines intersect the graph at most once.. is a function of. Vertical lines intersect the graph at most once.. ± is not a function of. Some vertical lines intersect the graph twice.. ±. is not a function of since there are two values of for some. is a function of since there is one value of for each.. ±. is not a function of since there are two values of for some. is a function of since there is one value of for each. 7. f is a horizontal shift 8. f is a vertical shift 9. f is a reflection in units to the left. Matches d. units downward. Matches b. the -ais, a reflection in the -ais, and a vertical shift downward units. Matches c.. f is a horizontal. f is a horizontal. f is a horizontal shift units to the right, followed b a reflection in the -ais. Matches a. shift to the left units, and a vertical shift upward units. Matches e. shift to the right unit, and a vertical shift upward units. Matches g.. (a) The graph is shifted units to the left. The graph is shifted unit to the right. The graph is shifted units upward. 8 CONTINUED

26 Chapter P Preparation for Calculus. CONTINUED (d) The graph is shifted units downward. (e) The graph is stretched verticall b a factor of. (f) The graph is stretched verticall b a factor of (a) g f g f g f g f g f Shift f right units 7 (, ) (, ) Shift f left units 7 (, ) (, ) Vertical shift upwards units (, ) (, ) (d) g f (e) g f (f) g f Vertical shift down unit g f g f g f g f (, ) (, ) (, ) ( ), ( ), (, ). (a) Vertical shift units upward Reflection about the -ais Horizontal shift units to the right. (a) h sin is a horizontal shift units to the left, followed b a vertical shift unit upwards. h sin is a horizontal shift unit to the right followed b a reflection about the -ais.

27 Section P. Functions and Their Graphs 7 7. (a) f g f g f g g f g (d) f g f (e) f g f (f ) g f g, 8. f sin, g (a) (d) (e) fg f sin fg f sin g f g g f gsin g fg f sin (f ) g f gsin sin 9. f, g. f g f g f, Domain:, g f g f g Domain:, No. Their domains are different. f g g f for. f, g cos f g fg f cos cos Domain:, g f g cos Domain:, No, f g g f.. f, g f g f g f Domain: all ± g f g f g 9 9 Domain: all No, f g g f.. f g f Domain:, g f g You can find the domain of g f b determining the intervals where and are both positive, or both negative Domain:,,,. (a) f g fg f g f g g f g, which is undefined (d) f g fg f (e) g f g f g (f ) fg f, which is undefined

28 8 Chapter P Preparation for Calculus. A rt Art A.t.t.t A rt represents the area of the circle at time t.. F Let h, g and f. Then, f g h fg f F. [Other answers possible]. F sin Let f, g sin and h. Then, f g h fg fsin sin F. [Other answers possible] 7. f f 8. Even f f Odd 9. f cos cos f Odd 7. f sin sin sin sin sin sin Even 7. (a) If f is even, then is on the graph., 7. (a) If f is even, then, 9 is on the graph. If f is odd, then, is on the graph. If f is odd, then, 9 is on the graph. 7. f is even because the graph is smmetric about the -ais. g is neither even nor odd. h is odd because the graph is smmetric about the origin. 7. (a) If f is even, then the graph is smmetric about the - ais. If f is odd, then the graph is smmetric about the origin. f f 7. Slope (, ) f, (, )

29 Section P. Functions and Their Graphs 9 7. Slope (, ) (, ) 77. f, f, 78. f, 79. Matches (ii). The function is g c. Since, 8. Matches (i). The function is f c. Since, satisfies the equation, c. Thus, g. satisfies the equation, c. Thus, f. 8. Matches (iv). The function is r c, since it must be 8. Matches (iii). The function is h c. Since, undefined at. Since, satisfies the equation, satisfies the equation, c. Thus, h. c. Thus, r. 8. (a) T, T If Ht Tt, then the program would turn on (and off) one hour later. If Ht Tt, then the overall temperature would be reduced degree. 8. (a) For each time t, there corresponds a depth d. 8. (a) A Domain: t Range: d d t A acresfarm t 8. (a) H

30 Chapter P Preparation for Calculus 87. f If <, then f. If <, then f. If, then f. Thus,, f,, < <. 88. p has one zero. p has three zeros. Ever cubic polnomial has at least one zero. Given p A B C D, we have p as and p as if A >. Furthermore, p as and p as if A <. Since the graph has no breaks, the graph must cross the -ais at least one time. p p 89. f a n n... a a a n n... a a f Odd 9. f a n n a n n... a a a n n a n n... a a f Even 9. Let F f g where f and g are even. Then F f g f g F. Thus, F is even. Let F f g where f and g are odd. Then Thus, F f g f g f g F. F is even. 9. Let F f g where f is even and g is odd. Then Thus, F f g f g f g F. F is odd. 9. (a) V Domain: < < The dimensions for maimum volume are cm. length and width volume The dimensions for maimum volume appear to be cm.

