Transcript Slide 1
Homework, Page 562
Let u 2, 1 , v 4, 2 , and w 1, 3 . Find the expression.
1. u v
u v 2 4 , 1 2 2, 3
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Slide 6- 1
Homework, Page 562
Let u 2, 1 , v 4, 2 , and w 1, 3 . Find the expression.
5.
uv
u v 2 4 1 2 8 2 6
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Slide 6- 2
Homework, Page 562
Let A = (2, –1), B = (3, 1), C = (–4, 2), and D = (1, –5). Find the
component form and magnitude of the vector.
9.
AC BD
AC 4 2 , 2 1 6,3
BD 1 3 , 5 1 2, 6
AC BD 6,3 2, 6 8, 3
AC BD
8
2
3 64 9
2
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73
Slide 6- 3
Homework, Page 562
Find a the direction angles of u and v and
b the angle between u and v.
13. u 4,3 , v 2,5
3
1 5
u tan
0.644 v tan
1.190
4
2
v u 1.190 0.644 0.546 OR
1
2
2
2
2
29
v
2
5
5
u
4
3
u v 4 2 3 5 23
cos
1
23
cos
0.5467
5 29
uv
uv
1
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Slide 6- 4
Homework, Page 562
Convert the polar coordinates to rectangular coordinates.
2,
17.
4
2,
4 2 cos 4 ,sin 4
2
2
2
, 2
2 2
2, 2
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Slide 6- 5
Homework, Page 562
Rectangular coordinates of point P are given. Find all polar coordinates of P
that satisfy: (a) 0 ≤θ ≤2π (b) –π ≤ θ ≤ π (c) 0 ≤ θ ≤ 4π
21. P 2, 3
3
r 2 3 13 tan 2 0.983
2.159 , 2 5.300 ,3 8.442 , 4 11.584
a P 13 cos 2.159,sin 2.159 , P 13 cos5.300,sin 5.300
2
2
1
b P 13 cos 0.983 ,sin 0.983 , P 13 cos 2.159,sin 2.159
c P 13 cos 2.159,sin 2.159 , P 13 cos5.300,sin 5.300
P 13 cos8.442,sin8.442 , P 13 cos11.584,sin11.584
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Slide 6- 6
Homework, Page 562
Eliminate the parameter t and identify the graph.
25.
x 3 5t , y 4 3t
3 x
x 3 5t , y 4 3t 5t 3 x t
5
3 x
y 4 3
5 y 20 3 3 x 5 y 20 9 3x
5
29 3
3
y
x The graph is a line with slope and
5 5
5
y -intercept 5.8.
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Slide 6- 7
Homework, Page 562
Eliminate the parameter t and identify the graph.
x e2t 1, y et
29.
x e 1, y e x e
2t
t
t
2
2
x
y
1
1
The graph is a parabola opening to the right.
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Slide 6- 8
Homework, Page 562
Imaginary Axis
Z1
4
Refer to the complex number shown in the figure.
33. If z1 = a + bi, find a, b, and |z1|.
a 3 , b 4 , z1
3
2
4 25 5
2
Real Axis
-3
a 3, b 4, z1 5
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Slide 6- 9
Homework, Page 562
Write the complex number in standard form.
37.
4
4
2.5 cos
i sin
3
3
4
4
5
2.5 cos
i sin
3
3
2
1 5
2 2
3
i
2
5 5 3
i
4
4
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Slide 6- 10
Homework, Page 562
Write the complex number in trigonometric form where 0 ≤ θ ≤ 2π.
Then write three other possible trigonometric forms for the number.
41. 3 5i
r 3 5
2
2
5
34 tan
1.030
3
1
3 5i 34 cos 1.030 i sin 1.030
34 cos 2.111 i sin 2.111
34 cos 5.253 i sin 5.253 2
34 cos 4.172 i sin 4.172
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Slide 6- 11
Homework, Page 562
Use DeMoivre’s Theorem to find the indicated power of the complex
number. Write the answer in (a) trigonometric form and (b) standard form.
45. 3 cos i sin
4
4
5
5
5
5
i sin
3 cos 4 i sin 4 243 cos
4
4
243 2 243 2
i
2
2
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Slide 6- 12
Homework, Page 562
Find and graph the nth roots of the complex number for the specified value
of n.
49. 3 3i, n 4
3
r 3 3 3 2 tan
3 4
z 3 2 cos i sin
4
4
2
2
1
Continued on next slide.
