
A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Textbook Question
Chapter 4.3, Problem 9E
(a)Prove that if
(b)Is the converse of part (a) true? Explain.
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7. The demand for a product, in dollars, is
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Find the consumer surplus when the sales level is 200.
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surplus can be calculated by foam (D(x) — pm) dx]
4. Find the general solution and the definite solution for the following differential equations:
(a)
+10y=15, y(0) = 0;
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(a) + 3y = 2, y(0) = 4;
(b) dy - 7y = 7, y(0) = 7;
(c) 3d+6y= 5, y(0) = 0
Chapter 4 Solutions
A Transition to Advanced Mathematics
Ch. 4.1 - Find two upper bounds (if any exits) for each of...Ch. 4.1 - Assign a grade of A (correct), C (partially...Ch. 4.1 - Prob. 3ECh. 4.1 - Prob. 4ECh. 4.1 - Let A and B be subsets of . Prove that if A is...Ch. 4.1 - Let x be an upper bound for A. Prove that if xy,...Ch. 4.1 - Let A. Prove that if A is bounded above, then Ac...Ch. 4.1 - Give an example of a set A for which both A and Ac...Ch. 4.1 - Let A. Prove that if sup(A) exists, then it is...Ch. 4.1 - Formulate and prove a characterization of greatest...
Ch. 4.1 - If possible, give an example of a set A such that...Ch. 4.1 - Let A. Prove that if sup(A) exists, then...Ch. 4.1 - Let A and B be subsets of . Prove that if sup(A)...Ch. 4.1 - (a)Give an example of sets A and B of real numbers...Ch. 4.1 - (a)Give an example of sets A and B of real numbers...Ch. 4.1 - An alternate version of the Archimedean Principle...Ch. 4.1 - Prob. 17ECh. 4.1 - Prove that an ordered field F is complete iff...Ch. 4.1 - Prove that every irrational number is "missing"...Ch. 4.2 - Let A and B be compact subsets of . Use the...Ch. 4.2 - Prob. 2ECh. 4.2 - Prob. 3ECh. 4.2 - Prob. 4ECh. 4.2 - Assign a grade of A (correct), C (partially...Ch. 4.2 - For real numbers x,1,2,...n, describe i=1nN(x,i)....Ch. 4.2 - State the definition of continuity of the function...Ch. 4.2 - Find the set of interior point for each of these...Ch. 4.2 - Suppose that x is an interior point of a set A....Ch. 4.2 - Let AB. Prove that if sup(A) and sup(B) both...Ch. 4.2 - Let Abe a nonempty collection of closed subsets of...Ch. 4.2 - Prob. 12ECh. 4.2 - Prob. 13ECh. 4.2 - Prob. 14ECh. 4.2 - Prob. 15ECh. 4.2 - Prob. 16ECh. 4.2 - Prove Lemma 7.2.4.Ch. 4.2 - Which of the following subsets of are compact? ...Ch. 4.2 - Give an example of a bounded subset of and a...Ch. 4.3 - Let A and F be sets of real numbers, and let F be...Ch. 4.3 - In the proof of Theorem 7.3.1 that =, it is...Ch. 4.3 - Assign a grade of A (correct), C (partially...Ch. 4.3 - Prove that 7 is an accumulation point for [3,7). 5...Ch. 4.3 - Find an example of an infinite subset of that has...Ch. 4.3 - Find the derived set of each of the following...Ch. 4.3 - Let S=(0,1]. Find S(Sc).Ch. 4.3 - Prob. 8ECh. 4.3 - (a)Prove that if AB, then AB. (b)Is the converse...Ch. 4.3 - Show by example that the intersection of...Ch. 4.3 - Prob. 11ECh. 4.3 - Prob. 12ECh. 4.3 - Let a, b. Prove that every closed interval [a,b]...Ch. 4.3 - Prob. 14ECh. 4.3 - Prob. 15ECh. 4.4 - Prob. 1ECh. 4.4 - Prove that if x is an interior point of the set A,...Ch. 4.4 - Recall from Exercise 11 of Section 4.6 that the...Ch. 4.4 - A sequence x of real numbers is a Cauchy* sequence...Ch. 4.4 - Prob. 5ECh. 4.4 - Assign a grade of A (correct), C (partially...Ch. 4.4 - Prob. 7ECh. 4.4 - Give an example of a bounded sequence that is not...Ch. 4.4 - Prob. 9ECh. 4.4 - Let A and B be subsets of . Prove that (AB)=AB....Ch. 4.5 - For the sequence y defined in the proof of Theorem...Ch. 4.5 - Prob. 2ECh. 4.5 - Prob. 3ECh. 4.5 - Let I be a sequence of intervals. Then for each...Ch. 4.5 - Prob. 5ECh. 4.5 - Prob. 6ECh. 4.5 - Find all divisors of zero in 14. 15. 10. 