Explanation Definition used: Taylor series : “If f is a function with derivatives of all order at a point a , p n ( x ) = ∑ k = 0 n a k ( x − a ) k is the n th-degree Taylor polynomial for f at a and a n = f ( n ) ( a ) n ! , n = 0 , 1 , 2 , 3 … , then ∑ k = 0 ∞ a k ( x − a ) k = a 0 + a 1 ( x − a ) + a 2 ( x − a ) 2 + a 3 ( x − a ) 3 + … + a n ( x − a ) n + ⋯ is the Taylor series for f at a. The Taylor series converges at x if lim n → ∞ p n ( x ) = lim n → ∞ ∑ k = 0 n a k ( x − a ) k exists and diverges at x if this limit does not exists. The set of numbers x for which this limit exists is called the interval of convergence”. Theorem used: Interval of Convergence: Let f be a function with derivatives of all order at a number a , let a n = f ( n ) ( a ) n ! , n = 0 , 1 , 2 , ... , and let ∑ k = 0 ∞ a k ( x − a ) k = a 0 + a 1 ( x − a ) + a 2 ( x − a ) 2 + ⋯ + a n ( x − a ) n + ⋯ be the Taylor series for f at a . If a n ≠ 0 for n ≥ n 0 , then: Case 1: If lim n → ∞ | a n + 1 a n | = L > 0 and R = 1 L , then the series converges for | x − a | < R and diverges for | x − a | > R . Case 2: If lim n → ∞ | a n + 1 a n | = 0 , then the series converges for all numbers x . Case 3: If lim n → ∞ | a n + 1 a n | = ∞ , then the series converges only at x = a . Calculation: Obtain the n th-degree Taylor polynomial of the function f ( x ) at the point 0 as follows. The n th-degree Taylor polynomial of the function f ( x ) at the point 0 is p n ( x ) = ∑ k = 0 n a k x k , where a k = f ( k ) ( 0 ) k ! , n = 0 , 1 , 2 , 3 … n . In order to find the coefficient of the Taylor polynomial, obtain the derivatives of the function f ( x ) = 2 2 − x . Obtain the first derivative of the function f ( x ) , f ′ ( x ) = d d x ( 2 2 − x ) = − 2 ( 2 − x ) 2 d d x ( 2 − x ) = − ( 1 ) 2 ( 2 − x ) 2 ( − 1 ) = ( 1 ! ) 2 ( 2 − x ) 2 Obtain the second derivative of the function f ( x ) , f ″ ( x ) = d d x ( d d x ( f ( x ) ) ) = d d x ( ( 1 ! ) 2 ( 2 − x ) 2 ) = − ( 1 ! ⋅ 2 ) ⋅ 2 ( 2 − x ) 3 d d x ( 2 − x ) = ( 2 ! ) 2 ( 2 − x ) 3 Obtain the third derivative of the function f ( x ) , f ‴ ( x ) = d d x ( d d x ( f ′ ( x ) ) ) = d d x ( ( 2 ! ) 2 ( 2 − x ) 3 ) = − ( 2 ! ⋅ 3 ) 2 ( 2 − x ) 4 d d x ( 2 − x ) = ( 3 ! ) 2 ( 2 − x ) 4 In general, the n th derivative of the function f ( x ) is f ( n ) ( x ) = ( n ! ) 2 ( 2 − x ) n + 1 . Obtain the value of f ( n ) ( 0 ) as follows. f ( n ) ( 0 ) = ( n ! ) 2 ( 2 − 0 ) n + 1 = ( n ! ) 2 2 n + 1 = ( n ! ) 2 n That is, f ( n ) ( 0 ) = ( n ! ) 2 n

Pearson eText for Calculus for Business, Economics, Life Sciences, and Social Sciences, Brief Version -- Instant Access (Pearson+)

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ISBN: 9780137400126

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

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Chapter 10.2, Problem 37E

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To find: The nth-degree Taylor polynomial and Taylor series of the of the function f(x)=22−x at zero and the values of x for which f(x)=limn→∞pn(x).

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Chapter 10.2, Problem 36E

Chapter 10.2, Problem 38E

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Question 1: If a barometer were built using oil (p = 0.92 g/cm³) instead of mercury (p = 13.6 g/cm³), would the column of oil be higher than, lower than, or the same as the column of mercury at 1.00 atm? If the level is different, by what factor? Explain. (5 pts) Solution: A barometer works based on the principle that the pressure exerted by the liquid column balances atmospheric pressure. The pressure is given by: P = pgh Since the atmospheric pressure remains constant (P = 1.00 atm), the height of the liquid column is inversely proportional to its density: Step 1: Given Data PHg hol=hgx Poil • Density of mercury: PHg = 13.6 g/cm³ Density of oil: Poil = 0.92 g/cm³ • Standard height of mercury at 1.00 atm: hμg Step 2: Compute Height of Oil = 760 mm = 0.760 m 13.6 hoil = 0.760 x 0.92 hoil = 0.760 × 14.78 hoil = 11.23 m Step 3: Compare Heights Since oil is less dense than mercury, the column of oil must be much taller than that of mercury. The factor by which it is taller is: Final…

