CHAPTER P Preparation for Calculus

Transcription

1 PART I CHAPTER P Preparation for Calulus Setion P. Graphs and Models Setion P. Linear Models and Rates of Change Setion P. Funtions and Their Graphs Setion P. Fitting Models to Data Review Eerises Problem Solving

2 CHAPTER P Preparation for Calulus Setion P. Graphs and Models Solutions to Odd-Numbered Eerises.. -interept:, -interept:, Mathes graph -interepts:,,, -interept:, Mathes graph (a) (, 7) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (9, ) (, ) (, )

3 Setion P. Graphs and Models. Xmin = - Xma = Xsl = Ymin = - Yma = Ysl = Note that when.. (., ) (a),,.7,, (,.7) interept: ;, -interepts:, ;,,, -interept: ;, -interepts:, ±;, ; ±,. -interept: None. annot equal. -interepts:. -interept: ;, -interept: ;, ;,. Smmetri with respet to the -ais sine. 7. Smmetri with respet to the -ais sine. 9. Smmetri with respet to the origin sine.. No smmetr with respet to either ais or the origin.. Smmetri with respet to the origin sine.. is smmetri with respet to the -ais sine. 7. Interepts:,,, (, ) Smmetr: none,

4 Chapter P Preparation for Calulus 9... Interepts: Interepts:,,, Smmetr: none,,,,, Smmetr: -ais Interepts:,,, 9 Smmetr: none (, ) (, ) (, ) (, ) (, ) (, 9) (, ) Interepts: Interepts: Interepts:,,,,,,, Smmetr: origin Smmetr: none Smmetr: none Domain: (, ) (, ) (, ) (, ) (, ).. Interepts: none Smmetr: origin Interepts:,,,,, Smmetr: -ais (, ) (, ) (, ) ± 9 Interepts:,,,, 9, Smmetr: -ais (, ) ( 9, ) (, ) ± Interepts:,,,,, Smmetr: -ais (, ) (, ) (, )

5 Setion P. Graphs and Models 9. (other answers possible). Some possible equations:.. The orresponding -value is. Point of intersetion:, The orresponding -value is. Point of intersetion:, 7. 9., The orresponding -values are (for ) and (for ). Points of intersetion:,,, or The orresponding -values are and. Points of intersetion:,,, 7. 7.,, or The orresponding -values are,, and. Points of intersetion:,,,,,,,,,, = + (, ) (, ) (, ),, = +

6 Chapter P Preparation for Calulus 7..,.9..9,..,,,.,.,, Use the Quadrati Formula. units The other root, 99, does not satisf the equation R C. This problem an also be solved b using a graphing utilit and finding the intersetion of the graphs of C and R. 77. (a) Using a graphing utilit, ou obtain.t.997t.9 () For the ear, t and 7. CPI. 79. If the diameter is doubled, the resistane is hanged b approimatel a fator of. For instane,. and... False; -ais smmetr means that if, is on the graph, then, is also on the graph.. True; the -interepts are b ± b a, a.. Distane to the origin K Distane to, K K K K Note: This is the equation of a irle! K, K K

7 Setion P. Linear Models and Rates of Change 7 Setion P. Linear Models and Rates of Change. m. m. m m = m. m (, ) m is undefined m = m = 7 (, ) (, ) undefined (, ) (, ). m (, ) (, ). Sine the slope is, the line is horizontal and its equation is. Therefore, three additional points are,,,, and,. 7. The equation of this line is 7. Therefore, three additional points are,,,, and,. 9. Given a line L, ou an use an two distint points to alulate its slope. Sine a line is straight, the ratio of the hange in -values to the hange in -values will alwas be the same. See Setion P. Eerise 9 for a proof.

8 Chapter P Preparation for Calulus. (a) Population (in millions) Year ( 99) The slopes of the line segments are The population inreased most rapidl from 99 to 99. m.9.. Therefore, the slope is m and the -interept is,. The line is vertial. Therefore, the slope is undefined and there is no -interept (, ) (, ) (, ). m. (, ) (, ) m (, ) (, ) 7. m (, ) (, )

9 Setion P. Linear Models and Rates of Change 9 9. m Undefined. Vertial line (, ) (, ). m 7 7 (, ) (, ). (, ). 7. a a a a a a 9....

