Thomas Calculus 13th [Solutions]

828 Chapter 11 Parametric Equations and Polar Coordinates
17.
r
r 1 1, , and
2 2
dr
d
Slope at
Slope at
sin ;
r 1 1, ;
2 2
Slope r sin r cos sin r cos
r cos r sin sin cos r sin
1, is
2
1, is
2
sin
2
( 1)cos
2 2
sin cos ( 1)sin
sin
2 2 2
2
( 1)cos
2 2
sin cos ( 1)sin
2 2 2
2
1;
1
18. 0 r 1 ( 1,0), and r 1 ( 1, );
r dr cos ; Slope r sin r cos cos sin r cos cos sin r cos
d
r cos r sin cos cos r sin 2
cos r sin
cos0sin 0 ( 1)cos0
Slope at ( 1, 0) is 1;
2
cos 0 ( 1)sin 0
cos sin ( 1) cos
Slope at ( 1, ) is
2
cos ( 1)sin
1
19.
r 1 1, ;
4 4
r 1 1, ;
4 4
3 r 1 1, 3 ; 3 r 1 1, 3 ;
4 4
4 4
r dr 2cos 2 ; Slope r sin r cos 2cos 2 sin r cos
d
r cos r sin 2cos 2 cos r sin
Slope at
Slope at
1, is
4
1, is
4
Slope at 1, 3 is
4
Slope at 1, 3 is
4
2cos sin (1)cos
2 4 4
2cos cos (1)sin
2 4 4
2cos sin ( 1)cos
2 4 4
2cos cos ( 1)sin
2 4 4
3 3 3
2 4 4
3 3 3
2 4 4
2cos sin ( 1)cos
2cos cos ( 1)sin
3 3 3
2 4 4
3 3 3
2 4 4
2cos sin (1) cos
2cos cos (1)sin
1;
1;
1;
1;
20. 0 r 1 (1, 0);
r 1 1, ;
2 2
r 1 1, ; r 1 (1, );
2 2
r dr 2sin 2 ; Slope r sin r cos 2sin 2 sin r cos
d
r cos r sin 2sin 2 cos r sin
Slope at (1,0) is
Slope at
Slope at
1, is
2
1, is
2
2sin 0sin 0 cos 0
, which is undefined;
2sin 0cos 0 sin 0
2sin 2 sin ( 1)cos
2sin 2 cos ( 1)sin
Slope at (1, ) is 2sin 2 sin cos
2sin 2 cos sin
2 2 2
2 2 2
2sin 2 sin ( 1) cos
2sin 2 cos ( 1)sin
2 2 2
2 2 2
0;
0;
, which is undefined
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