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Discrete Mathematics With ApplicationsExercises 49 and 50 use L’Hôpital’s rule from calculus. 49. a. Let b be any real number greater than 1. Use L’Hôpital’s rule and mathematical induction to prove that for every integer n ≥ 1 , lim x → ∞ x a b x = 0. b. Use the result of part (a) and the definitions of limit and of O -notation to prove that x n is O ( b x ) for any integer n ≥ 1 .BuyFind*arrow_forward*

5th Edition

EPP + 1 other

Publisher: Cengage Learning,

ISBN: 9781337694193

Chapter 11.4, Problem 49ES

Textbook Problem

Exercises 49 and 50 use L’Hôpital’s rule from calculus.

49. a. Let *b *be any real number greater than 1. Use L’Hôpital’s rule and mathematical induction to prove that for every integer

b. Use the result of part (a) and the definitions of limit and of *O*-notation to prove that

Discrete Mathematics With Applications

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Ch. 11.1 - If f is a real-valued function of a real variable,...Ch. 11.1 - A point (x,y) lies on the graph of a real-valued...Ch. 11.1 - If a is any nonnegative real number, then the...Ch. 11.1 - Given a function f:RR and a real number M, the...Ch. 11.1 - Given a function f:RR , to prove that f is...Ch. 11.1 - Given a function f:RR , to prove that f is...Ch. 11.1 - The graph of a function f is shown below. a. Is...Ch. 11.1 - The graph of a function g is shown below. a. Is...Ch. 11.1 - Sketch the graphs of the power functions p1/3and...Ch. 11.1 - Sketch the graphs of the power functions p3 and p4...

