Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
Chapter 9.1, Problem 3P
To determine
The components of the
To determine
The value of
To determine
The unit vector
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7.
Define the sequence {b} by
bo = 0
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= 2
8.
bn=4bn-1-4bn-2 for n ≥ 2
(a) Give the first five terms of this sequence.
(b) Prove: For all n = N, bn = 2nn.
Let a Rsuch that a 1, and let nЄ N. We're going to derive a formula for
Σoa without needing to prove it by induction. Tip: it can be helpful to use C1+C2+...+Cn
notation instead of summation notation when working this out on scratch paper.
(a) Take a a² and manipulate it until it is in the form Σ.a.
i=0
(b) Using this, calculate the difference between a Σ0 a² and Σ0 a², simplifying away the
summation notation.
i=0
(c) Now that you know what (a – 1) Σ0 a² equals, divide both sides by a − 1 to derive the
formula for
a².
(d) (Optional, just for induction practice) Prove this formula using induction.
3.
Let A, B, and C be sets and let f: A B and g BC be functions. For
each of the following, draw arrow diagrams that illustrate the situation, and then prove the
proposition.
(a) If ƒ and g are injective, then go f is injective.
(b) If ƒ and g are surjective, then go f is surjective.
(c) If gof is injective then f is injective. Make sure your arrow diagram shows that 9 does
not need to be injective!
(d) If gof is surjective then g is surjective. Make sure your arrow diagram shows that f
does not need to be surjective!
4.
5.
6.
Let X be a set and let f: XX be a function. We say that f is an involution if
fof idx and that f is idempotent if f f = f.
(a) If f is an involution, must it be invertible? Why or why not?2
(b) If f is idempotent, must it be invertible? Why or why not?
(c) If f is idempotent and x E range(f), prove that f(x) = x.
Prove that [log3 536] 5. You proof must be verifiable by someone who does not
have access to a scientific calculator or a logarithm table (you cannot use log3 536≈ 5.7).
Define the sequence {a} by a = 2-i for i≥ 1.
(a) Give the first five terms of the sequence.
(b) Prove that the sequence is increasing.
Chapter 9 Solutions
Advanced Engineering Mathematics
Ch. 9.1 - Prob. 1PCh. 9.1 - Find the components of the vector v with initial...Ch. 9.1 - Prob. 3PCh. 9.1 - Prob. 4PCh. 9.1 - Find the components of the vector v with initial...Ch. 9.1 - Find the terminal point Q of the vector v with...Ch. 9.1 - Find the terminal point Q of the vector v with...Ch. 9.1 - Find the terminal point Q of the vector v with...Ch. 9.1 - Find the terminal point Q of the vector v with...Ch. 9.1 - Find the terminal point Q of the vector v with...
Ch. 9.1 - Let a = [3, 2, 0] = 3i + 2j; b = [−4, 6, 0] = 4i +...Ch. 9.1 - Let a = [3, 2, 0] = 3i + 2j; b = [−4, 6, 0] = 4i +...Ch. 9.1 - Prob. 13PCh. 9.1 - Let a = [3, 2, 0] = 3i + 2j; b = [−4, 6, 0] = 4i +...Ch. 9.1 - Let a = [3, 2, 0] = 3i + 2j; b = [−4, 6, 0] = 4i +...Ch. 9.1 - Let a = [3, 2, 0] = 3i + 2j; b = [−4, 6, 0] = 4i +...Ch. 9.1 - Prob. 17PCh. 9.1 - Let a = [3, 2, 0] = 3i + 2j; b = [−4, 6, 0] = 4i +...Ch. 9.1 - What laws do Probs. 12–16 illustrate?
