Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
8th Edition
ISBN: 9781259676512
Author: Kenneth H Rosen
Publisher: McGraw-Hill Education
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Chapter 12, Problem 6CAE
To determine
The average number of steps required to minimize the
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Exercises
Evaluate the following limits.
1. lim cot x/ln x
+01x
2. lim x² In x
+014
3. lim x*
x0+
4. lim (cos√√x)1/x
+014
5. lim x2/(1-cos x)
x10
6. lim e*/*
818
7. lim (secx - tan x)
x-x/2-
8. lim [1+(3/x)]*
x→∞0
In Exercises 1 through 3, let xo =
O and calculate P7(x) and R7(x).
1. f(x)=sin x, x in R.
2. f(x) = cos x, x in R.
3. f(x) = In(1+x), x≥0.
4. In Exercises 1, 2, and 3, for |x| 1, calculate a value of n such that P(x)
approximates f(x) to within 10-6.
5. Let (an)neN be a sequence of positive real numbers such that L =
lim (an+1/an) exists in R. If L < 1, show that an → 0. [Hint: Let
1111
L
iation
7. Let f be continuous on [a, b] and differentiable on (a, b). If lim f'(x)
xia
exists in R, show that f is differentiable at a and f'(a) = lim f'(x). A
similar result holds for b.
x-a
8. In reference to Corollary 5.4, give an example of a uniformly continuous
function on [0, 1] that is differentiable on (0, 1] but whose derivative is not
bounded there.
9. Recall that a fixed point of a function f is a point c such that f(c) = c.
(a) Show that if f is differentiable on R and f'(x)| x if x 1 and hence In(1+x) 0.
12. For 0 л/2. (Thus,
as x л/2 from the left, cos x is never large enough for x+cosx to be
greater than л/2 and cot x is never small enough for x + cot x to be less
than x/2.)
Chapter 12 Solutions
Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
Ch. 12.1 - Prob. 1ECh. 12.1 - Find the values, if any, of the Boolean...Ch. 12.1 - a) Show that(1.1)+(0.1+0)=1 . b) Translate the...Ch. 12.1 - a) Show that(10)+(10)=1 . b) Translate the...Ch. 12.1 - Use a table to express the values of each of these...Ch. 12.1 - Use a table to express the values of each of these...Ch. 12.1 - Use a 3-cubeQ3to represent each of the Boolean...Ch. 12.1 - Use a 3-cubeQ3to represent each of the Boolean...Ch. 12.1 - What values of the Boolean...Ch. 12.1 - How many different Boolean functions are there of...
Ch. 12.1 - Prove the absorption lawx+xy=x using the other...Ch. 12.1 - Show thatF(x,y,z)=xy+xz+yz has the value 1 if and...Ch. 12.1 - Show thatxy+yz+xz=xy+yz+xz .Ch. 12.1 - 3Exercises 14-23 deal the Boolean algebra {0, 1}...Ch. 12.1 - Exercises 14-23 deal with the Boolean algebra {0,...Ch. 12.1 - Prob. 16ECh. 12.1 - Exercises 14-23 deal with the Boolean algebra {0,...Ch. 12.1 - Prob. 18ECh. 12.1 - Prob. 19ECh. 12.1 - Prob. 20ECh. 12.1 - Prob. 21ECh. 12.1 - Prob. 22ECh. 12.1 - Exercises 4-3 deal with the Boolean algebra {0, 1}...Ch. 12.1 - Prob. 24ECh. 12.1 - Prob. 25ECh. 12.1 - Prob. 26ECh. 12.1 - Prove or disprove these equalities. a)x(yz)=(xy)z...Ch. 12.1 - Find the duals of these Boolean expressions. a)x+y...Ch. 12.1 - Prob. 29ECh. 12.1 - Show that ifFandGare Boolean functions represented...Ch. 12.1 - How many different Boolean functionsF(x,y,z) are...Ch. 