Thomas Calculus 13th [Solutions]

60 Chapter 2 Limits and Continuity
13. (a)
3 3
(1 ) 12(1 ) (1 12(1))
y h h
x
h
h
2
2 3
1 3 3 12 12 ( 11) 2 3
9 3
h h h h h h h
h
h
As 0, 9 3h h 9 at P (1, 11) the slope is 9.
(b) y ( 11) ( 9)( x 1) y 11 9x 9 y 9x
2.
2
9 3 h h .
14. (a)
3 2 3 2
(2 ) 3(2 ) 4 (2 3(2) 4)
y h h
x
h
2
As 0, 3 0
y x y
h h h at P (2, 0) the slope is 0.
(b) 0 0( 2) 0.
2 3 2 2 3
8 12 6 12 12 3 4 0 3
h h h h h h h
h
h
2
3 h h .
15. (a) Q Slope of PQ
Q 1 (10,225) 650 225
20 10
Q 2 (14,375) 650 375
20 14
Q 3 (16.5, 475) 650 475
20 16.5
Q 4 (18,550) 650 550
20 18
p
t
42.5 m/sec
45.83 m/sec
50.00 m/sec
50.00 m/sec
(b) At t 20, the sportscar was traveling approximately 50 m/sec or 180 km/h.
16. (a) Q Slope of PQ
Q 1 (5, 20) 80 20
10 5
Q 2 (7,39) 80 39
10 7
Q 3 (8.5,58) 80 58
10 8.5
Q 4 (9.5,72) 80 72
10 9.5
(b) Approximately 16 m/sec
12 m/sec
p
t
13.7 m/sec
14.7 m/sec
16 m/sec
17. (a)
200
p
160
120
80
40
0
p
(b)
174 62 112
t 2014 2012 2
56 thousand dollars per year
(c) The average rate of change from 2011 to 2012 is
62 27
p
t
p
t
2012 2011
35 thousand dollars per year.
The average rate of change from 2012 to 2013 is
111 62
49 thousand dollars per year.
2013 2012
So, the rate at which profits were changing in 2012 is approximately 1 (35 49) 42 thousand dollars
2
per year.
18. (a) F( x) ( x 2)/( x 2)
x 1.2 1.1 1.01 1.001 1.0001 1
F( x ) 4.0 3.4 3.04 3.004 3.0004 3
F 4.0 ( 3)
5.0;
F 3.4 ( 3)
4.4;
x
F
x
2010 2011 2012 2013 2014
Ye ar
1.2 1
3.04 ( 3)
1.01 1
4.04;
F 3.0004 ( 3)
4.0004;
x 1.0001 1
(b) The rate of change of F( x ) at x 1 is 4.
t
x
F
x
1.1 1
3.004 ( 3)
1.001 1
4.004;
Copyright
2014 Pearson Education, Inc.