12. (a + b)...Ch. 9.1 - Prob. 20PCh. 9.1 - Find the resultant in terms of components and its...Ch. 9.1 - Prob. 22PCh. 9.1 - Find the resultant in terms of components and its...Ch. 9.1 - Find the resultant in terms of components and its...Ch. 9.1 - Find the resultant in terms of components and its...Ch. 9.1 - Equilibrium. Find v such that p, q, u in Prob. 21...Ch. 9.1 - Find p such that u, v, w in Prob. 23 and p are in...Ch. 9.1 - Unit vector. Find the unit vector in the direction...Ch. 9.1 - Restricted resultant. Find all v such that the...Ch. 9.1 - Prob. 30PCh. 9.1 - For what k is the resultant of [2, 0, −7], [1, 2,...Ch. 9.1 - If |p| = 6 and |q| = 4, what can you say about the...Ch. 9.1 - Same question as in Prob. 32 if |p| = 9, |q| = 6,...Ch. 9.1 - Relative velocity. If airplanes A and B are moving...Ch. 9.1 - Same question as in Prob. 34 for two ships moving...Ch. 9.1 - Prob. 36PCh. 9.1 - Prob. 37PCh. 9.2 - Let a = [1, −3, 5], b = [4, 0, 8], c = [−2, 9, 1]....Ch. 9.2 - Let a = [1, −3, 5], b = [4, 0, 8], c = [−2, 9, 1]....Ch. 9.2 - Let a = [1, −3, 5], b = [4, 0, 8], c = [−2, 9, 1]....Ch. 9.2 - Let a = [1, −3, 5], b = [4, 0, 8], c = [−2, 9, 1]....Ch. 9.2 - Let a = [1, −3, 5], b = [4, 0, 8], c = [−2, 9, 1]....Ch. 9.2 - Let a = [1, −3, 5], b = [4, 0, 8], c = [−2, 9, 1]....Ch. 9.2 - Let a = [1, −3, 5], b = [4, 0, 8], c = [−2, 9, 1]....Ch. 9.2 - Prob. 8PCh. 9.2 - Prob. 9PCh. 9.2 - Let a = [1, −3, 5], b = [4, 0, 8], c = [−2, 9, 1]....Ch. 9.2 - Prob. 11PCh. 9.2 - What does u • v = u • w imply if u = 0? If u ≠...Ch. 9.2 - Prove the Cauchy–Schwarz inequality.
Ch. 9.2 - Verify the Cauchy–Schwarz and triangle...Ch. 9.2 - Prob. 15PCh. 9.2 - Triangle inequality. Prove Eq. (7). Hint. Use Eq....Ch. 9.2 - Prob. 17PCh. 9.2 - Prob. 18PCh. 9.2 - Prob. 19PCh. 9.2 - Prob. 20PCh. 9.2 - Prob. 21PCh. 9.2 - Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0,...Ch. 9.2 - Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0,...Ch. 9.2 - Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0,...Ch. 9.2 - What will happen to the angle in Prob. 24 if we...Ch. 9.2 - Prob. 26PCh. 9.2 - Addition law. cos (α − β) = cos α cos β + sin α...Ch. 9.2 - Prob. 28PCh. 9.2 - Prob. 29PCh. 9.2 - Prob. 30PCh. 9.2 - Prob. 31PCh. 9.2 - Prob. 32PCh. 9.2 - Prob. 33PCh. 9.2 - Prob. 34PCh. 9.2 - Prob. 35PCh. 9.2 - Prob. 36PCh. 9.2 - Prob. 37PCh. 9.2 - Prob. 38PCh. 9.2 - Prob. 39PCh. 9.2 - Prob. 40PCh. 9.3 - Prob. 1PCh. 9.3 - Prob. 2PCh. 9.3 - Prob. 3PCh. 9.3 - Prob. 4PCh. 9.3 - Prob. 5PCh. 9.3 - Prob. 6PCh. 9.3 - Prob. 7PCh. 9.3 - Prob. 8PCh. 9.3 - Prob. 9PCh. 9.3 - Prob. 11PCh. 9.3 - Prob. 12PCh. 9.3 - Prob. 13PCh. 9.3 - Prob. 14PCh. 9.3 - Prob. 15PCh. 9.3 - Prob. 16PCh. 9.3 - Prob. 17PCh. 9.3 - Prob. 18PCh. 9.3 - Prob. 19PCh. 9.3 - Prob. 20PCh. 9.3 - Prob. 21PCh. 9.3 - Prob. 22PCh. 9.3 - Prob. 23PCh. 9.3 - Prob. 25PCh. 9.3 - Prob. 26PCh. 9.3 - Prob. 27PCh. 9.3 - Prob. 28PCh. 9.3 - Prob. 29PCh. 9.3 - Prob. 30PCh. 9.3 - Prob. 31PCh. 9.3 - Prob. 32PCh. 9.3 - Prob. 33PCh. 9.3 - Prob. 34PCh. 9.4 - Prob. 1PCh. 9.4 - Prob. 2PCh. 9.4 - Prob. 3PCh. 9.4 - Prob. 4PCh. 9.4 - Prob. 5PCh. 9.4 - Prob. 6PCh. 9.4 - Prob. 7PCh. 9.4 - Prob. 9PCh. 9.4 - Prob. 10PCh. 9.4 - Prob. 11PCh. 9.4 - Prob. 12PCh. 9.4 - Prob. 13PCh. 9.4 - Prob. 14PCh. 9.4 - Prob. 15PCh. 9.4 - Prob. 16PCh. 9.4 - Prob. 17PCh. 9.4 - Prob. 18PCh. 9.4 - Prob. 19PCh. 9.4 - Prob. 20PCh. 9.4 - Prob. 22PCh. 9.4 - Prob. 23PCh. 9.4 - Prob. 24PCh. 9.5 - Prob. 1PCh. 9.5 - Prob. 2PCh. 9.5 - Prob. 3PCh. 9.5 - Prob. 4PCh. 9.5 - Prob. 5PCh. 9.5 - Prob. 6PCh. 9.5 - Prob. 7PCh. 9.5 - Prob. 8PCh. 9.5 - Prob. 9PCh. 9.5 - Prob. 10PCh. 9.5 - Prob. 11PCh. 9.5 - Prob. 12PCh. 9.5 - Prob. 13PCh. 9.5 - Prob. 14PCh. 9.5 - Prob. 15PCh. 9.5 - Prob. 16PCh. 9.5 - Prob. 17PCh. 9.5 - Prob. 18PCh. 9.5 - Prob. 19PCh. 9.5 - Prob. 20PCh. 9.5 - Prob. 21PCh. 9.5 - r(t) = [10 cos t, 1, 10 sin t], P: (6, 1, 8)Ch. 9.5 - r(t) = [cos t, sin t, 9t], P: (1, 0, 18)Ch. 9.5 - Prob. 27PCh. 9.5 - Prob. 29PCh. 9.5 - Prob. 30PCh. 9.5 - Prob. 31PCh. 9.5 - Prob. 32PCh. 9.5 - Prob. 33PCh. 9.5 - Prob. 34PCh. 9.5 - Prob. 35PCh. 9.5 - Prob. 36PCh. 9.5 - Prob. 37PCh. 9.5 - Prob. 38PCh. 9.5 - Prob. 43PCh. 9.5 - Prob. 44PCh. 9.5 - Prob. 45PCh. 9.5 - Prob. 46PCh. 9.5 - CURVATURE AND TORSION