12.1 - How many different Boolean functionsF(x,y,z) are...Ch. 12.1 - Show that you obtain De Morgan’s laws for...Ch. 12.1 - Show that you obtain the ab,sorption laws for...Ch. 12.1 - In Exercises 35-42, use the laws in Definition 1...Ch. 12.1 - In Exercises 35-42, use the laws in Definition to...Ch. 12.1 - Prob. 37ECh. 12.1 - Prob. 38ECh. 12.1 - In Exercises 35-42, use the laws in Definition 1...Ch. 12.1 - Prob. 40ECh. 12.1 - Prob. 41ECh. 12.1 - Prob. 42ECh. 12.1 - Prob. 43ECh. 12.2 - Find a Boolean product of the Boolean...Ch. 12.2 - Find the sum of products expansions of these...Ch. 12.2 - Find the sum-of-products expansions of these...Ch. 12.2 - Find the sum-of-products expansions of the Boolean...Ch. 12.2 - Find the sum-of -products expansion of the Boolean...Ch. 12.2 - Find the sum-of-products expansion of the Boolean...Ch. 12.2 - Another way to find a Boolean expression that...Ch. 12.2 - Prob. 8ECh. 12.2 - Prob. 9ECh. 12.2 - Another way to find a Boolean expression that...Ch. 12.2 - Prob. 11ECh. 12.2 - Express each of these Boolean functions using the...Ch. 12.2 - Express each of the Boolean functions in...Ch. 12.2 - Show that a)x=xx . b)xy=(xy)(xy) . c)x+y=(xx)(yy)...Ch. 12.2 - Prob. 15ECh. 12.2 - Show that{} is functionally complete using...Ch. 12.2 - Express each of the Boolean functions in Exercise...Ch. 12.2 - Express each of the Boolean functions in Exercise...Ch. 12.2 - Show that the set of operators{+,} is not...Ch. 12.2 - Are these sets of operators functionally complete?...Ch. 12.3 - In Exercises 1—5 find the output of the given...Ch. 12.3 - In Exercises 1—5 find the output of the given...Ch. 12.3 - In Exercises 1—5 find the output of the given...Ch. 12.3 - In Exercises 1—5 find the output of the given...Ch. 12.3 - In Exercises 1—5 find the output of the given...Ch. 12.3 - Construct circuits from inverters, AND gates, and...Ch. 12.3 - Design a circuit that implements majority voting...Ch. 12.3 - Design a circuit for a light fixture controlled by...Ch. 12.3 - Show how the sum of two five-bit integers can be...Ch. 12.3 - Construct a circuit for a half subtractor using...Ch. 12.3 - Construct a circuit for a full subtractor using...Ch. 12.3 - Use the circuits from Exercises 10 and 11 to find...Ch. 12.3 - Construct a circuit that compares the two-bit...Ch. 12.3 - Construct a circuit that computes the product of...Ch. 12.3 - Use NAND gates to construct circuits with these...Ch. 12.3 - Use NOR gates to construct circuits for the...Ch. 12.3 - Construct a half adder using NAND gates.Ch. 12.3 - Construct a half adder using NOR gates.Ch. 12.3 - Construct a multiplexer using AND gates, OR gates,...Ch. 12.3 - Find the depth of a) the circuit constructed in...Ch. 12.4 - Prob. 1ECh. 12.4 - Find the sum-of-products expansions represented by...Ch. 12.4 - Draw the K-maps of these sum-of-products...Ch. 12.4 - Use a K-map to find a minimal expansion as a...Ch. 12.4 - a) Draw a K-map for a function in three variables....Ch. 12.4 - Use K-maps to find simpler circuits with the same...Ch. 12.4 - Prob. 7ECh. 12.4 - Prob. 8ECh. 12.4 - Construct a K-map for F(x,y,z) =xz + yz+y