Ch. 11.1 - Sketch the graphs of y=2x and y=2x for each real...Ch. 11.1 - Sketch a graph for each of the functions defined...Ch. 11.1 - Sketch a graph for each of the functions defined...Ch. 11.1 - Sketch a graph for each of the functions defined...Ch. 11.1 - Sketch a graph for each of the functions defined...Ch. 11.1 - In each of 10—13 a function is defined on a set of...Ch. 11.1 - In each of 10—13 a function is defined on a set of...Ch. 11.1 - In each of 10—13 a function is defined on a set of...Ch. 11.1 - In each of 10—13 a function is defined on a set of...Ch. 11.1 - The graph of a function f is shown below. Find the...Ch. 11.1 - Show that the function f:RR defined by the formula...Ch. 11.1 - Show that the function g:RR defined by the formula...Ch. 11.1 - Let h be the function from R to R defined by the...Ch. 11.1 - Let k:RR be the function defined by the formula...Ch. 11.1 - Show that if a function f:RRis increasing, then f...Ch. 11.1 - Given real-valued functions f and g with the same...Ch. 11.1 - a. Let m be any positive integer, and define...Ch. 11.1 - Let f be the function whose graph follows. Sketch...Ch. 11.1 - Let h be the function whose graph is shown below....Ch. 11.1 - Let f be a real-valued function of a real...Ch. 11.1 - Let f be a real-valued function of a real...Ch. 11.1 - Let f be a real-valued function of a real...Ch. 11.1 - In 27 and 28, functions f and g are defined. In...Ch. 11.1 - In 27 and 28, functions f and g are defined. In...Ch. 11.2 - A sentence of the form Ag(n)f(n) for every na...Ch. 11.2 - A sentence of the tirm “ 0f(n)Bg(n) for every nb ”...Ch. 11.2 - A sentence of the form “ Ag(n)f(n)Bg(n)for every...Ch. 11.2 - When n1,n n2 and n2 n5__________.Ch. 11.2 - According to the theorem on polynomial orders, if...Ch. 11.2 - If n is a positive integer, then 1+2+3++n has...Ch. 11.2 - The following is a formal definition for ...Ch. 11.2 - The following is a formal definition for...Ch. 11.2 - The following is a formal definition for ...Ch. 11.2 - In 4—9, express each statement using -, O-, or ...Ch. 11.2 - In 4—9, express each statement using -, O-, or ...Ch. 11.2 - In 4—9, express each statement using -, O-, or ...Ch. 11.2 - In 4—9, express each statement using -, O-, or ...Ch. 11.2 - In 4—9, express each statement using -, O-, or ...Ch. 11.2 - In 4—9, express each statement using -, O-, or ...Ch. 11.2 - a. Show that for any integer n1,02n2+15n+421n2 ....Ch. 11.2 - a. Show that for any integer n1,023n4+8n2+4n35n4 ....Ch. 11.2 - a. Show that for any integer n1,07n3+10n2+320n3 ....Ch. 11.2 - Use the definition of -notation to show that...Ch. 11.2 - Use the definition of -notation to show that...Ch. 11.2 - Use the definition of -notation to show that...Ch. 11.2 - Use the definition of -notation to show that...Ch. 11.2 - Use the definition of -notation to show that n2is...Ch. 11.2 - Prove Theorem 11.2.7(b): If f and g are...Ch. 11.2 - Prove Theorem 11.2.1(b): If f and g are...Ch. 11.2 - Without using Theorem 11.2.4 prove that n5 is not...Ch. 11.2 - Prove Theorem 11.2.4: If f is a real-valued...Ch. 11.2 - a. Use one of the methods of Example 11.2.4 to...Ch. 11.2 - a. Use one of the methods of Example 11.2.4 to...Ch. 11.2 - a. Use one of the methods of Example 11.2.4 to...Ch. 11.2 - Suppose P(n)=amnm+am1nm1++a2n2+a1n+a0 , where all...Ch. 11.2 - Use the theorem on polynomial orders to prove each...Ch. 11.2 - Use the theorem on polynomial orders to prove each...Ch. 11.2 - Use the theorem on polynomial orders to prove each...Ch. 11.2 - Use the theorem on polynomial orders to prove each...Ch. 11.2 - Use the theorem on polynomial orders to prove each...Ch. 11.2 - Use the theorem on polynomial orders to prove each...Ch. 11.2 - Prove each of the statements in 32—39. Use the...Ch. 11.2 - Prove each of the statements in 32—39. Use the...Ch. 11.2 - Prove each of the statements in 32—39. Use the...Ch. 11.2 - Prove each of the statements in 32—39. Use the...Ch. 11.2 - Prove each of the statements in 32—39. Use the...Ch. 11.2 - Prove each of the statements in 32—39. Use the...Ch. 11.2 - Prove each of the statements