Calculo 2 De dos variables_9na Edición - Ron Larson

730 CAPÍTULO 10 Cónicas, ecuaciones paramétricas y coordenadas polares
730 Chapter 10 Conics, Parametric Equations, and Polar Coordinates
730 Chapter 10 Conics, Parametric Equations, and Polar Coordinates
y
91. Reloj de arena: 0 ≤ t ≤ 2 92. Lágrima: 0 ≤ t ≤ 2
91. Hourglass: 0 t 2 92. Teardrop: 0 ≤ t ≤ 2
y
91. Hourglass: sen sin 2t 0 t 2 92. Teardrop: x 2a cos 0 ≤ t t a ≤ sen sin 2 2t
y
x a sin 2t
x 2a cos t a sin 2t
1
1
x a sen sin t2t
x y 2a b cos sen sin t a sin 2t
y b sin t
y b sin t
1
y b sin yt
y b sin y
r
y
yt
r θ
x
b y
y
rr
θ
b
b
P x
b
rθ
b
P x
b
r P
x
x
Figura para 99 Figura para 100
a
a
x
a
a
Figure for 99 Figure for 100
x 100. Evolvente o involuta de un círculo La figura muestra un segmento
de of cuerda a Circle sujeto The a un figure círculo shows de a radio piece 1. of La string cuerda tiedes
100. Involute justo a circle lo suficientemente of with a Circle a radius The larga of figure one para unit. shows llegar The a piece al string lado of is opuesto string just long tied del
a
a
Figure for 99 Figure for 100
100. Involute
to enough círculo. a circle to Encontrar reach with a the radius el opposite área of que one side se unit. of cubre the The circle. cuando string Find is la just cuerda the long arease
Centroide
In En Exercises los ejercicios 93 and 9394, y find 94, the hallar centroid el centroide of the region de la enough that desenrolla is covered to reach en sentido when the opposite the contrario string side is al unwound de of las the manecillas circle. counterclockwise.
Find del the reloj. area
región Centroid bounded limitada by In the Exercises por graph la gráfica 93 of and the de 94, parametric las find ecuaciones the centroid equations paramétricas of the and region they
that is covered when the string is unwound counterclockwise.
101. (a) Usar Use a una graphing herramienta utility to de graph graficación the curve para given trazar by la curva
los bounded coordinate ejes de by coordenadas. axes. the (Use graph the (Usar of result the el of parametric resultado Exercise del 83.) equations ejercicio and 83.) the
101. (a) dada Use pora graphing utility to graph the curve given by
coordinate axes. (Use the result of Exercise 83.)
1 t
93. 94.
x
2
93. 94.
x
20 ≤ ≤ Volumen En los ejercicios 95 y 96, hallar el volumen del sólido
t 20.
1 t , y 2t
2 1 t , 20 t 20.
1 t , y 2t
x t, y 4 t
x 4 t, y t
x t, y 4 t
x 4 t, y t
2 1 t , 2 2 In Exercises 95 and 96, find the of the solid (b) Describir Describe la the gráfica graph y and confirmar la your respuesta result analytically. en forma analítica.
generado Volume formed by por In revolving Exercises revolución the 95 en region and torno 96, bounded al find eje the x by de volume la the región graphs of the limitada of solid the (b) Describe the graph and confirm your result analytically.
por formed la gráfica by revolving de las the ecuaciones region bounded dadas. (Usar by the el graphs resultado of the del
(c) Discuss the speed at which the curve is traced as t
given equations about the x-axis. (Use the result of Exercise 83.) (c)
ejercicio 83.)
Analizar Discuss la the velocidad speed at a which la cual the se traza curve la is curva traced cuando as t
given equations about the x-axis. (Use the result of Exercise 83.)
increases from 20 to 20.
t
95. x 6 cos , y 6 sen
aumenta increases de from 20 a 20 20. to 20.
102. Tractrix A person moves from the origin along the positive
95.
