34 CHAPTER R Review
Figure 22
It is not necessary to verify that all three angles are equal and all three sides are
proportional to determine whether two triangles are congruent.
Determining Similar
54. Minimizing Surface Area United Parcel Service has
contracted you to design an open box with a square base
that has a volume of 513013 cubic inches. See the illustration.
(a) Express the surface
Step 2: Determine the real zeros
(XviHWFGBPtL- ofthe graph) of fanel
the real numbere for which fie
undened.
Step 5: Uee the zerce anel undefined
valuee fcu nel in Step 2 to eliviele the
real number l
353 CHAPTERS Polynomialand Rational Functions
STEP 3: There is no y-intercept. Since 1:4 + 1 = 0 has no real solutions, there are no
xintercepts.
STEP 4: R is in lowest terms, so .1: = 0 (the y-axis)
SECTION 5.3 The Graph of a Ratinnal Functinn 365
5.3 Assess Your Understanding
'Are YOU PI'E pa red? The answer is given at the end af these exercises. Ifyaa get a wrang answer. read the pages listed
SECTION 5.3 The Graph of a Rational Function 363
Figure 4D
E Exploration
2x2 5x + 2
ERROR at X = 2? Are you convinced that an algebraic analysis of a rational function is required in order
to accura
EXAMPLE 2
Step-by-Step Solution
Step 1: Write the inequality ee
that a polynomial expreeeien fie
en the left aide and zero ie on the
right eide.
Step 2: Determine the reel zeree
(sq-intercepts: oi the
356 CHAPTERS Polynomialand Rational Functions
Use the zeros of the numerator and denominator of R to divide the .r-axis into intervals. Determine where
the graph of R is above or below the x-axis by c
SECTION 5.3 The Graph of a Rational Function 361
E Exploration
Graph Rbr =
Result Figure 37 shows the graph in connected mode,and Figure 38(a) shows it in dot mode. Neither
graph displays clearly th
362 CHAPTERS Polynomialand Rational Functions
STEP 4: Since .1: + 2 is the only factor of the denominator of R(.v) in lowest terms, the
graph has one vertical asymptote, x = 2. However, the rational f
SECTION 5.3 The Graph of a Rational Function 359
Seeing the Concept
x4 + 1
X2
turning points. Enter Y2 = x2 and ZOOM-OUT. What do you see? I
_-*Ii|nw Wlll'li mouse: 1 3
Graph Rot) =
and compare what
374 CHAPTERS Polynomialand Rational Functions
74. Gravitational Force According to Newtons Law of
universal gravitation, the attractive force F between two
bodies is given by
mimz
F2
F=G
where m1, m
364 CHAPTERS Polynomialand Rational Functions
q(.r). A possibility for the denominator is q(.r) = (x + 5)(.r 2?. So far we have
(x + 2)2(.r 5)
Z (r: + 5)(r 2)?
The horizontal asymptote of the graph gi
SECTION 5.3 The Graph of a Rational Function 355
5133!? 5: USE: til FEE-Ult Ubtilll Figure 320:) ShUWS the graph 0f R
in Stripe 1 through 7 to graph E.
Figure 32
ll
re
re
X=2 y X
E Exploration
366 CHAPTERS Pelynemialand HatienalFunctiens
Applications and Extensions
49. Drug Cencentratien The cencentratien C ef a certain
drug in a patients bleedstream I heurs after injectien is
given by
1
370 CHAPTERS Polynomialand Rational Functions
EXAMPLE 3
approach to help us understand the algebraic procedure for solving inequalities
involving rational expressions.
Solution
Figure 4?
EXAMPLE 4
S
50 CHAPTER R Review
COMMENT Over the real numbers,
5x + 4 factors into 5(s: + g). It is the
noninteger % that caueee 5s + 4 to be
prime over the integers. in most
instances, we will be factoring over
38 CHAPTER R Review
51. Hew Tall Is the Great Pyramid? The ancient Greek
philesepher Thales ef Miletus is reperted en ene eccasien te
have visited Egypt and calculated the height ef the Great
Pyramid
52 CHAPTER It Review
If a trinomial is not a perfect square, it may be possible to factor it using the
technique discussed next.
Factor a Second-Degree Polynomialm2 + Bx + C
The idea behind factoring
40 CHAPTER R Review
DEFINITION
r
,n In Words Mal
A poiynornial is a euro of
'- monomials.
EXAMPLE 3
2 Recognize Polynomials
Two monomials with the same variable raised to the same power are called lik
48 CHAPTER R Review
Skill Building
In Prcbierns 2716, teii whether the expressicn is e rncnctniei. If it is. nerne the veriehie(s) end the ccefcient end give the degree cf the
rncncrnini. I f it is ne
SECTION 11.4 Polynomials 41
1
are not polynomials. The first is not a polynomial because = f1 has an exponent
x
that is not a nonnegative integer. Although the second expression is the quotient
of two
SECTION 11.4 Polynomials 47
The process of dividing two polynomials leads to the following result:
THEOREM Let Q be a polynomial of positive degree and let P be a polynomial whose
degree is greater th
36 CHAPTER R Review
R.3 Assess Your Understanding
Concepts and Vocabulary
1. A.(n) triangle is one that contains an angle of 9. True or False The triangles shown are similar.
90 degrees The longest si
32 CHAPTER R Review
EXAMPLE 4
Solution
Figure 19
In Words
Two trianglee are congruent if I
theyr have the same size and
DEFINITION
For a circle of radius r (diameter d = 2r),
2
Circumference =
SECTION [1.5 Factoring Polynomials 51
m Factoring the Difference of Two Cubes
Factor completely: x3 1
Solution Because 1:3 1 is the difference of two cubes, 1:3 and 13,
x31=(x1)(x2+x+1) .J
EXAMPLE 4 F
SECTION 11.3 Geometry Essentials 31
Figure 17 Notice that the sum of the first two squares (25 and 144) equals the third square
(169). Hence. the triangle is a right triangle. The longest side, 13, is
2.2. DEFINIT E INTEGRALS 71
integral ef e in terms ef the elementary funetiens ef ealeulus. In this ease, the
best we eeuld de is leek fer geed appreximatiens fer the pesitien funetien 2:.
Seeend, thi
2.5. APPLICATIONS OF DEF INITE INTEGRALS 89
0.5 y=sr
Figure 2.5.11: Graph ef y = 2:2 ever [0,1]
New we sheuld expect the apprexiniatien in (2.5.24) te beeenie exact when
N is innite. That is, fer N
78 CHAPTER 2. INTEGRALS
We will leek at several applieatiens ef denite integrals in the next seetien.
Fer new, we nete hew this theereni prevides a methed fer evaluating integrals.
Namely, given a fun
2.5. APPLICATIONS OF DEFINITE INTEGRALS 87
Figure 2.5.8: Region T rotated about zaxis to create solid body B
T about the saxis. See Figure 2.5.9. If R(s) is a cross section of B perpendicular
to the
2.3. PROPERTIES OF DEF INITE INTEGRALS 73
That is, the denite integral of a constant is the constant times the length of
the interval. In particular, the integral of 1 over an interval is simpl,r the