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Slide 6- 13
Homework, Page 562
49. Cont’d
z1 4 3
z2 4 3
z3 4 3
z4 4 3
4
2 cos 4 i sin 4 3 2 cos 16 i sin 16
4
4
2
2 4
9
9
3
2
cos
i
sin
2 cos 4
i sin 4
16
16
4
4
4
4 4
17
17
3 2 cos 16 i sin 16
2 cos 4
i sin 4
4
4
6
6 4
25
25
3 2 cos 16 i sin 16
2 cos 4
i sin 4
4
4
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Slide 6- 14
Homework, Page 562
49. Continued
4
z1 18 cos i sin
16
16
9
9
z2 18 cos
i sin
16
16
17
17
4
z3 18 cos
i sin
16
16
25
25
4
z4 18 cos
i sin
16
16
y
4
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x
Slide 6- 15
Homework, Page 562
Decide whether the graph of the polar function appears among the
four.
53. r 3sin 4
b.
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Slide 6- 16
Homework, Page 562
Decide whether the graph of the polar function appears among the
four.
57. r 2 2sin
Not shown.
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Slide 6- 17
Homework, Page 562
Convert the polar equation to rectangular form and identify the
graph.
r 2
r 2 r 2 2 2 x2 y 2 4
The graph is a circle of radius 2 centered at the origin.
61.
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Slide 6- 18
Homework, Page 562
Convert the rectangular equation to polar form and graph the polar
equation.
y 4
65.
y 4 r sin 4
4
r
r 4csc
sin
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Slide 6- 19
Homework, Page 562
Analyze the graph of the polar curve.
69. r 2 5sin
Domain: All real numbers
Range: –3 ≤ r ≤ 7
Continuity: Continuous
Symmetry: Symmetric about the y-axis.
Boundedness: Bounded
Maximum r-value: 7
Asymptotes: None
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Slide 6- 20
Homework, Page 562
73. a Explain why r a sec
is a polar form of x a.
Multiply both sides of r a sec by cos and the equation
becomes r cos a and r cos x.
b Explain why r a csc is a polar form of y a.
Multiply both sides of r a csc by sin and the equation
becomes r sin a and r sin y.
b
is a polar form
sin m cos
for the line. What is the domain of r ?
c Let y mx b. Prove that r
Substituting y mx b becomes r sin mr cos b
r sin mr cos b r sin m cos b
b
r
sin m cos Domain : : m 1
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Slide 6- 21
Homework, Page 562
73. d Illustrate the result of part c by graphing the line
y 2 x 3 using the polar form from part c .
3
y 2x 3 r
sin 2cos
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Slide 6- 22
Homework, Page 562
77. A 3,000-lb car is parked on a street that makes an angle
of 16º With the horizontal.
(a) Find the force required to keep the car from rolling
down the hill.
F 3000 sin16 lb
(b) Find the component of the force perpendicular to the
ground.
F 3000 cos16 lb
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Slide 6- 23
Homework, Page 562
81. The lowest point on a Ferris wheel of radius 40-ft is
10-ft above the ground, and the center is on the y-axis.
Find the parametric equation for Henry’s position as a
function of time t in seconds, if his starting position (t = 0)
is the point (0, 10) and the wheel turns at a rate of one
revolution every 15 sec.
2
2
2
15 b
x 40sin bt p
x 40sin
t
b
15
15
2
y 50 40cos bt y 50 40cos
t
15
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Slide 6- 24
Homework, Page 562
85. Diego releases a baseball 3.5-ft above the ground with
an initial velocity of 66-fps at an angle of 12º with the
horizontal. How many seconds after the ball is thrown
will it hit the ground? How far from Diego will the ball be
when it hits the ground?
x vo cos t , y 3.5 vo sin t 16t 2
x 66cos12 t , y 3.5 66sin12 t 16t 2
The ball hits the ground about
1.06 secs after it is thrown,
68.431 ft from Diego.
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Slide 6- 25
Homework, Page 562
89. A 60-ft radius Ferris wheel turns counterclockwise
one revolution every 12 sec. Sam stands at a point 80 ft to
the left of the bottom (6 o’clock) of the wheel. At the
instant Kathy is at 3 o’clock, Sam throws a ball with an
initial velocity of 100 fps and an angle of 70º with the
horizontal. He releases the ball at the same height as the
bottom of the Ferris wheel. Find the minimum distance
between the ball and Kathy. xball 80 100 cos 70 t
360
yball 100sin 70 t 16t 2
p 12 b
30
12
xKathy 60 cos 30t yKathy 60 60sin 30t
distance
x
ball
xKathy yball yKathy
2
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2
Slide 6- 26
Homework, Page 562
89. Continued
Minimum distance between the ball
and Kathy is about 17.654 feet
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Slide 6- 27