in 32—39. Use the...Ch. 11.2 - Prove each of the statements in 32—39. Use the...Ch. 11.2 - a. Prove: If c is a positive real number and if f...Ch. 11.2 - Prove: If c is a positive real number and...Ch. 11.2 - What can you say about a function f with the...Ch. 11.2 - Use Theorems 11.2.5-11.2.9 and the results of...Ch. 11.2 - Use Theorems 11.2.5-11.2.9 and the results of...Ch. 11.2 - Use Theorems 11.2.5-11.2.9 and the results of...Ch. 11.2 - a. Use mathematical induction to prove that if n...Ch. 11.2 - a. Let x be any positive real number. Use...Ch. 11.2 - Prove Theorem 11.2.6(b): If f and g are...Ch. 11.2 - Prove Theorem 11.2.7(a): If f is a real-valued...Ch. 11.2 - Prove Theorem 11.2.8: a. Let f and g be...Ch. 11.2 - Prove Theorem 11.2.9: a. Let f1,f2 , and g be...Ch. 11.3 - When an algorithm segment contains a nested...Ch. 11.3 - In the worst case for an input array of length n,...Ch. 11.3 - The worst-case order of the insertion sort...Ch. 11.3 - Suppose a computer takes 1 nanosecond ( =109...Ch. 11.3 - Suppose an algorithm requires cn2operations when...Ch. 11.3 - Suppose an algorithm requires cn3operations when...Ch. 11.3 - Exercises 4—5 explore the fact that for relatively...Ch. 11.3 - Exercises 4—5 explore the fact that for relatively...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - For each of the algorithm segments in 6—19, assume...Ch. 11.3 - Construct a table showing the result of each step...Ch. 11.3 - Construct a table showing the result of each step...Ch. 11.3 - Construct a trace table showing the action of...Ch. 11.3 - Construct a trace table showing the action of...Ch. 11.3 - How many comparisons between values of a[j] and x...Ch. 11.3 - How many comparisons between values of a[j] and x...Ch. 11.3 - According to Example 11.3.6. the maximum number of...Ch. 11.3 - Consider the recurrence relation that arose in...Ch. 11.3 - Exercises 28—35 refer to selection sort, which is...Ch. 11.3 - Exercises 28—35 refer to selection sort, which is...Ch. 11.3 - Exercises 28—35 refer to selection sort, which is...Ch. 11.3 - Exercises 28—35 refer to selection sort, which is...Ch. 11.3 - Exercises 28—35 refer to selection sort, which is...Ch. 11.3 - Exercises 28—35 refer to selection sort, which is...Ch. 11.3 - Exercises 28—35 refer to selection sort, which is...Ch. 11.3 - Exercises 28—35 refer to selection sort, which is...Ch. 11.3 - Exercises 36—39 refer to the following algorithm...Ch. 11.3 - Exercises 36—39 refer to the following algorithm...Ch. 11.3 - Exercises 36—39 refer to the following algorithm...Ch. 11.3 - Exercises 36—39 refer to the following algorithm...Ch. 11.3 - Exercises 40—43 refer to another algorithm, known...Ch. 11.3 - Exercises 40—43 refer to another algorithm, known...Ch. 11.3 - Exercises 40—43 refer to another algorithm, known...Ch. 11.3 - Exercises 40—43 refer to another algorithm, known...Ch. 11.4 - The domain of any exponential function is , and...Ch. 11.4 - The domain of any logarithmic function is and its...Ch. 11.4 - If k is an integer and 2kx2k+1 then...Ch. 11.4 - If b is a real number with b1 , then there is a...Ch. 11.4 - If n is a positive integer, then 1+12+13++1nhas...Ch. 11.4 - Graph each function defined in 1-8. 1. f(x)=3x for...Ch. 11.4 - Graph each function defined in 1—8. 2. g(x)=(13)x...Ch. 11.4 - Graph each function defined in 1—8. 3. h(x)=log10x...Ch. 11.4 - Graph each function defined in 1—8. 4. k(x)=log2x...Ch. 11.4 - Graph each function defined in 1—8. 5. F(x)=log2x...Ch. 11.4 - Graph each function defined in 1—8. 6. G(x)=log2x...Ch. 11.4 - Graph each function defined in 1—8. 7. H(x)=xlog2x...Ch. 11.4 - Graph each function defined in 1—8. 