SOLUTION MANUAL
THOMAS' CALCULUS, SI UNITS, 15TH EDITION JOEL R. HASS
CHRISTOPHER E. HEIL MAURICE D. WEIR
CHAPTER 1-19
CHAPTER 1 FUNCTIONS
1.1 FUNCTIONS AND THEIR GRAPHS
1. domain (, ); range [1, ) 2. domain [0, ); range (, 1]
3. domain [2, ); y in range and y 5 x 10 0 y can be any positive real number range [0, ).
4. domain (, 0] [3, ); y in range and y x 2 3x 0 y can be any positive real number
range [0, ).
5. domain (, 3) (3, ); y in range and y 3 4 t , now if t 3 3 t 0 3 4 t 0, or if t 3
3 t 0 3 4 t 0 y can be any nonzero real number range (, 0) (0, ).
6. domain (, 4) ( 4, 4) (4, ); y in range and y 2 , now if t 4 t 2 16 0 2 0, or if
t 2 16 t 2 16
4 t 4 16 t 2 16 0 16
2 2 , or if t 4 t 16 0
2 2 0 y can be any nonzero
t 2 16 t 2 16
real number range (, 18 ] (0, ).
7. (a) Not the graph of a function of x since it fails the vertical line test.
(b) Is the graph of a function of x since any vertical line intersects the graph at most once.
8. (a) Not the graph of a function of x since it fails the vertical line test.
(b) Not the graph of a function of x since it fails the vertical line test.
2
9. base x; (height)2 2x x 2 height 23 x; area is a( x) 12 (base)(height) 12 ( x) 23 x 43 x 2 ;
perimeter is p( x) x x x 3x.
10. s side length s 2 s 2 d 2 s d ; and area is a s 2 a 12 d 2 .
2
11. Let D diagonal length of a face of the cube and the length of an edge. Then 2 D2 d 2 and
2 2 3/2 3
D2 2 2 3 2 d 2 d . The surface area is 6 2
6d3 2d 2 and the volume is 3
d3 d .
3 3 3
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1
,2 Chapter 1 Functions
12. The coordinates of P are x, x so the slope of the line joining P to the origin is m x
x
1 ( x 0).
x
Thus, x, x 1
m2
, 1
m .
13. 2 x 4 y 5 y 12 x 54 ; L ( x 0)2 ( y 0)2 x 2 ( 12 x 54 )2 x 2 14 x 2 54 x 16
25
20 x 2 20 x 25 20 x 2 20 x 25
54 x 2 54 x 16
25
16
4
14. y x 3 y 2 3 x; L ( x 4) 2 ( y 0) 2 ( y 2 3 4) 2 y 2 ( y 2 1) 2 y 2
y4 2 y2 1 y2 y4 y2 1
15. The domain is ( , ). 16. The domain is ( , ).
17. The domain is ( , ). 18. The domain is ( , 0].
19. The domain is (, 0) (0, ). 20. The domain is (, 0) (0, ).
21. The domain is 22. The range is [5, ) .
(, 5) (5, 3] [3, 5) (5, ).
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, Section 1.1 Functions and Their Graphs 3
23. Neither graph passes the vertical line test.
(a) (b)
24. Neither graph passes the vertical line test.
(a) (b)
x y 1 y 1 x
x y 1 or or
x y 1 y 1 x
25. x 0 1 2 26. x 0 1 2
y 0 1 0 y 1 0 0
4 x , x 1 , x 0
2 1
27. F ( x) 28. G ( x) x
x 2 x, x 1
2
x, 0 x
29. (a) Line through (0, 0) and (1, 1): y x; Line through (1, 1) and (2, 0): y x 2
x, 0 x 1
f ( x)
x 2, 1 x 2
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, 4 Chapter 1 Functions
2, 0 x 1
0, 1 x 2
(b) f ( x)
2, 2 x3
0, 3 x 4
30. (a) Line through (0, 2) and (2, 0): y x 2
0 1
Line through (2, 1) and (5, 0): m 5 2 31 13 , so y 1 ( x 2) 1 1 x 5
3 3 3
x 2, 0 x 2
f ( x) 1
3 x 3 , 2 x 5
5
3 0
(b) Line through (1, 0) and (0, 3): m 0 (1) 3, so y 3x 3
1 3
Line through (0, 3) and (2, 1) : m 2 0 24 2, so y 2 x 3
3x 3, 1 x 0
f ( x)
2 x 3, 0 x 2
31. (a) Line through (1, 1) and (0, 0): y x
Line through (0, 1) and (1, 1): y 1
0 1
Line through (1, 1) and (3, 0): m 3 1 21 12 , so y 12 ( x 1) 1 12 x 23
x 1 x 0
f ( x) 1 0 x 1
1 1 x 3
2 x 2
3 1x 2 x 0
2
(b) Line through ( 2, 1) and (0, 0): y 12 x f ( x) 2 x 2 0 x 1
Line through (0, 2) and (1, 0): y 2 x 2 1 1 x 3
Line through (1, 1) and (3, 1): y 1
1 0
32. (a) Line through T2 , 0 and (T, 1): m T (T /2) T2 , so y T2 x T2 0 T2 x 1
0, 0 x T2
f ( x)
T x 1, 2 x T
2 T
A, 0 x T
2
A, T x T
2
(b) f ( x)
A, T x 32T
A, 32T x 2T
33. (a) x 0 for x [0, 1) (b) x 0 for x (1, 0]
34. x x only when x is an integer.
35. For any real number x, n x n 1, where n is an integer. Now: n x n 1 (n 1) x n.
By definition: x n and x n x n. So x x for all real x.
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