31 Section P. Fitting Models to Data 9. B equating slopes,, L. 9. False; let f. Then f f 9, but. 9. True 97. True, the function is even. 98. False; let f. Then f 9 and f. Thus, f f. 99. First consider the portion of R in the first quadrant:, and ; shown below. B smmetr, ou obtain the entire region R: (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) The area of this region is. The area of R is. [9th competition, Problem A, 988]. Let g c be a constant polnomial. Then fg fc and g f c. Thus, fc c. Since this is true for all real numbers c, f is the identit function: f. Section P. Fitting Models to Data. Quadratic function. Trigonometric function. Linear function. No relationship. (a),. (a) 9 Yes. The cancer mortalit increases linearl with increased eposure to the carcinogenic substance. If, then. No, the relationship does not appear to be linear. Quiz scores are dependent on several variables such as stud time, class attendance, etc. These variables ma change from one quiz to the net.

32 Chapter P Preparation for Calculus 7. (a) d.f or F.d. 8. (a) s 9.7t. F =. d+. The model fits well. If F, then d.. cm. The model fits well. If t., s. meterssecond. 9. (a) Using a graphing utilit,..8. r.88 correlation coefficient = (a) Linear model: H.t.9 8 The data indicates that greater per capita electricit consumption tends to correspond to greater per capita gross national product. The data for Hong Kong, Venezuela and South Korea differ most from the linear model. (d) Removing the data 8,.9,,.7 and 7, 7., ou obtain the model..8 with r.98. The fit is ver good. When t, H (a).t.t.887t..9t.7.97t.797. (a) S When, S 8.98 pounds. For t,. centsmile.. (a) Linear:.8t 8. Cubic:.89t.t.8t. 9 (d).8t.8t.7 9 The cubic model is better. (e) For t : Linear model 9. million Cubic model. million (f ) Answers will var.

33 Section P. Fitting Models to Data. (a) t.7s.9s.7. (a) The curve levels off for s <. If., horsepower. (d) t.s.s.8 (e) The model is better for low speeds.. (a) T.98 p. p.8p. 7. (a) Yes, is a function of t. At each time t, there is one and onl one displacement. The amplitude is approimatel.... For T F, p 8.9 pounds per square inch. (d) The model is based on data up to pounds per square inch. The period is approimatel.7... One model is. sint. (d) (.,.) (.7,.).9 8. (a) Ht 8..8 sin t.8 One model is Ct 8 7 sin t.. (d) The average in Honolulu is 8.. The average in Chicago is 8. (e) The period is months ( ear). (f ) Chicago has greater variabilit 7 > Answers will var.. Answers will var.

34 Chapter P Preparation for Calculus Review Eercises for Chapter P.,, -intercept -intercept., -intercept,,,, -intercepts., -intercept. and are both impossible. No intercepts., -intercept. Smmetric with respect to -ais since.. Smmetric with respect to -ais since Slope: -intercept:, Slope: -intercept:..... Slope: -intercept:, 8 7

35 Review Eercises for Chapter P... Domain:, Xmin = - Xma = Xscl = Ymin = - Yma = Yscl = Xmin = - Xma = Xscl = Ymin = - Yma = Yscl = 7 8 Point:, 8. 7 No real solution No points of intersection The graphs of and 7 do not intersect. 9. You need factors and.. Multipl b to obtain origin smmetr. k (a) k k and k k and 8 8 k an k will do! (d) k k. ( ), ( ),. 8 (7, ) (7, ). t t t 7 Slope 7 7 The line is vertical and has no slope.