z. Use...Ch. 12.4 - Draw the 3-cube Q3 and label each vertex with the...Ch. 12.4 - Prob. 11ECh. 12.4 - Use a K-map to find a minimal expansion as a...Ch. 12.4 - a) Draw a K-map for a function in four variables....Ch. 12.4 - Use a K-map to find a minimal expansion as a...Ch. 12.4 - Find the cells in a K-map for Boolean functions...Ch. 12.4 - How many cells in a K-map for Boolean functions...Ch. 12.4 - a) How many cells does a K-map in six variables...Ch. 12.4 - Show that cells in a K-map for Boolean functions...Ch. 12.4 - Which rows and which columns of a 4 x 16 map for...Ch. 12.4 - Prob. 20ECh. 12.4 - Prob. 21ECh. 12.4 - Use the Quine-McCluskey method to simplify the...Ch. 12.4 - Use the Quine—McCluskey method to simp1i’ the...Ch. 12.4 - Prob. 24ECh. 12.4 - Use the Quine—McCluskey method to simplify the...Ch. 12.4 - Prob. 26ECh. 12.4 - Prob. 27ECh. 12.4 - Prob. 28ECh. 12.4 - Prob. 29ECh. 12.4 - Prob. 30ECh. 12.4 - Prob. 31ECh. 12.4 - Prob. 32ECh. 12.4 - show that products of k literals correspond to...Ch. 12 - Define a Boolean function of degreen.Ch. 12 - Prob. 2RQCh. 12 - Prob. 3RQCh. 12 - Prob. 4RQCh. 12 - Prob. 5RQCh. 12 - Prob. 6RQCh. 12 - Explain how to build a circuit for a light...Ch. 12 - Prob. 8RQCh. 12 - Is there a single type of logic gate that can be...Ch. 12 - a) Explain how K-maps can be used to simplify...Ch. 12 - a) Explain how K-maps can be used to simplify...Ch. 12 - a) What is a don’t care condition? b) Explain how...Ch. 12 - a) Explain how to use the Quine-McCluskev method...Ch. 12 - Prob. 1SECh. 12 - Prob. 2SECh. 12 - Prob. 3SECh. 12 - Prob. 4SECh. 12 - Prob. 5SECh. 12 - Prob. 6SECh. 12 - Prob. 7SECh. 12 - Prob. 8SECh. 12 - Prob. 9SECh. 12 - Prob. 10SECh. 12 - Prob. 11SECh. 12 - Prob. 12SECh. 12 - Prob. 13SECh. 12 - Prob. 14SECh. 12 - Prob. 15SECh. 12 - Prob. 16SECh. 12 - How many of the 16 Boolean functions in two...Ch. 12 - Prob. 18SECh. 12 - Prob. 19SECh. 12 - Design a circuit that determines whether three or...Ch. 12 - Prob. 21SECh. 12 - A Boolean function that can be represented by a...Ch. 12 - Prob. 23SECh. 12 - Prob. 24SECh. 12 - Given the values of two Boolean variablesxandy,...Ch. 12 - Prob. 2CPCh. 12 - Prob. 3CPCh. 12 - Prob. 4CPCh. 12 - Prob. 5CPCh. 12 - Prob. 6CPCh. 12 - Prob. 7CPCh. 12 - Prob. 8CPCh. 12 - Prob. 9CPCh. 12 - Given the table of values of a Boolean function,...Ch. 12 - Prob. 11CPCh. 12 - Prob. 12CPCh. 12 - Prob. 1CAECh. 12 - Prob. 2CAECh. 12 - Prob. 3CAECh. 12 - Prob. 4CAECh. 12 - Prob. 5CAECh. 12 - Prob. 6CAECh. 12 - Prob. 7CAECh. 12 - Describe some of the early machines devised to...Ch. 12 - Explain the difference between combinational...Ch. 12 - Prob. 3WPCh. 12 - Prob. 4WPCh. 12 - Find out how logic gates are physically...Ch. 12 - Explain howdependency notationcan be used to...Ch. 12 - Describe how multiplexers are used to build...Ch. 12 - Explain the advantages of using threshold gates to...Ch. 12 - Describe the concept ofhazard-free switching...Ch. 12 - Explain how to use K-maps to minimize functions of...Ch. 12 - Prob. 11WPCh. 12 - Describe what is meant by the functional...
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