MTH207 Lab 1 Homework

7/30/2019 MTH207 Lab 1 Homework

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Cnaprpn 1 . FuNcrroNs*,8 13-20. Domain and range Graph eachfunctionwith a graphing utility

using the given window. Then state the domnin and range of the function.13. f(x) : 3x4 - to; l-z,zl x [-to, ts]v*lA. eU) : 0 . ,------t -)(y )i l-4.01 x [-:.:]ls. /(x) : \/q - *' l-+,+) x l-t,+)16. F(w) : (z -i; l-t,zl x lo,z)17. h(u) : t/" -1; l-t,g) xl-z,z)18. s(x) : (*2 - q\/x + s; [-s,s] x [-to,so]te. f(x) - (s - x13/2; [-+, +] x [0, :o]

120. g(t) :-r l-t.tl

x [0, t.s]t+t'2l-24.Domain in context Determine an appropriate domain of eachfunction. Identify the independent and dependent variables.21. A stone is thrown vertically upward from the ground at a speed

of 40 m/s attime t : 0. Its distance d (in meters) above theground (neglecting air resistance) is approximated by the functionf(t) : 4ot - stz '22. A stone is dropped off a bridge from a height of 20 m above ariver. If / represents the elapsed time (in seconds) after the stone is

released, then its distance d (in meters) above the river is approxi-matedbythefunction f(t) : 20 - St2.23. A cylindrical water tower with a radius of l0 m and a height of

50 m is fil1ed to a height of h. The volume V of water (in cubicmeters) is given by the tunction g(h) : lOonh.

24. The volume V of a balloon of radius r (in meters) filled withhelium is given by the function f(r) : tor3. Assu-e the ballooncan hold up to 1 m3 of helium.

25-36. Composite functions and notation let f(x) : x2 - 4,s(x) : x3, and F(x) : I I Q - 3). Simplifu or evaluate the followingexpressions.2s. f(to)28. F(ya)3r. g(f(u))sn. g@(f(x)))

f(p')r(s(v))f(2+h)-f(2)

s0lz).f(s(,))

41-48. More composite functions Det f(x) : lrl, S(*) : x2 - 4F(r) : \/i, and G(*) : I lQ - 2). Determine the followingcomposite.functions and give their domains.42. g"745. G" 8"f48. G. G49-54. Missing piece Let g(x) : x2 +

produces the given composition.s0. (.f.s)(x) : -+'+3

51. (.f.s)(r): ra+6*2+g 52. (f"s)(x): xa +6*2+2s3. (s.,f)(r) : xa + 3 5a. (s..f)(r) : ,213 + 355. Composite functions from graphs Use the graphs of / and gthe figure to determine the following function values.

l0

41. 7. r44. 7" g" 647. g. g

4e. U" d@) : *2

a. flsQ))a. s(fls))

43. 7'646. Fogog

3. Find afunction f that

c. "f(s(+))f./(/(8))a.sUQ))e. f(sQ))

56. Composite functions from tables Use the table to evaluate thgiven compositions.

27.30.

26.29.32.

5/.59.61.

38.39.

.ralg(r)h(*)

2-133

-1 03l-1 00-1I020

34-3 -l45o43740. Working with composite functions Find possible choices forouter and inner functions f and g such that the given function h equals7" g. Give the domain of h.37. h(x): (x3 - 5)10

r.s(fl+)) c. h(h(o))e.f(f(f(r))) t. h(h(h(o)))h. s(f(h(4))) i s(s(s(t)))

58. f(x) : ax -32-3x+l0. f(x) : 2x4

57-66. Working with difference quotients Simplify the dffirenceIG + h) - f(x) ."t(*) - I@)quoients ----'i-r ann --- _; for rhe following funcrif(*) : *'f(x): zlxxf(*) : --1- I

ts. |l'/-x +)33. r(r(x))*"(q#) a. a(s(o))d.8(h(f(4)))e.f(nGQ)))i. fu&(3)))

h(*) :h(r) :h(*) :

