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Unformatted text preview: 200 CHAPTER 2 Functions The maxi mum and minimum commands on a TI—82 or TI—83 calculator
another method for ﬁnding extreme values of functions. We use this metho next example. A Model for the Food Price index A model‘for the food price index (the price of a i'egaresentative “basket” of feud:
between 1990 and 2000 is given by the function 1(t) = "0.0113z3 + 0.06811“? + 0.198: + 99.1 where r is measured in years since midyear 1990, so 0 S I :5 10, and [(t) is
so that [(3) m 100. Estimate the time when food was most expensive during'th period 1990—2000. Solution The graph of I as a function of t is shown in Figure 8(a). There ap
to be a maximum between t 2 4 and t = 7. Using the ma x i mu in command, as
shown in Figure 8(b), we see that the maximum value of I is about 100.38, at
occurs when t m 5.15, which corresponds to August 1995. Qaéii‘éi’tgg; I: 9!" '15}: 382M
K..." :.:;‘..:,u,ws;n‘_w'._;_4:. : I 96 ¢ 03) Figure 3 _  Exercises 1—4 B The graph of a quadratic function f is given.
(3) Find the coordinates of the vertex.
0;) Find the maximum or minimum value of f. i 1. ﬁx) 2 Wxz + 6x  5 2. f(x) = —%x2 w 2x + 6 SECTiON 2.5 Quadratic Functions; Maxima and Minima 201 is a Aquadratic function is given. 45, ﬁx) 5 x2 + 17995 .— 321
,) Express the quadratic function in standard form. 46_ ﬁx) m 1 + x W WK; (1,} Find its vertex and its x and y—intercept(s). \ ) sketch its graph. f(x) = J:2 ._ 6x 6_ ﬁx) m x2 in 3x 47—59 Find alt local maximum and minimum values of the
function whose graph is shown. " . ﬁx) = 2x2 «1— 6x 8. ﬁx) 2 mxt + 10x'
1 'f(x).«:x2+4x+3 10.f(x)=x2~2x+2
a {6" ﬁx) 2 —Jc2 + 6x + 4 12. ﬁx) a "x2  4x + 4
’(tiis scat ﬁx) = 2x2 + 4x + 3 14. ﬁx} 2 "—33? + 6x m 2
911‘” the ﬁx) = 2x2 m 20x + 57 15. f(x} = 2x2 + x  6
game a x} e —~4x2 — 16x + 3 18. ﬁx) a 6x2 + 12x — 5
1,2" PP 19.23 m A quadratic function is given.
) Express the quadratic fenction in standard form. } Sketch its graph.
c) Find its maximnm or minimum vaiue. .ﬂx) ﬂZx"xz 20. ﬁx) =x +Jt2
',f(x)wx2+2xm1 22.f{x):x2"8x+8
:,'(x)=—~x2—3x+3 24.f{:c)$1—6xx2
'; (x) =3x2 12x+13 26. g(x) = 2x3+8x+1i
.{x)$1—x—~x2 28.h(x)33m4x—4x2 0.38244 Mme. 10. _.(x =x2+x+1 30.f()c)mi~%m3xwx2
. (r) x 100 — 49: — 7:2 32. m} = 10:2 + 40: + 113
s) m .8 m 1.2s + 16 34. 90:} a 10019 ~ 1500;: mi 2 _ ,3 ﬁg 51—58 Find thelocai maximum and minimum values of
5' km W 5" + 21‘ W 6 36' ﬁx) "“ _—3' + 2x + 7 the function and the vaiue ofx at which each occers. State each
i 2 answer correct to two decimal places.
f(x)$3~wxm§x ‘ 38.g(x)w2x(x—4)+7
I _ . 51.f(x)wx3wx 52.f{x)=3+x+x2—x3
. Find a function whose graph is a parabola With vertex 4 3 2 5 3
(E, “2) and that passes through the point {4, 16). 53‘ 59¢“) 2 x W 2" W “X 54' WC) 3 x ﬂ 8" + 20x
0. Find a function whose graph is a paraboia with vertex {3, 4t) 55 [1(1) 2 x 5 "“ x 56 Uh”) 5 x V x _ x2
and that passes through the point (1, ~8}. 1 _ x2 I
57. V = 5 . V x m
(x) x3 (x) xzwtvxnt 1 IIﬂx) m —x?— 4» 4x w 3 42. ﬁx) : 2:2 ~— 2): — 3 = 2x2 + 6x — 7 44. m ""3262 4* 5x + 4 Appﬁcations :6 I A quadratic function is given, 59. Height of a Ball If a bail is thrcwri directly upward with a
veiocity of 40 ftls, its height (in feet) after t seconds is given
by y = 401‘  16:2. What is the maximum height attained by
the bali? Use a graphing device toﬁnd the maximum or minimum nd the exact maximum or minimum value of f, and 66. Path of 3 Bali A bail is thrown across a playing fieid.
mPare with your answer to part {21). Its oath is given by the equation y : w0.005252 + x + 5, 202 61. 62. 63. 64. 65. CHAPTER 2 Functions where x is the distance the ball has traveled horizontally,
and y is its height above ground level, both measured
in feet. (a) What is the maximum height attained by the ball? (b) How far has it traveled horizontally when it hits the
ground? Revenue A manufacturer ﬁnds that the revenue generated
by selling x units of a certain commodity is given by the
function R(x) ~= 80x — 04x2, where the revenue R05) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this
maximum? Sales A softdrink vendor at a popular beach analyzes his
sales records, and ﬁnds that if he selis x cans of soda pep in
one day, his proﬁt (in doliars) is given by P(x) 2 macaw a» 3x m 1300 What is his maximum proﬁt per day, and how many cans
must he soil for maximum proﬁt? Advertising The effectiveness of a television com~
mercial depends on how many times a viewer watches it.