6 cos , y 6 sen
96. x cos , y 3 sin , a > 0
102. y- Tractrix Tractriz axis pulling Una A person a persona weight moves at se the mueve from end the of desde a origin 12-meter el origen along rope. a the lo Initially, positive largo del
96.
x cos , , y 3 sen
sin , , a > 0
y- the eje axis weight y positivo pulling is located a tirando weight at un at the peso the point end atado of 12, a al 012-meter extremo . rope. de una Initially, cuerda
97. Cycloid Use the parametric equations
the de 12 weight metros is located de largo. at Inicialmente, the point 12, el 0 peso . está situado en el
97. 97. Cicloide Cycloid Use Emplear the parametric las ecuaciones equations paramétricas
(a) In Exercise 96 of Section 8.7, it was shown that the path
(a) punto In 12, Exercise 0. 96 of Section 8.7, it was shown that the path
x a sin and y a 1 cos , a > 0
of the weight is modeled by the rectangular equation
x a a sen sin sin yand
y y a1a 1 cos cos , a , > a 0> 0
a) En of the el ejercicio weight is 96 modeled la sección by the 8.7 rectangular se mostró equation que la trayectoria
y del 12 peso ln 12
12 144 x2
para to answer responder the following. lo siguiente.
se describe
144
mediante la 144 siguiente x ecuación
to answer the following.
rectangular y 12 ln x
2
x2
a) (a) Hallar Find
dydx dx and y d 2 dydx 2 y dx 2 .
2 .
144 x
x
2
(a) Find dy dx and d 2 y dx 2 .
b) (b) Hallar Find the las ecuaciones equation of de the la tangent recta tangente line at en the el point punto where
where 0 < x
en el
y 12 ln
que
12 12. 144 Use a graphing x 2 utility to graph the
x 144 x2
(b) Find the equation of the tangent line at the point where
where 0 < x 12. Use a graphing utility to graph the
6.
rectangular equation.
6.
rectangular equation.
c) (c) Localizar Find all points todos (if los any) puntos of horizontal (si los hay) tangency.
(b)
de tangencia horizontal.
donde Use a graphing 0 < x ≤ utility 12. Usar to graph una herramienta the parametric de equations graficación
(c) Find all points (if any) of horizontal tangency.
(b) para Use representar a graphing utility la ecuación to graph rectangular. the parametric equations
(d) Determine where the curve is concave upward or concave
b) Usar una herramienta de graficación para trazar la gráfica
d)
(d)
Calcular
Determine
dónde
where
es la
the
curva
curve
cóncava
is concave
hacia
upward
arriba
or
y dónde
concave
x 12 sech
es
de
x
las
12
ecuaciones
sech t
t and y t 12 tanh
paramétricas
and y t 12 tanh t
t
downward.
12
12
cóncava
downward.
hacia abajo.
12
12
(e) Find the length of one arc of the curve.
where t 0. How
e)
(e)
Hallar
Find the
la longitud
length of
de
one
un
arc
arco
of
de
the
la
curve.
curva.
where x 12 sech How t does this graph compare with
does y this ygraph t compare 12 tanh with t the graph
98. Use the parametric equations
in part t (a)? 0.
the graph
12 Which graph (if either) do you 12 think is a
98.
98.
Emplear
Use the parametric
las ecuaciones
equations
paramétricas
in part (a)? Which graph (if either) do you think is a
donde
better representation
t ≥ 0. Comparar
of the
esta
path?
1
gráfica con la del inciso a).
x t 2 3 and y 3t
¿Qué gráfica (si hay alguna) representa mejor la trayectoria?
y y 3t 1 better representation of the path?
