Exam (elaborations)
Solution Manual for Applied Calculus for The Managerial, Life and Social Sciences, 1st Edition
- Course
- Institution
Solution Manual for Applied Calculus for The Managerial, Life and Social Sciences, 1st Edition
[Show more]
Content preview
CHAPTER 1EXERCISES 1.1, page 9
1. The statement is false because −3 is greater than −20. (See the number line that
follows).
2. The statement is true because −5 is equal to −5.
3. The statement is false because 2/3 [which is equal to (4/6)] is less than 5/6.
4. The statement is false because −5/6 (which is −10/12) is greater than −11/12.
5. The interval (3,6) is shown on the number line that follows. Note that this is an
open interval indicated by ( and ).
x
0 3 6
6. The interval (−2,5] is shown on the number line that follows.
7. The interval [−1,4) is shown on the number line that follows. Note that this is a
half-open interval indicated by [ (closed) and ) (open).
x
–1 4
8. The closed interval [−6/5, −1/2] is shown on the number line that follows.
9. The infinite interval (0,∞) is shown on the number line that follows.
x
0
1 1 Preliminaries
,10. The infinite interval (−∞,5] is shown in the number line that follows.
11. First, 2x + 4 < 8 (Add −4 to each side of the inequality.)
Next, 2x < 4 (Multiply each side of the inequality by 1/2)
and x < 2.
We write this in interval notation as ( −∞,2).
12. −6 > 4 + 5x ⇒ −6 − 4 > 5x ⇒ −10 > 5x ⇒ −2 > x or x < −2.
We write this in interval notation as (−∞,−2).
13. We are given the inequality −4x > 20.
Then x < −5. (Multiply both sides of the inequality by −1/4
and reverse the sign of the inequality.)
We write this in interval notation as (−∞,−5].
14. −12 < −3x ⇒ 4 > x, or x < 4. We write this in interval notation as (−∞,4].
15. We are given the inequality −6 < x − 2 < 4.
First −6 + 2 < x < 4 + 2 (Add +2 to each member of the inequality.)
and −4 < x < 6,
so the solution set is the open interval (−4,6).
16. We add −1 to each member of the given double inequality 0 < x + 1 < 4 to
obtain
−1 < x < 3,
and the solution set is [−1,3].
17. We want to find the values of x that satisfy the inequalities
x + 1 > 4 or x + 2 < −1.
Adding −1 to both sides of the first inequality, we obtain
x + 1 − 1 > 4 − 1,
or x > 3.
Similarly, adding −2 to both sides of the second inequality, we obtain
x + 2 − 2 < −1 − 2,
or x < −3.
Therefore, the solution set is (−∞,−3) ∪ (3,∞).
18. We want to find the values of x that satisfy the inequalities
x + 1 > 2 or x − 1 < −2.
Solving these inequalities, we find that
1 Preliminaries 2
, x > 1 or x < −1,
and the solution set is (−∞,−1) ∪ (1,∞).
19. We want to find the values of x that satisfy the inequalities
x + 3 > 1 and x − 2 < 1.
Adding −3 to both sides of the first inequality, we obtain
x + 3 − 3 > 1 − 3,
or x > −2.
Similarly, adding 2 to each side of the second inequality, we obtain
x < 3,
and the solution set is (−2,3).
20. We want to find the values of x that satisfy the inequalities
x − 4 < 1 and x + 3 > 2.
Solving these inequalities, we find that
x < 5 and x > −1, and the solution set is (−1,5].
21. −6 + 2 = 4. 22. 4 + −4 = 4 + 4 = 8.
−12 + 4 −8 0.2 − 14
. −12
.
23. = = 2. 24. = = 15
..
16 − 12 4 . − 2.4 −0.8
16
25. 3 −2 + 3 − 3 = 3 ( 2 ) + 3 3 = 5 3 . 26. −1 + 2 −2 = 1 + 2 2 .
27. π − 1 + 2 = π − 1 + 2 = π + 1. 28. π − 6 − 3 = 6 − π − 3 = 3 − π .
29. 2 − 1 + 3 − 2 = 2 − 1 + 3 − 2 = 2.
30. 2 3 − 3 − 3 − 4 = 2 3 − 3 − (4 − 3 ) = 3 3 − 7.
31. False. If a > b, then −a < − b, −a + b < −b + b, and b − a < 0.
32. False. Let a = −2 and b = −3. Then a/b = −2/−3 < 1.
33. False. Let a = −2 and b = −3. Then a2 = 4 and b2 = 9, and 4 < 9. Note that we
only need to provide a counterexample to show that the statement is not always
true.
34. False. Let a = −2 and b = −3. Then 1/a = −1/2 and 1/b = −1/3 and −1/2 < −1/3.
3 1 Preliminaries
, 35. True. There are three possible cases.
Case 1 If a > 0, b > 0, then a3 > b3, since a3 − b3 = (a − b)(a2 + ab + b2) > 0.
Case 2 If a > 0, b < 0, then a3 > 0 and b3 < 0 and it follows that a3 > b3.
Case 3 If a < 0 and b < 0, then a3 − b3 = (a − b)(a2 + ab + b2) > 0, and we see
that a3 > b3. (Note that (a − b) > 0 and ab > 0.)
36. True. If a > b, then it follows that −a < −b because an inequality symbol is
reversed when both sides of the inequality are multiplied by a negative number.
37. False. Take a = −2, then − a = − ( −2) = 2 = 2 ≠ a.
38. True. If b < 0, then b2 > 0, and b 2 = b 2 .
39. True. If a − 4 < 0, then a − 4 = 4 − a = 4 − a . If a − 4 > 0, then
4 − a = a − 4 = a − 4.
40. False. Let a = −2, then a + 1 = −2 + 1 = −1 = 1 ≠ −2 + 1 = 3.
41. False. Take a = 3, b = −1. Then a + b = 3 − 1 = 2 ≠ a + b = 3 + 1 = 4.
42. False. Take a = 3, b = −1. Then a − b = 4 ≠ a − b = 3 − (1) = 2.
43. 272/3 = (33)2/3 = 32 = 9.
FG 1 IJ = 1 = 1 .
44. 8− =
H 8 K 2 16
4/3 4
FG 1 IJ 0
45.
H 3K = 1. Recall that any number raised to the zero power is 1.
46. (71/2)4 = 74/2 = 72 = 49.
LMFG 1IJ OP
1/ 3 −2
F 1I
=G J
−2
47.
MNH 8K PQ H 2K = (2 2 ) = 4.
LMFG − 1IJ OP
2 −3
FG 1 IJ −3
48.
MNH 3K PQ =
H 9K = (9) 3 = 729.
1 Preliminaries 4
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller solutions. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $17.99. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
81531 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now