Introductory Combinatorics
5th Edition
ISBN: 9780134689616
Author: Brualdi, Richard A.
Publisher: Pearson,
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Chapter 7, Problem 2E
To determine
To prove: That thee nth Fibonacci number
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4. Direction Fields/Phase Portraits. Use the given direction fields to plot solution curves
to each of the given initial value problems.
(a)
x = x+2y
1111
y = -3x+y
with x(0) = 1, y(0) = -1
(b) Consider the initial value problem corresponding to the given phase portrait.
x = y
y' = 3x + 2y
Draw two "straight line solutions"
passing through (0,0)
(c) Make guesses for the equations of the straight line solutions: y = ax.
It was homework
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Chapter 7 Solutions
Introductory Combinatorics
Ch. 7 - Prob. 1ECh. 7 - Prove that the nth Fibonacci number fn is the...Ch. 7 - Prove the following about the Fibonacci...Ch. 7 - 4. Prove that the Fibonacci sequence is the...Ch. 7 - By examining the Fibonacci sequence, make a...Ch. 7 - * Let m and n be positive integers. Prove that if...Ch. 7 - * Let m and n be positive integers whose greatest...Ch. 7 - Consider a 1-by-n chessboard. Suppose we color...Ch. 7 - Prob. 9ECh. 7 - Prob. 10E
Ch. 7 - Prob. 11ECh. 7 - Prob. 12ECh. 7 - 13. Determine the generating function for each of...Ch. 7 - 14. Let S be the multiset {∞ · e1, ∞ · e2, ∞ · e3,...Ch. 7 - 15. Determine the generating function for the...Ch. 7 - 16. Formulate a combinatorial problem for which...Ch. 7 - 17. Determine the generating function for the...Ch. 7 - 18. Determine the generating function for the...Ch. 7 - 19. Let h0, h1, h2, …, hn, … be the sequence...Ch. 7 - Prob. 20ECh. 7 - 21. * Let hn denote the number of regions into...Ch. 7 - 22. Determine the exponential generating function...Ch. 7 - 23. Let α be a real number. Let the sequence h0,...Ch. 7 - 24. Let S be the multiset {∞ · e1, ∞ · e2, · , ∞ ·...Ch. 7 - 25. Let hn denote the number of ways to color the...Ch. 7 - Determine the number of ways to color the squares...Ch. 7 - Determine the number of n-digit numbers with all...Ch. 7 - Determine the number of n-digit numbers with all...Ch. 7 - We have used exponential generating functions to...Ch. 7 - Prob. 30ECh. 7 - Solve the recurrence relation hn = 4hn−2, (n ≥ 2)...Ch. 7 - Prob. 32ECh. 7 - Solve the recurrence relation hn = hn−1 + 9hn−2 −...Ch. 7 - Solve the recurrence relation hn = 8hn−1 − 16hn−2,...Ch. 7 - Solve the recurrence relation hn = 3hn − 2 − 2hn −...Ch. 7 - Prob. 36ECh. 7 - Determine a recurrence relation for the number an...Ch. 7 - Prob. 38ECh. 7 - Let hn denote the number of ways to perfectly...Ch. 7 - Let an equal the number of ternary strings of...Ch. 7 - * Let 2n equally spaced points be chosen on a...Ch. 7 - Solve the nonhomogeneous recurrence relation
Ch. 7 - Solve the nonhomogeneous recurrence relation
hn =...Ch. 7 - Solve the nonhomogeneous recurrence relation
Ch. 7 - Prob. 45ECh. 7 - Solve the nonhomogeneous recurrence relation
Ch. 7 - Solve the nonhomogeneous recurrence relation
Ch. 7 - Solve the following recurrence relations by using...Ch. 7 - (q-binomial theorem) Prove that
where
is the...Ch. 7 - Call a subset S of the integers {1, 2, …, n}...Ch. 7 - Solve the recurrence relation
from Section 7.6...Ch. 7 - Prob. 52ECh. 7 - Suppose you deposit $500 in a bank account that...
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- (7) (12 points) Let F(x, y, z) = (y, x+z cos yz, y cos yz). Ꮖ (a) (4 points) Show that V x F = 0. (b) (4 points) Find a potential f for the vector field F. (c) (4 points) Let S be a surface in R3 for which the Stokes' Theorem is valid. Use Stokes' Theorem to calculate the line integral Jos F.ds; as denotes the boundary of S. Explain your answer.arrow_forward(3) (16 points) Consider z = uv, u = x+y, v=x-y. (a) (4 points) Express z in the form z = fog where g: R² R² and f: R² → R. (b) (4 points) Use the chain rule to calculate Vz = (2, 2). Show all intermediate steps otherwise no credit. (c) (4 points) Let S be the surface parametrized by T(x, y) = (x, y, ƒ (g(x, y)) (x, y) = R². Give a parametric description of the tangent plane to S at the point p = T(x, y). (d) (4 points) Calculate the second Taylor polynomial Q(x, y) (i.e. the quadratic approximation) of F = (fog) at a point (a, b). Verify that Q(x,y) F(a+x,b+y). =arrow_forward(6) (8 points) Change the order of integration and evaluate (z +4ry)drdy . So S√ ² 0arrow_forward
- (10) (16 points) Let R>0. Consider the truncated sphere S given as x² + y² + (z = √15R)² = R², z ≥0. where F(x, y, z) = −yi + xj . (a) (8 points) Consider the vector field V (x, y, z) = (▼ × F)(x, y, z) Think of S as a hot-air balloon where the vector field V is the velocity vector field measuring the hot gasses escaping through the porous surface S. The flux of V across S gives the volume flow rate of the gasses through S. Calculate this flux. Hint: Parametrize the boundary OS. Then use Stokes' Theorem. (b) (8 points) Calculate the surface area of the balloon. To calculate the surface area, do the following: Translate the balloon surface S by the vector (-15)k. The translated surface, call it S+ is part of the sphere x² + y²+z² = R². Why do S and S+ have the same area? ⚫ Calculate the area of S+. What is the natural spherical parametrization of S+?arrow_forward(1) (8 points) Let c(t) = (et, et sint, et cost). Reparametrize c as a unit speed curve starting from the point (1,0,1).arrow_forward(9) (16 points) Let F(x, y, z) = (x² + y − 4)i + 3xyj + (2x2 +z²)k = - = (x²+y4,3xy, 2x2 + 2²). (a) (4 points) Calculate the divergence and curl of F. (b) (6 points) Find the flux of V x F across the surface S given by x² + y²+2² = 16, z ≥ 0. (c) (6 points) Find the flux of F across the boundary of the unit cube E = [0,1] × [0,1] x [0,1].arrow_forward
- (8) (12 points) (a) (8 points) Let C be the circle x² + y² = 4. Let F(x, y) = (2y + e²)i + (x + sin(y²))j. Evaluate the line integral JF. F.ds. Hint: First calculate V x F. (b) (4 points) Let S be the surface r² + y² + z² = 4, z ≤0. Calculate the flux integral √(V × F) F).dS. Justify your answer.arrow_forwardDetermine whether the Law of Sines or the Law of Cosines can be used to find another measure of the triangle. a = 13, b = 15, C = 68° Law of Sines Law of Cosines Then solve the triangle. (Round your answers to four decimal places.) C = 15.7449 A = 49.9288 B = 62.0712 × Need Help? Read It Watch Itarrow_forward(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]¸arrow_forward
- (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.arrow_forwardFind 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 darrow_forwardAnswer the following questions related to the following matrix A = 3 ³).arrow_forward
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