10 Chapter P Preparation for Calulus 7. The lines do not appear perpendiular. The lines appear perpendiular. The lines are perpendiular beause their slopes and are negative reiproals of eah other. You must use a square setting in order for perpendiular lines to appear perpendiular. 9.. (a) m m (a) (a). The slope is. Hene, V t t 7. The slope is. Hene, V t, t, 9. (, ) (, ) You an use the graphing utilit to determine that the points of intersetion are, and,. Analtiall, The slope of the line joining, and, is m. Hene, an equation of the line is.,,.

11 Setion P. Linear Models and Rates of Change 7. m m m m The points are not ollinear. 7. Equations of perpendiular bisetors: a b a b Letting in either equation gives the point of intersetion:, a b. a b ( b, ) b a This point lies on the third perpendiular bisetor,. ( ) (, ) b a, a b ( a, ) ( a, ) 7. Equations of altitudes: a b a b ( b, ) a b a ( a, ) ( a, ) Solving simultaneousl, the point of intersetion is b, a b. 77. Find the equation of the line through the points, and,. m 9 F 9 C F 9 C F 9C For F 7, C (a) W.7. W. 9. () Both jobs pa $7 per hour if units are produed. For someone who an produe more than units per hour, the seond offer would pa more. For a worker who produes less than units per hour, the first offer pas more. (, 7) Using a graphing utilit, the point of intersetion is approimatel, 7. Analtiall,

12 Chapter P Preparation for Calulus. (a) Two points are, and 7,. The slope is m 7. p p 7 or p If p, units. () If p 9, 9 9 units.. d. d 7. A point on the line is,. The distane from the point, to is d. 9. If A, then B C is the horizontal line CB. The distane to, is d C B B C B A B C. A B If B, then A C is the vertial line CA. The distane to, is d C A A C A A B C. A B (Note that A and B annot both be zero.) The slope of the line A B C is AB. The equation of the line through, perpendiular to A B C is: B A A A B B B A B A The point of intersetion 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 CONTINUED

13 Setion P. Linear Models and Rates of Change 9. CONTINUED AC B AB, BC AB A point of intersetion A B A B The distane between, and this point gives us the distane 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. For simpliit, let the verties of the rhombus be,, a,, b,, and a b,, as shown in the figure. The slopes of the diagonals are then ( b, ) ( a + b, ) m a b and m b a. Sine the sides of the Rhombus are equal, and we have a b, (, ) ( a, ) m m a b b a b a. Therefore, the diagonals are perpendiular. 9. Consider the figure below in whih the four points are ollinear. Sine the triangles are similar, the result immediatel follows. 9. True. a b a b b m a b b a b a a m b a (, ) ( *, * ) (, ) ( *, * ) m m

14 Chapter P Preparation for Calulus Setion P. Funtions and Their Graphs. (a) f f 9 () f b b (d) f. (a) () g g g (d) gt t t t. (a) f os os () f os os f os os 7. f f, 9. f f,. h. Domain:, Range:, t f t se t k t k Domain: all t k, k an integer Range:,,,. f Domain:,,, Range:,,, 7. f, <, 9. <, f, (a) f (a) f f f () f () f (d) f t t t (Note: t for all t) (d) f b b b Domain:, Domain:, Range:,, Range:,,,

15 Setion P. Funtions and Their Graphs. f. h Domain:, Domain:, Range:, Range:,. f 9 7. Domain:, Range:, gt sin t Domain:, Range:, t 9. ± is not a funtion of. Some vertial lines interset the graph twie.. is a funtion of. Vertial lines interset the graph at most one.. 7. ±. is not a funtion of sine there are two values of for some. f If <, then f. If <, then f. If, then f. ± is not a funtion of sine there are two values of for some. Thus,, f,, < <.. 9. The funtion is g. Sine, satisfies the. The funtion is r, sine it must be undefined at equation,. Thus, g.. Sine, satisfies the equation,. Thus, r.. (a) For eah time t, there orresponds a depth d.. d Domain: t 7 Range: d () d 9 t t t t t

16 Chapter P Preparation for Calulus 7. (a) The graph is shifted units to the left. The graph is shifted unit to the right. () The graph is shifted units upward. (d) The graph is shifted units downward. (e) The graph is strethed vertiall b a fator of. (f) The graph is strethed vertiall b a fator of. 9. (a) () Vertial shift units upward Refletion about the -ais Horizontal shift units to the right. (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 redued degree.. f, g. 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 f g f Domain: all ± g f g f g 9 9 Domain: all No, f g g f.