After some experiments an advertising agency found that if the effectiveness E is measured on a scale of
0 to 10, then an) =% as where n is the number of times a viewer watches a given
commercial. For a commercial to have maximum effective“
ness, how many times should a viewer watch it? Pharmaceuticals When a certain drug is taken orally,
the concentration of the drug in the patient’s bioodstream
after tminutes is given by C(t) m 0.06: m 0.000212, where
0 5 rs 240 and the concentration is measured in nag/L.
When is the maximum serum concentration reached, and
what is that maximum concentration? Agrieuiture The number of apples produced by each tree
in an appie orchard depends on how densely the trees are
planted. If n trees are planted on an acre of land, then each
tree produces 900 m 911 apples. So the number of apples
produced per acre is A011) E 11(930 W 9n) HE ﬁ 67. 68. . Migrating Fish How many trees shouid be pianted per acre in order to
obtain the maximum yield of appics? water, against a current of 5 mi/h. Using a mathematical
model of energy expenditure, it can be shown that the tot '
energy E required to swim a distance of 10 mi is given by ' 10 E $2.733
00 UPS Biologists believe that migrating ﬁsh try to minimize the
total energy required to swim a ﬁxed distance. Find the
vaiue of u that minimizes energy required. NOTE This resuit has been veriﬁed; migrating ﬁsh swim '
against a current at a speed 50% greater than the speed of the 
current. Highway Engineering A highway engineer wants to
estimate the maximum number of cars that can safely trave
a particular highway at a given speed. She assumes that 83?
car is 17 ft long, traveis at a speed s, and follows the car in
front of it at the “safe following distance” for that speed.
She finds that the number N of cars that can pass a given
point per minute is modeled by the function 88.? 2
s
.i. _.
17 i7 ( 20)
At what speed can the greatest number of cars travel ﬁle I highway safeiy? Voiume of Water BetWecn 0°C and 30°C, the volume V
(in cubic centimeters) of 1 kg of water at a temperature T1?
given hy the formula V = 9199.87 — 0.06426?" + 0.0085043? ~ 000006723” Find the temperature at which the volume of i kg of mici
a minimum. Nisl m SECTFON 2.6? Modeiing with Functions 293 ,69 Coughing When a foreign object lodged in the trachea 71. Minimizing a Distance When we seek a minimum or order to
(wiadpipe} forces a person to cough, the diaphragm thrusts maximum value of a function, it is sometimes easier to work
upward causing an increase in pressure in the lungs. At the with a simpler function instead.
same time, the trachea contracts, causing the expelied air to (a) Sugpose 9(x) a m, where ﬁx) a 0 for a}; x_
" movo faster and increasing the pressure on the foreign 0b“ Bxpiain why the local minima and maxima of f and g jest. According to a mathematical} model of coughing, the wear at the same values of x‘  velocity}; of the airstream through an averagesized person’s
trachea is related to the radius r of the trachea (in ccntirne~
mg) by the function (1’)) Let g(x) be the distance between the point (3,0) and
the point (x,x2) on the graph of the paraboia y = x2.
Express g as a function of x. Emil: t 0(r} a 3.2(1 "" 1")”, i S r 5 1 (c) Find the minimum value of the function 9 that you
heuristics: Determine the value of r for which I) is a maximum. TiganéfgWii: the PYmCiple described 111 53a“ (a)
{pat the total p y y ' gas given by. 72. Maximum of a Fourth—Degree Polynomial Find the Discovery ° Discussaen maximum vaiue of the function at L, . Maximo and Minima in Example 5 we saw a real—world ﬁx) = 3 __ 4x2 _ x4
situation in which the maximum value of a function is in» .nimize Eh portent. Name several other everyday situations in which a {HZmo“ L3” 3 1623 ' Find the maximum or minimum value is important. 3g ﬁsh swirrr speed of the: ' Modeling suitcases Many of the prooesses studied in the physical and social sciences involve under—
standing how one quantity varies with respect to another. Finding a function that de«
scribes the dependence of one quantity on another is called modeling. For example,
a bioiogist observes that the number of bacteria in a certain culture increases with
time. He tries to model this phenomenon by ﬁnding the precise function (or rule) that
relates the bacteria population to the elapsed time. In this section we will learn how to ﬁnd models that can be constructed using geo metric or algebraic properties of the object under study. (Finding models from data is 1“ Safely “3 studied in the Focus on Modeling at the end of this chapter.) Once the model is found,
“mes that? Mi:  we use it to analyze and predict properties of the object or process being studied.
ms the can % ' that speed.
)ass a given er wants to . Modeling with Functions I We begin with a simple real—life situation that iiiustrates the modeling process. Modeling the Voiume of a Box A breakfast cereal company manufactures boxes to package their product. For
aesthetic reasons, the box must have the following proportions: Its Width is 3 times
its depth and its height is 5 times its depth. (a) Find a function that models the volume of the box in terms of its depth.
(b) Find the volume of the box if the depth is 1.5 in.
(c) For what depth is the volume 90mg? (d) For What depth is the volume greater than 60 in3? ...
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