(x6+x2+l)2fliI

-\/r3 - t

v10987.654321

0 t-z 3 4 5 6 7 8 9 x

40. 62. f(x) : x


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63. f(x) : x3 - 2x46s. /(x) x-

6a. fQ):4-4x-x266. f(x):i-,,

67-70. Interpreting the slope of secant lines 1z each exercise, afunction and an interval of its independent variable are given. Theendpoints of the interval are associated with the points P and Q on thegraph of the function.a. Sketch a graph of the function and the secant line through P and Q.b. Find the slope of the secant line in part (a), and interpret your an-

swer in terms of an ayerage rate of change over the interval.Include units in your answer.

67. Alter / seconds, an object dropped from rest falls a distanced : 16t2, where d is measured in feet and 2 = t < 5.68. After , seconds, the second hand on a clock moves through anangle D : 6r, whereD is measuredin degrees and 5 < t < 20.

69. The volume V of an ideal gas in cubic centimeters is givenby V : 2 f p, where p is the pressure in atmospheres and0.5=p


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t2 Cne.prnx. I . FuNcrroNs

95- E+O 96. E.O99. E" E l00.O"O

89-92. Difference quotients Simplifu the dffirence quotientsf(x + h) - f(x) . f(*) - f(o) .and-

by rationalizing the numerator.h x-ase. f(x): t/x s0. f(x): \/t - b;

92. f(x) - "'/*4 1t. f(x) :Applications

#H 93. Launchlng a rocket A small rocket is launched verticallyupward from the edge of a cliff 80 ft off the ground at a speedof 96 ft/s. Its height in feet above the ground is given byh(t) : -6rz + 96t + 80, where / represents time measuredin seconds.a. Assuming the rocket is launched at t : 0, what is an

appropriate domain for ft?b. Graph ft and determine the time at which the rocket reaches itshighest point. What is the height at that time?

94. Draining a tank (Torricelli's law) A cylindrical tankwith a cross-sectional area of 100 cm2 is filled to a depth of100 cm with water. At / : 0, a drain in the bottom of the tankwith an area of 10 cm2 is opened, allowing water to flow outof the tank. The depth of water in the tank at time , > 0 isd(t):(r0-2.2t)2.a. Check that d(0) : 100, as specified.b. At what time is the tank empty?c. What is an appropriate domain for d?

Additional Exercises95-101. Combinlng even and odd functions Let E be an evenfunc-tion and O be an odd function. Determine the symmetry, if any, of thefollowing functions.

102. Composite even and odd functions from tables Assume / is aeven function and g is an odd function. Use the table to evaluatethe given compositions.

f(*)g(r)t22-l-3 -1

343-4-4 -2--\/x a."f(g(-t))d. fls?z))e. fkkez)))103. Composite even and odd functions from graphs Assume / is aeven function and g is an odd funcdon. Use the (incomplete) grap

of / and g in the figure to determine the following function value

u. s(fl-+))e.s(s(-t))h.s(flfl-+)))c. /(s(-:))r. /(s(o) - t)i g(s(g(-t)))

a.flsez))0. g(fls) - a) t.sjez)) c. fls(-+))e. s(s(-z)) f. /(1 -/(8))

e7. Elol0l.O"E

98. E. O QulcK cHEcK:ANSWERSl. 3,x4 - 2x2,P - 2t,p2 - 4p + 3 2. Domain is all realnumbers;rangeis {y,0 < y < 1}. 3. (f"s)(r) : xa + Imd (S " fl@) : 1x2 + L)2. 4. ffthe graph were symmetricwith respect to the x-axis, it would not pass the vertical line test

1 .2 Representing Functions

One version of the Fundamental Theoremof Algebra states that a nonconstantpolynomial of degree n has exactlyn (possibly complex) roots, countingeach root up to its multiplicity.