17 Setion P. Funtions and Their Graphs 7 7. A rt Art A.t.t.t 9. A rt represents the area of the irle at time t. f f Even. f os os f Odd. (a) If f is even, then, is on the graph. If f is odd, then, is on the graph.. f a n n... a a a n n... a a f Odd 7. 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 F f g f g f g F. Thus, F is even. 9. f and g are even. f is odd and g is even. f g is even. f g is odd. 7. (a) length and width volume V (d) Yes, V is a funtion of. The maimum volume appears to be m. () V Domain: < < Maimum volume is V m for bo having dimensions m. 7. False; let f. 7. True, the funtion is even. Then f f 9, but.

18 Chapter P Preparation for Calulus Setion P. Fitting Models to Data. Quadrati funtion. Linear funtion. (a), 7. (a) d.f or F.d. 9 Yes. The aner mortalit inreases linearl with inreased eposure to the arinogeni substane. () If, then. F =. d+. The model fits well. () If F, then d.. m. 9. (a) Let per apita energ usage (in millions of Btu) per apita gross national produt (in thousands) r.7. (a).t.t.7t..9t.7.97t =. +. () Denmark, Japan, and Canada (d) Deleting the data for the three ountries above,.99.9 ( r.9 is muh loser to.) For, t and. entsmile. (a).7t.9.99t.t.99t. 7 =.t +.9 (d).97t.99t.97 7 =.t +.t +.t +. () The ubi model is better. (e) The slope represents the average inrease per ear in the number of people (in millions) in HMOs. (f) For, t, and 9. million. (linear). million (ubi)

19 Review Eerises for Chapter P 9. (a) (a) Yes, is a funtion of t. At eah time t, there is one and onl one displaement. The amplitude is approimatel.... () If., horsepower. 7 The period is approimatel.7... () One model is. sint. (d).9 9. Answers will var. Review Eerises for Chapter P.,, -interept -interept., -interept. Smmetri with respet to -ais sine., -interept Slope: -interept:

20 Chapter P Preparation for Calulus.. Domain:, 7. Xmin = - Xma = Xsl = Ymin = - Yma = Ysl = 7 Point:, 9. You need fators and. Multipl b to obtain origin smmetr.... t ( ), t ( ), t 7 Slope (, ) (, )

21 Review Eerises for Chapter P 9. (a) () 7 Slope of line is. 7 m 7 7 (d). The slope is. V t,. V, $99.. Not a funtion of sine there are two values of for some. ± Funtion of sine there is one value of for eah. 7. f (a) f does not eist. f f,, 9. (a) Domain: or, Range:, Domain: all or,,, Range: all or,,, () Domain: all or, Range: all or,. (a) f,,, f,,, CONTINUED

22 Chapter P Preparation for Calulus. CONTINUED () f,,, (d) 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 inreases, the graph rises and falls more steepl. All three graphs pass through the points,,,, and,. 7 will look like h, but rise and fall even more steepl. will look like h, but rise even more steepl. The graphs of f, g, and h all rise to the left and to the right. As the degree inreases, the graph rises more steepl. All three graphs pass through the points,,,, and,.. (a) Domain: < < A () Maimum area is A. In general, the maimum area is attained when the retangle is a square. In this ase,. 7. (a) (ubi), negative leading oeffiient (quarti), positive leading oeffiient () (quadrati), negative leading oeffiient (d), positive leading oeffiient 9. (a) Yes, is a funtion of t. At eah time t, there is one and onl one displaement. The amplitude is approimatel.... The period is approimatel.. () One model is (d). os. t os.7t..