Introductory Combinatorics
5th Edition
ISBN: 9780134689616
Author: Brualdi, Richard A.
Publisher: Pearson,
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Chapter 8, Problem 36E
To determine
To prove: The Catalan number
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(4) (10 points) Evaluate
√(x² + y² + z²)¹⁄² exp[}(x² + y² + z²)²] dV
where D is the region defined by 1< x² + y²+ z² ≤4 and √√3(x² + y²) ≤ z.
Note: exp(x² + y²+ 2²)²] means el (x²+ y²+=²)²]¸
(2) (12 points) Let f(x,y) = x²e¯.
(a) (4 points) Calculate Vf.
(b) (4 points) Given x
directional derivative
0, find the line of vectors u =
D₁f(x, y) = 0.
(u1, 2) such that the
-
(c) (4 points) Let u= (1+3√3). Show that
Duƒ(1, 0) = ¦|▼ƒ(1,0)| .
What is the angle between Vf(1,0) and the vector u? Explain.
Find the missing values by solving the parallelogram shown in the figure. (The lengths of the diagonals are given by c and d. Round your answers to two decimal places.)
a
b
29
39
66.50
C
17.40
d
0
54.0
126°
a
Ꮎ
b
d
Chapter 8 Solutions
Introductory Combinatorics
Ch. 8 - Let 2n(equally spaced) points on a circle be...Ch. 8 - Prove that the number of 2-by-n arrays
that can...Ch. 8 - Write out all of the multiplication schemes for...Ch. 8 - 5. * Let m and n be nonnegative integers with n ≥...Ch. 8 - 6. Let the sequence h0, h1, … , hn, … be defined...Ch. 8 - 7. The general term hn of a sequence is a...Ch. 8 - 8. Find the sum of the fifth powers of the first n...Ch. 8 - 9. Prove that the following formula holds for the...Ch. 8 - 10. If hn is a polynomial in n of degree m, prove...Ch. 8 - 11. Compute the Stirling numbers of the second...
Ch. 8 - 12. Prove that the Stirling numbers of the second...Ch. 8 - 13. Let X be a p-element set and let Y be a...Ch. 8 - Prob. 14ECh. 8 - 15. The number of partitions of a set of n...Ch. 8 - 11. Compute the Stirling numbers of the second...Ch. 8 - 17. Compute the triangle of Stirling numbers of...Ch. 8 - Write [n]k as a polynomial in n for k = 5, 6, and...Ch. 8 - Prove that the Stirling numbers of the first kind...Ch. 8 - Verify that [n]n = n!, and write n! as a...Ch. 8 - For each integer n = 1, 2, 3, 4, 5, construct the...Ch. 8 - Prob. 22ECh. 8 - Prob. 23ECh. 8 - Prob. 24ECh. 8 - Prob. 25ECh. 8 - Determine the conjugate of each of the following...Ch. 8 - For each integer n > 2, determine a self-conjugate...Ch. 8 - Prove that conjugation reverses the order of...Ch. 8 - Prove that the number of partitions of the...Ch. 8 - Prove that the partition function satisfies
Ch. 8 - Prob. 32ECh. 8 - Prob. 33ECh. 8 - Prob. 34ECh. 8 - Prob. 35ECh. 8 - 36. Prove that the Catalan number Cn equals the...
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- Determine whether the Law of Sines or the Law of Cosines can be used to find another measure of the triangle. B 13 cm 97° Law of Sines Law of Cosines A 43° Then solve the triangle. (Round your answers to two decimal places.) b = x C = A = 40.00arrow_forward13' - 3π 2 < u <- π and tan v 5 14) Find the exact value given that sin u = a) sin(u-v) b) cos(u+v) c) tan(u-v) d) cos(2u) == √³, 1 < v < π.arrow_forward13) Find all solutions in [0, 2π): sec²x + 6 tan x + 4 = 0arrow_forward
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