36 Chapter P Preparation for Calculus. t 8 t.. Horizontal line 9 t t t (, ) 8 8 (, ) 8 7. (, ) m is undefined. Line is vertical. (, ) 8 9. (a) Slope of line is (d) 7. (a) 7 m Slope of perpendicular line is. (d) m 9. The slope is 8. V 8t,. V 8, $99

37 Review Eercises for Chapter P 7. (a) C 9.t.t,..7t, R t t.7t, 7.t, t.8 hours to break even Not a function of since there are two values of for some. ±... 9 Function of since there is one value for for each. Function of since there is one value of for each. Not a function of since there are two values of for some f (a) f does not eist. f f,, 8. (a) f 8 (because ) f < f 9. (a) Domain: or,. f and g Range:, (a) f g Domain: all or,, Range: all or,, f g g f g Domain: all or, Range: all or,

38 8 Chapter P Preparation for Calculus. (a) f c, c,, f c, c,, c c c c c c f c, c,, (d) f c, c,, c c c c c c. f (, ) (, ) (a) The graph of g is obtained from f b a vertical shift down unit, followed b a reflection in the -ais: g f The graph of g is obtained from f b a vertical shift upwards of and a horizontal shift of to the right. g f. (a) Odd powers: f, g, h Even powers: f, g, h h g h g f f The graphs of f, g, and h all rise to the right and fall to the left. As the degree increases, the graph rises and falls more steepl. All three graphs pass through the points,,,, and,. The graphs of f, g, and h all rise to the left and to the right. As the degree increases, the graph rises more steepl. All three graphs pass through the points,,,, and,. 7 will look like h, but rise and fall even more steepl. 8 will look like h, but rise even more steepl.. (a) f g h 8

39 Problem Solving for Chapter P 9. (a) Domain: < < A Maimum area is A. In general, the maimum area is attained when the rectangle is a square. In this case,.. For compan (a) the profit rose rapidl for the first ear, and then leveled off. For the second compan, the profit dropped, and then rose again later. 7. (a) (cubic), negative leading coefficient (quartic), positive leading coefficient (quadratic), negative leading coefficient (d), positive leading coefficient 8. (a)..7 7 The data point 7, is probabl an error. Without this point, the new model is (a) Yes, is a function of t. At each time t, there is one and onl one displacement. The amplitude is approimatel.... The period is approimatel.. One model is (d). (.,.) cos. t cos.7t.. (.,.) Problem Solving for Chapter P. (a) 8 Slope of line from, to, is Slope of tangent line is. Hence,. Center:, Radius: Slope of line from, to, is. Slope of tangent line is Hence,. 9 Tangent line (d) 9 9 Intersection:, 9 Tangent line

40 Chapter P Preparation for Calculus. Let m be a tangent line to the circle from the point,. Then m m m Setting the discriminant b ac equal to zero, m m m m m m ± Tangent lines: and.. H,, < (a) H H H (d) H (e) H (f ) H

41 Problem Solving for Chapter P. (a) f f f (d) f (e) f f (f ) (g) f. (a) Domain: < < A A A m is the maimum. m, m Maimum of m at m, m.

42 Chapter P Preparation for Calculus. (a) A Domain: < < 7 Maimum of 7 ft at ft, 7. ft. A 7 7 A 7 square feet is the maimum area, where ft and 7. ft. 7. The length of the trip in the water is, and the length of the trip over land is. Hence, the total time is T hours. 8. Let d be the distance from the starting point to the beach. Average speed distance time d d d 8 kmhr 9. (a) Slope 9. Slope of tangent line is less than. Slope. Slope of tangent line is greater than.. Slope.. Slope of tangent line is less than... (d) Slope h h f h f h h h h h, h (e) Letting h get closer and closer to, the slope approaches. Hence, the slope at, is.

43 Problem Solving for Chapter P. (, ) (a) (d) (e) Slope Slope Slope h h Slope of tangent line is greater than 9. Slope of tangent line is less than.. Slope.. f h f h h h h h h h h Slope of tangent line is greater than h, h h h. As h gets closer to, the slope gets closer to The slope is at the point,..... (a) 9 I I 9 ± I I ± 8., Circle of radius 8 and center,. 8

44 Chapter P Preparation for Calculus. (a) I ki k k 8 k If k, then is a vertical line. Assume k. 8 k k k k 8 k k k k k, Circle As k k becomes ver large, and k k. The center of the circle gets closer to,, and its radius approaches. If k,. d d (, ) (, ) (, ) Let. Then or. Thus,,,, and, are on the curve.. f (a) Domain: all Range: all f f f Domain: all, f f f f Domain: all, (d) The graph is not a line. It has holes at, and,.