Capítulo II: Modelo Basado en Casos para planificación de proyectos de
2.3 Principales fases de los SBC
2.3.1 Representación del conocimiento
2.3.1.3 Estructura de los casos
Addition of Ordinates
New functions can be formed by adding or subtract-New functions can be formed by adding or subtract-ing other functions. A functioning other functions. A function formed by combining two other functions,formed by combining two other functions, such assuch as
has historically been graphed using a method known as
has historically been graphed using a method known as addition of ordinatesaddition of ordinates..
(The
(The x x-value of a point is sometimes called its-value of a point is sometimes called its abscissa,abscissa, while itswhile its y y-value is-value is called its
called its ordinateordinate.) T.) To apply this meto apply this method to this functihod to this function,on, we graph the func-we graph the func-tions
tions and and Then,Then, for for selected selected values values of of x x,, we awe add cdd cosos x x andand sin
sin x x,, and and plot plot the the points points Joining Joining the the resulting resulting points points with with aa sinusoidal curve gives the graph of the desired function. While this method sinusoidal curve gives the graph of the desired function. While this method illustrates some v
illustrates some valuable concepts inaluable concepts involving the aritvolving the arithmetic of functions,hmetic of functions, it isit is time-consuming.
time-consuming.
With
With graphing calculgraphing calculators,ators, this techniquthis technique is ease is easily illustrily illustrated. Letated. Let and
and FigureFigure 54 54 shows shows the the result result when when andand are
are graphed graphed in in thin thin graph graph style,style, and and is is graphed graphed in in thick thick graph
graph style. style. Notice Notice that that for for XX 66
.52359878,.52359878,YY11YY22 YY33..Y
Y33 cos Xcos X sin Xsin X Y
Y22
Y Y11
Y
Y33YY11YY22..
Y
Y22 sin X,sin X, cos X,
cos X,
Y Y11
x x, cos, cos x x sinsin x x
..y
y sinsin x x..
y
y cos xcos x
y
y coscos x x sinsin x x,,
3344 ,, 44
,,
22 ,,33
,,
3344 ,,22
,,
,,11
4
4 x x5544 x
x 4
4 00,, ssoo x x 4
4 andand x x 4
4 ,, soso x x 55 4 4 ..
x x 44
..
4 4
cc 22 b
b 1,1, y
y 22 cotcot
x x 44
..y y
0 0
– –33
1 1
cc == ––22 x x
y
y== ––22––cotcot
( (
x x–– 44) )
5 5
4 4 3 3
4 4
2 2
4 4 –
– 2 2
– 3 – 3
4 – 4 –
4 4
Figure 53 Figure 53
Figure 54 Figure 54
Now try Exercise 61.
Now try Exercise 61.
2 2
– –22 –
–22 22
2 2
– –22 –
–22 22
2 2
– –22 –
–22 22
Concept Check
Concept Check In Exercises 1 In Exercises 1 – –6, match each function with its graph from choices A6, match each function with its graph from choices A – – F.F.
1
Graph each function over a one-period interval. See Examples 1 and 2.
Graph each function over a one-period interval. See Examples 1 and 2.
7
Graph each function over a one-period interval. See Examples 3 Graph each function over a one-period interval. See Examples 3 – – 5.5.
2
Graph each function over a two-period interval. See Examples 6 and 7.
Graph each function over a two-period interval. See Examples 6 and 7.
3
6.4 Graphs of the Other Circular FunctionsGraphs of the Other Circular Functions 585585
1.
6.4 Exercises Exercises
3
Concept Check
Concept Check In Exercises 47 In Exercises 47 – – 52, tell whether each statement is52, tell whether each statement is truetrue or or false.false. If If false, tell why.
false, tell why.
47.
47. The smallest positive numberThe smallest positive numberk k for for which which is is an an asymptote asymptote for for the the tangent tangent func- func-tion
tion is is ..
48.
48. The smallest positive numberThe smallest positive number k k for for which which is is an an asymptote asymptote for for the the cotangentcotangent function
function is is ..
49.
49. The tangent and secant functions are undeThe tangent and secant functions are unde fifined for the same values.ned for the same values.
50.
50. The secant and cosecant functions are undeThe secant and cosecant functions are unde fifined for the same values.ned for the same values.
51.
51. The The graph graph of of in in Figure Figure 45 45 suggests suggests that that for for allall x xin thein the domain of tan
domain of tan x x..
52.
52. The The graph graph of of inin Figure Figure 40 40 suggests suggests that that for for allall x xin thein the domain of sec
domain of sec x x..
53.
53. Concept CheckConcept Check If If cc is any number, then how many solutions does the equationis any number, then how many solutions does the equation have
have in in the the interval interval ??
54.
54. Concept CheckConcept Check If If cc is is any any number number such such that that , , then then how how many many solu- solu-tions
tions does does the the equation equation have have over over the the entire entire domain domain of of the the secant secant function?function?
55.
55. Consider the function deConsider the function defifined ned by by .. What What is is the the domain domain of of f f ??
What is its range?
What is its range?
56.
56. Consider the function deConsider the function defifined ned by by .. What What is is the the domain domain of of gg??
What is its range?
What is its range?
(Modeling)
(Modeling) Solve each problem.Solve each problem.
57.
57. Distance of a Rotating Beacon Distance of a Rotating Beacon A rotating beacon is located at pointA rotating beacon is located at point A Anext to anext to a long wall. (See the figure.) The beacon is 4 m from the wall. The distance
wheret t is time is time measured measured in in seconds seconds since since the the beacon beacon started started rotating. rotating. (When (When ,, the beacon is aimed at point
the beacon is aimed at point R R. When the beacon is aimed to the right of . When the beacon is aimed to the right of R R, the, the value of
value of d d is positive;is positive; d d is negative if the beacon is aimed to the left of is negative if the beacon is aimed to the left of R R.) Find.) Findd d for each time.
for each time.
(a)
(e) Why is .25 a meaningless valueWhy is .25 a meaningless value for
586 CHAPTER 6CHAPTER 6 The Circular Functions and Their Graphs The Circular Functions and Their Graphs
1
Answer graphs for odd-numbered Answer graphs for odd-numbered Exercises 33
Exercises 33––45 are included on45 are included on page A-39 of the answer section page A-39 of the answer section at the back of the text.
at the back of the text.
x
6.4
6.4 Graphs of the Other Circular FunctionsGraphs of the Other Circular Functions 587587
3
47. truetrue 48.48. false; The smallestfalse; The smallest such
suchk k is is .. 49.49. truetrue 50.50. false;false;
Secant values are unde
Secant values are undefifined whenned when , while cosecant , while cosecant values are unde
values are undefifined whenned when .
. 51.51. false;false;
for all
for all x xin thein the domain.
domain. 52.52. truetrue 53.53. fourfour 54.
54. nonenone tan
Findaafor each time.for each time.
((aa)) ((bb)) ((cc))
59.
59. Simultaneously Simultaneously graph graph and and in in the the window window by by with with aa graphing calculator. Write a sentence or two describing the
graphing calculator. Write a sentence or two describing the relationship of tanrelationship of tan x xandand x
xfor smallfor small x x-values.-values.
60.
60. Between Between each each pair pair of of successive successive asymptotes, asymptotes, a a portion portion of of the the graph graph of of oror resembles a parabola. Can each of these portions actually be a parabola?
resembles a parabola. Can each of these portions actually be a parabola?
Explain.
See the discussion on addition of ordinates.
See the discussion on addition of ordinates.
61.
61. ,, 62.62. ,,