47. Circle. Show that a...Ch. 9.5 - Prob. 48PCh. 9.5 - Prob. 49PCh. 9.5 - Prob. 50PCh. 9.5 - Prob. 51PCh. 9.5 - Prob. 52PCh. 9.5 - Prob. 53PCh. 9.5 - Prob. 54PCh. 9.5 - Prob. 55PCh. 9.7 - Prob. 1PCh. 9.7 - Prob. 2PCh. 9.7 - Prob. 3PCh. 9.7 - Prob. 4PCh. 9.7 - Prob. 5PCh. 9.7 - Prob. 6PCh. 9.7 - Prob. 7PCh. 9.7 - Prob. 8PCh. 9.7 - Prob. 9PCh. 9.7 - Prob. 10PCh. 9.7 - Prob. 11PCh. 9.7 - Prob. 12PCh. 9.7 - Prob. 13PCh. 9.7 - Prob. 14PCh. 9.7 - Prob. 15PCh. 9.7 - Prob. 16PCh. 9.7 - Prob. 17PCh. 9.7 - Prob. 18PCh. 9.7 - Prob. 19PCh. 9.7 - Prob. 20PCh. 9.7 - Prob. 21PCh. 9.7 - Prob. 22PCh. 9.7 - Prob. 23PCh. 9.7 - Prob. 24PCh. 9.7 - Prob. 25PCh. 9.7 - Prob. 26PCh. 9.7 - Prob. 28PCh. 9.7 - Prob. 29PCh. 9.8 - Prob. 1PCh. 9.8 - Prob. 2PCh. 9.8 - Prob. 3PCh. 9.8 - Prob. 4PCh. 9.8 - Prob. 5PCh. 9.8 - Prob. 6PCh. 9.8 - Prob. 7PCh. 9.8 - Prob. 8PCh. 9.8 - CAS EXPERIMENT. Visualizing the Divergence. Graph...Ch. 9.8 - Prob. 11PCh. 9.8 - Prob. 12PCh. 9.8 - Prob. 13PCh. 9.8 - Prob. 14PCh. 9.8 - Prob. 15PCh. 9.8 - Prob. 16PCh. 9.8 - Prob. 17PCh. 9.8 - Prob. 18PCh. 9.8 - Prob. 19PCh. 9.8 - Prob. 20PCh. 9.9 - Prob. 1PCh. 9.9 - Prob. 2PCh. 9.9 - Prob. 3PCh. 9.9 - Prob. 4PCh. 9.9 - Prob. 5PCh. 9.9 - Prob. 6PCh. 9.9 - Prob. 7PCh. 9.9 - Prob. 8PCh. 9.9 - Prob. 9PCh. 9.9 - Prob. 10PCh. 9.9 - Prob. 11PCh. 9.9 - Prob. 12PCh. 9.9 - Prob. 13PCh. 9.9 - Prob. 15PCh. 9.9 - Prob. 16PCh. 9.9 - Prob. 17PCh. 9.9 - Prob. 18PCh. 9.9 - Prob. 19PCh. 9.9 - Prob. 20PCh. 9 - Prob. 1RQCh. 9 - Prob. 2RQCh. 9 - Prob. 3RQCh. 9 - Prob. 4RQCh. 9 - Prob. 5RQCh. 9 - Prob. 6RQCh. 9 - Prob. 7RQCh. 9 - Prob. 8RQCh. 9 - Prob. 9RQCh. 9 - Prob. 11RQCh. 9 - Prob. 12RQCh. 9 - Prob. 13RQCh. 9 - Prob. 14RQCh. 9 - Prob. 15RQCh. 9 - Prob. 16RQCh. 9 - Prob. 17RQCh. 9 - Prob. 18RQCh. 9 - Prob. 19RQCh. 9 - Prob. 20RQCh. 9 - Prob. 21RQCh. 9 - Prob. 22RQCh. 9 - Prob. 23RQCh. 9 - Prob. 24RQCh. 9 - Prob. 25RQCh. 9 - Prob. 26RQCh. 9 - Prob. 27RQCh. 9 - Prob. 28RQCh. 9 - Prob. 29RQCh. 9 - Prob. 30RQCh. 9 - Prob. 31RQCh. 9 - Prob. 32RQCh. 9 - Prob. 33RQCh. 9 - Prob. 34RQCh. 9 - Prob. 35RQCh. 9 - Prob. 36RQCh. 9 - Prob. 37RQCh. 9 - Prob. 38RQCh. 9 - Prob. 39RQCh. 9 - Prob. 40RQ
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