1
x t 2 33
and y 3t 3 t3
(c) Use the parametric equations for the tractrix to verify that
3 t3 3 t3
(c) Use the parametric equations for the tractrix to verify that
c) Emplear the distance las ecuaciones from the y-paramétricas intercept of the de tangent la tractriz line para to the verificar
point que of tangency la distancia independent la intersección of the con location el eje of y de thela
para to answer los incisos the following.
to answer the following.
the distance from the y-intercept of the tangent line to (a) Use a graphing
siguientes.
utility to graph the curve on the interval
is independent of the location of the
recta point tangente of tangency. al punto de tangencia es independiente de
(a) Emplear Use 3 a graphing tuna 3. herramienta utility to de graph graficación the curve para on trazar the la interval curva
point of tangency.
la ubicación del punto de tangencia.
en el 3 intervalo t 3. 3 ≤ t ≤ 3.
True or False? In Exercises 103 and 104, determine whether
(b) Find dy dx and d 2 y dx 2 .
True (b) Hallar Find dydx and y d 2 ydx 2 .
¿Verdadero or False? o falso? In Exercises En los ejercicios 103 and 103 104, y 104, determine determinar whether
dx d 2 y dx 2 .
the statement is true or false. If it is false, explain why or give si la
(c) Find the equation of the tangent line at the point 3, 8 the afirmación statement es is verdadera true or false. o falsa. If it Si is es false, falsa, explain explicar why por or qué giveo
(c) Hallar Find the la ecuación equation de of la the recta tangent tangente line at en the el punto point 3, 8 3 . an example that shows it is false.
3 3. .
(d) Find the length of the curve.
an dar example un ejemplo that que shows demuestre it is false. que es falsa.
(d) Hallar Find the la longitud length of de the la curve.
(e) Find the surface area
curva.
103. If x ft and y gt, then d 2 y dx 2 g t f t .
generated by revolving the curve 103. If Si x
ft
f tand y y y gt, gt, entonces then d 2 yddx 2 ydx 2 2 g tgtft.
f t .
(e) Hallar Find about el the área x-axis. surface de la area superficie generated generada by revolving por revolución the curve de la
104. The curve given by x t 3 , y t 2 has a horizontal tangent at
104. The La curva curve dada given por by x t tiene una tangente horizontal
en origin el origen because puesto dy dt que dydt 0 when t0
cuando 0. t 0.
99. Evolvente Involute the endpoint of o involuta a PCircle of a string de The círculo that involute is held La of evolvente taut a circle as it is o is unwound involuta described from by un 105. Recording Cinta de grabación Tape Another Otro method método you que could se puede use usar to solve para
curva about en the torno x-axis. al eje x.
the origin because dy dt 3 , y0
when t 2
has t a 0. horizontal tangent at
99. Involute of a Circle The involute of a circle is described by the
círculo the a spool endpoint está that descrita does P of a not string por turn that el (see extremo is figure). held taut PShow de as una it that is unwound cuerda a parametric que fromse
105. Recording Example Tape Another method you could use to solve
solucionar 5 el is ejemplo to find 5 es area encontrar of the reel el área with del an carrete inner radius con un
mantiene a representation spool that tensa does of mientras the not involute turn se (see desenrolla isfigure). de Show un carrete that a que parametric no gira Example of 5 is to find the area of the reel with inner radius
radio 0.5 interior inch and de an 0.5 outer pulgadas radius y un of radio 2 inches, exterior and de then 2 pulgadas, use the
(ver representation la figura). of Mostrar the involute que la issiguiente es una representación of 0.5 inch and an outer radius of 2 inches, and then use the
y después usar la fórmula para el área del rectángulo cuyo
paramétrica
x r cos
de la
sin
evolvente o
and
formula for the area of the rectangle where the width is 0.001
involuta
y r sin cos .
x r cos sin and y r sin cos .
formula inch. for the area of the rectangle where the width is 0.001
ancho Use es de this 0.001 method pulgadas. to determine Utilizar how este much método tape para is required determinar
fill cuánta the reel. cinta se necesita para llenar el carrete.
inch. Use this method to determine how much tape is required
x rcos sen y y rsin sen
to
to fill the reel.
6.
sin
cos .