We consider four different approaches to defining and representing functions: formulagraphs, tables, and words.Using FormulasThe following list is a brief catalog of the families of functions that are introduced in tchapter and studied systematically throughout this book; they are all defined by formula1. Polynomials are functions of the form

f(*) : a,{' + an-1x'-1 + "' + afi * as'where the coeffici[tS as, at,... ,an ate real numbers with an * 0 and tnonnegative integer n is the degree of the polynomial. The domain of any polynomis the set of all real numbers. An nth-degree polynomial can have as many as n r


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3467.

Crup:rrx I . FuxcrroNsLaw of sines Use the figure to prove the law of sines:sinA sin B sin Cabc

REVIEW EXERCISES1. Explain why or why not Determine whether the following state-

ments are true and give an explanation or counterexample.a. A function could have the property that /(-x) : /(x), forall x.b. cos (a + b) - cos a * cos D, for all a ard b inl},2lr).c. lt f is a linear function of the form f(r) : mx * b, tbenf(u + v) : f(") + flv),for alluandy.d. The function f(x) : I - x has the property thatf(f(*)) : *.e. Theset {x:lx + 3l > 4} canbedrawnonthenumberlinewithout lifting your pencil.2. Domain and range Find the domain and range of the following

functions.a. f(x) : x5 + t/i b. s(y) -- -+)-2c. h(z):\/*-22-3

3. Equations of lines Find an equation of the lines with the followingproperties. Graph the lines.a. The line passing through the points (2,-3) and @,2)b. The line with slope J and r-intercept (-4, 0)c. The line with intercepts (4, 0) and (0, -2)4. Piecewise linear functions The parking costs in a city garageare $2.00 for the first half hour and $1.00 for each additional halfhour. Graph the function C : f(t) that gives the cost ofparkingforlhours, where0 < t < 3.

5. Graphing absolute value Consider the functionf(*) : 2 (x - lxl). Express the function in two pieceswithout using the absolute value. Then graph the function byhand. Use a graphing utility only to check your work.

6. Function from words Suppose you plan to take a 500-mile trip ina car that gets 35 mi/gal. Find the function C : f(p) that givesthe cost of gasoline for the trip when gasoline costs $p per gallon.7. Graphing equations Graph the following equations. Use a graph-ing utility only to check your work.a.2x-3y+10:0b.Y:x2+2x-3c. *2+2*ty2 +4y+1:od.x2-2**y2-8y+5:0

quicr tHEcK Aruiwiisl. 3rr12;225" 2. \/112)-rtpsin2o + coszo: lbysin2g.-*: 3. Divide both sides o

Root functions Graph the functions f(x) : xr/3 ands(x) : xlla.Findall points where the two graphs intersect. For-r > 1, is f(*) > s(x) oris g(x) > f(x)?Root functions Find the domain and range of the functionsf(r) : xUl andg(x) : xtl4.Intersection points Graph the equations ! : x2 andx2 + y2 - 7y + 8 : 0. At what point(s) do the curves intersectBoiling-point function Water bolls at2l2o F at sea level and at2O0o F at an elevation of 6000 ft. Assume that the boiling pointvaries linearly with altitude a. Find the function B : f(a) thatdescribes the dependence. Comment on whether a linear functiogives a realistic model.Publishing costs A small publisher plans to spend $1000 foradvertising a paperback book and estimates the printing cost is$2.50 per book. The publisher will receive $7 for each book sola. Find the function C : f(x) that gives the cost of producing

x books.b. Find the function R : S(x) that gives the revenue from sellix books.c. Graph the cost and revenue functions, and find the number obooks that must be sold for the publisher to break even.

Shifting and scaling Starting with the graph of /(x) : x2, plothe following functions. Use a graphing calculator only to checkyour work.a. f(x + 3) b.2fQ - a) c. -f(3x) d. f(2(x - 3

14. Shifting and scaling The graph of/is shown in the figure. Grapthe following functions.a. /(x + 1) b. 2f(x - t) c. -[(xl2) d. /(2(x - t

"..i..,1 { i.'.rfu i3-,_Fi,!3r{}r';. . .;,..,--.1,i., ,,.;a.' t'11

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MTH207 Lab 1 Homework