8....Ch. 11.4 - The scale of the graph shown in Figure 11.4.1 is...Ch. 11.4 - a. Use the definition of logarithm to show that...Ch. 11.4 - Let b1 . a. Use the fact that u=logbvv=bu to show...Ch. 11.4 - Give a graphical interpretation for property...Ch. 11.4 - Suppose a positive real number x satisfies the...Ch. 11.4 - a. Prove that if x is a positive real number and k...Ch. 11.4 - If n is an odd integer and n1 ,is log2(n1)=log2(n)...Ch. 11.4 - If, n is an odd integer and n1 , is...Ch. 11.4 - If n is an odd integer and n1 , is...Ch. 11.4 - In 18 and 19, indicate how many binary digits are...Ch. 11.4 - In 18 and 19, indicate how many binary digits are...Ch. 11.4 - It was shown in the text that the number of binary...Ch. 11.4 - In each of 21 and 22, a sequence is specified by a...Ch. 11.4 - In each of 21 and 22, a sequence is specified by a...Ch. 11.4 - Define a sequence c1,c2,c3,recursively as follows:...Ch. 11.4 - Use strong mathematical induction to show that for...Ch. 11.4 - Exercises 25 and 26 refer to properties 11.4.9 and...Ch. 11.4 - Exercises 25 and 26 refer to properties 11.4.9 and...Ch. 11.4 - Use Theorems 11.2.7-11.2.9 and properties 11.4.11,...Ch. 11.4 - Use Theorems 11.2.7-11.2.9 and properties 11.4.11,...Ch. 11.4 - Use Theorems 11.2.7—11.2.9 and properties 11.4.11,...Ch. 11.4 - Use Theorems 11.2.7—11.2.9 and properties 11.4.11,...Ch. 11.4 - Show that 4n is not O(2n) .Ch. 11.4 - Prove each of the statements in 32—37, assuming n...Ch. 11.4 - Prove each of the statements in 32—37, assuming n...Ch. 11.4 - Prove each of the statements in 32—37, assuming n...Ch. 11.4 - Prove each of the statements in 32—37, assuming n...Ch. 11.4 - Prove each of the statements in 32—37, assuming n...Ch. 11.4 - Prove each of the statements in 32—37, assuming n...Ch. 11.4 - Quantities of the form k1n+k2nlognfor positive...Ch. 11.4 - Calculate the values of the harmonic sums...Ch. 11.4 - Use part (d) of Example 11.4.7 to show that...Ch. 11.4 - Show that log2n is (log2n) .Ch. 11.4 - Show that log2n is (log2n) .Ch. 11.4 - Prove by mathematical induction that n10n for...Ch. 11.4 - Prove by mathematical induction that log2nn for...Ch. 11.4 - Show that if n is a variable that takes positive...Ch. 11.4 - Let n be a variable that takes positive integer...Ch. 11.4 - For each positive real number u,log2uuUse this...Ch. 11.4 - Use the result of exercise 47 above to prove the...Ch. 11.4 - Exercises 49 and 50 use L’Hôpital’s rule from...Ch. 11.4 - Exercises 49 and 50 use L’Hôpital’s rule from...Ch. 11.4 - Complete the proof in Example 11.4.4.Ch. 11.5 - To solve a problem using a divide-and-conquer...Ch. 11.5 - To search an array using the binary search...Ch. 11.5 - The worst-case order of the binary search...Ch. 11.5 - To sort an array using the merge sort algorithm,...Ch. 11.5 - The worst-case order of the merge sort algorithm...Ch. 11.5 - Use the facts that log2103.32 and that for each...Ch. 11.5 - Suppose an algorithm requires clog2n operations...Ch. 11.5 - Exercises 3 and 4 illustrate that for relatively...Ch. 11.5 - Exercises 3 and 4 illustrate that for relatively...Ch. 11.5 - In 5 and 6, trace the action of the binary search...Ch. 11.5 - In 5 and 6, trace the action of the binary search...Ch. 11.5 - Suppose bot and top are positive integers with...Ch. 11.5 - Exercises 8—11 refer to the following algorithm...Ch. 11.5 - Exercises 8—11 refer to the following algorithm...Ch. 11.5 - Exercises 8—11 refer to the following algorithm...Ch. 11.5 - Exercises 8—11 refer to the following algorithm...Ch. 11.5 - Exercises 12—15 refer to the following algorithm...Ch. 11.5 - Exercises 12—15 refer to the following algorithm...Ch. 11.5 - Exercises 12—15 refer to the following algorithm...Ch. 11.5 - Exercises 12—15 refer to the following algorithm...Ch. 11.5 - Complete the proof of case 2 of the strong...Ch. 11.5 - Trace the modified binary search algorithm for the...Ch. 11.5 - Suppose an array of length k is input to the while...Ch. 11.5 - Let wnbe the number of iterations of the while...Ch. 11.5 - In 20 and 21, draw a diagram like Figure 11.5.4 to...Ch. 11.5 - In 20 and 21, draw a diagram like Figure 11.5.4 to...Ch. 11.5 - In 22 and 23, draw a diagram like Figure 11.5.5 to...Ch. 11.5 - In 22 and 23, draw a diagram like Figure 11.5.5 to...Ch. 11.5 - Show that given an array a[bot],a[bot+1],,a[top]of...Ch. 11.5 - The recurrence relation for m1,m2,m3,,which arises...Ch. 11.5 - It might seem that n1 multiplications are needed...

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