101.Ch. 4.5 - Prob. 8ECh. 4.5 - Suppose m and m2. Prove that 1 and m1 are distinct...Ch. 4.5 - Prob. 10ECh. 4.5 - Prob. 11ECh. 4.5 - Determine whether each sequence is monotone. For...Ch. 4.5 - Prob. 13ECh. 4.5 - Complete the proof that xn=(1+1n)n is increasing...Ch. 4.5 - Prob. 15ECh. 4.5 - Prob. 16ECh. 4.5 - Prob. 17ECh. 4.6 - Prob. 1ECh. 4.6 - Repeat Exercise 2 with the operation * given by...Ch. 4.6 - Prob. 3ECh. 4.6 - Let m,n and M=A:A is an mn matrix with real number...Ch. 4.6 - Let be an associative operation on nonempty set A...Ch. 4.6 - Let be an associative operation on nonempty set A...Ch. 4.6 - Suppose that (A,*) is an algebraic system and * is...Ch. 4.6 - Let (A,o) be an algebra structure. An element lA...Ch. 4.6 - Let G be a group. Prove that if a2=e for all aG,...Ch. 4.6 - Prob. 10ECh. 4.6 - Complete the proof of Theorem 6.1.4. First, show...Ch. 4.6 - Prob. 12ECh. 4.6 - Prob. 13ECh. 4.7 - Give an example of an algebraic structure of order...Ch. 4.7 - Let G be a group. Prove that G is abelian if and...Ch. 4.7 - Prob. 3ECh. 4.7 - (a)In the group G of Exercise 2, find x such that...Ch. 4.7 - Show that (,), with operation # defined by...Ch. 4.7 - Let m be a prime natural number and a(Um,). Prove...Ch. 4.7 - Prob. 7ECh. 4.7 - Prob. 8ECh. 4.7 - Prob. 9E
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- 1. Evaluate the following improper integrals: (a) fe-rt dt; (b) fert dt; (c) fi da dxarrow_forward8. Given the rate of net investment I(t) = 9t¹/2, find the level of capital formation in (i) 16 years and (ii) between the 4th and the 8th years.arrow_forward9. If the marginal revenue function of a firm in the production of output is MR = 40 - 10q² where q is the level of output, and total revenue is 120 at 3 units of output, find the total revenue function. [Hints: TR = √ MRdq]arrow_forward
- 6. Solve the following first-order linear differential equations; if an initial condition is given, definitize the arbitrary constant: (a) 2 + 12y + 2et = 0, y(0) = /; (b) dy+y=tarrow_forward4. Let A = {a, b, c, d, e, f}, B = {e, f, g, h} and C = {a, e, h,i}. Let U = {a, b, c, d, e, f, g, h, i, j, k}. • Draw a Venn Diagram to describe the relationships between these sets Find (AB) NC • Find (AC) UB Find AUBUC • Find (BC) N (A - C)arrow_forward7. A consumer lives on an island where she produces two goods x and y according to the production possibility frontier x² + y² < 200 and she consumes all the goods. Her utility function is U(x, y) = x y³. She faces an environmental constraint on her total output of both goods. The environmental constraint is given by x + y ≤20. • (a) Write down the consumer's optimization problem. (b) Write out the Kuhn-Tucker first order conditions. (c) Find the consumer's optimal consumption bundle (x*, y*).arrow_forward
- 3. Answer the following questions: (a) Given the marginal propensity to import M'(Y) = 0.1 and the information that M = 20 when Y = 0, find the import function M(Y). (b) Given a continuous income stream at the constant rate of $1,000 per year, what will be the present value II if the income stream terminates after exactly 3 years and the discount rate is 0.04? (c) What is the present value of a perpetual cash flow of $2,460 per year, discounted at r = 8%?arrow_forward5. Let A and B be arbitrary sets. Prove AnB = AUB.arrow_forward2. Answer the following questions: (a) Given the marginal-revenue function R'(Q) = 28Q - €0.3Q, find the total-revenue function R(Q). What initial condition can you introduce to definitize the constant of integration? = (b) Given the marginal propensity to consume C'(Y) 0.80.1Y-1/2 and the information that C = Y when Y = 100, find the consumption function C(Y).arrow_forward
- 7. Let X, A, and B be arbitrary sets such that ACX and BC X. Prove AUB CX.arrow_forward1. Write out the following sets as a list of elements. If necessary you may use ... in your description. {x EZ: |x|< 10 A x < 0} {x ЄN: x ≤ 20 A x = 2y for some y = N} {n EN: 3 | n^ 1 < n < 20} {y Є Z: y² <0}arrow_forward3. For each statement below, write an equivalent statement using the justification given. = y Є A or yЄ B by the definition of union = y Є A or y Є B by the definition of set complement = x = C and x & D by DeMorgan's Law =Vx (x EnFxЄEUF) by definition of subset. = (X CYUZ)A (YUZ CX) by definition of set equalityarrow_forward
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