Question 3: A sealed flask at room temperature contains a mixture of neon (Ne) and nitrogen (N2) gases. Ne has a mass of 3.25 g and exerts a pressure of 48.2 torr. . N2 contributes a pressure of 142 torr. • What is the mass of the N2 in the flask? • Atomic mass of Ne = 20.1797 g/mol • Atomic mass of N = 14.0067 g/mol Solution: We will use the Ideal Gas Law to determine the number of moles of each gas and calculate the mass of N2. PV = nRT where: • P = total pressure • V volume of the flask (same for both gases) n = number of moles of gas • R 0.0821 L atm/mol K • T = Room temperature (assume 298 K) Since both gases are in the same flask, their partial pressures correspond to their mole fractions. Step 1: Convert Pressures to Atmospheres 48.2 PNe = 0.0634 atm 760 142 PN2 = = 0.1868 atm 760 Step 2: Determine Moles of Ne nNe = mass molar mass 3.25 nNe 20.1797 nne 0.1611 mol Step 3: Use Partial Pressure Ratio to Find n

Chapter 10 Solutions

Pearson eText for Calculus for Business, Economics, Life Sciences, and Social Sciences, Brief Version -- Instant Access (Pearson+)

Ch. 10.1 - Find the nth derivative of f(x)=lnx.Ch. 10.1 - Prob. 2MP Ch. 10.1 - Prob. 3MP Ch. 10.1 - Find the second-degree Taylor polynomial at a = 8...Ch. 10.1 - Prob. 5MP Ch. 10.1 - Prob. 1ED Ch. 10.1 - (A)Let p(x) be a polynomial of degree n 1....Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E

Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - In Problems 1316, find f(3)(x). 15.f(x)=ex Ch. 10.1 - In Problems 1316, find f(3)(x). 16.f(x)=x Ch. 10.1 - Prob. 17E Ch. 10.1 - In Problems 1720, find f4(x). 18.f(x)=e5x Ch. 10.1 - Prob. 19E Ch. 10.1 - In Problems 1720, find f4(x). 20.f(x)=12+x Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - In Problems 2128, find the indicated Taylor...Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - Prob. 32E Ch. 10.1 - Prob. 33E Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.1 - Use the third-degree Taylor polynomial at 0 for...Ch. 10.1 - Prob. 41E Ch. 10.1 - Use the third-degree Taylor polynomial at 4 for...Ch. 10.1 - Prob. 43E Ch. 10.1 - Prob. 44E Ch. 10.1 - Prob. 45E Ch. 10.1 - Prob. 46E Ch. 10.1 - Prob. 47E Ch. 10.1 - Prob. 48E Ch. 10.1 - Prob. 49E Ch. 10.1 - Prob. 50E Ch. 10.1 - Prob. 51E Ch. 10.1 - Prob. 52E Ch. 10.1 - Prob. 53E Ch. 10.1 - Prob. 54E Ch. 10.1 - Prob. 55E Ch. 10.1 - Prob. 56E Ch. 10.1 - Prob. 57E Ch. 10.1 - Prob. 58E Ch. 10.1 - Prob. 59E Ch. 10.1 - Prob. 60E Ch. 10.1 - Prob. 61E Ch. 10.1 - Prob. 62E Ch. 10.1 - Prob. 63E Ch. 10.1 - Prob. 64E Ch. 10.1 - Prob. 65E Ch. 10.1 - Prob. 66E Ch. 10.1 - Prob. 67E Ch. 10.1 - Prob. 68E Ch. 10.1 - Prob. 69E Ch. 10.1 - Prob. 70E Ch. 10.1 - Prob. 71E Ch. 10.1 - Consider f(x) = ln (1 + x) and its third-degree...Ch. 10.1 - Prob. 73E Ch. 10.1 - Prob. 74E Ch. 10.1 - Prob. 75E Ch. 10.1 - Prob. 76E Ch. 10.1 - Prob. 77E Ch. 10.1 - Prob. 78E Ch. 10.1 - Prob. 79E Ch. 10.1 - Prob. 80E Ch. 10.1 - Prob. 81E Ch. 10.1 - Average price. Given the demand equation...Ch. 10.1 - Prob. 83E Ch. 10.1 - Prob. 84E Ch. 10.1 - Prob. 85E Ch. 10.1 - Prob. 86E Ch. 10.1 - Prob. 87E Ch. 10.1 - Prob. 88E Ch. 10.1 - Prob. 89E Ch. 10.1 - Prob. 90E Ch. 10.1 - Prob. 91E Ch. 10.1 - Prob. 92E Ch. 10.1 - Prob. 93E Ch. 10.1 - Prob. 94E Ch. 10.1 - Prob. 95E Ch. 10.1 - Prob. 96E Ch. 10.1 - Prob. 97E Ch. 10.1 - Prob. 98E Ch. 10.2 - Prob. 1MP Ch. 10.2 - Prob. 2MP Ch. 10.2 - Prob. 3MP Ch. 10.2 - Prob. 1ED Ch. 10.2 - (A)The six functions pn(x)=1+x++xn, n = 1, 2, , 6,...Ch. 10.2 - Prob. 1E Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - Prob. 5E Ch. 10.2 - Prob. 6E Ch. 10.2 - Prob. 7E Ch. 10.2 - Prob. 8E Ch. 10.2 - Prob. 9E Ch. 10.2 - Prob. 10E Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Prob. 14E Ch. 10.2 - Prob. 15E Ch. 10.2 - Prob. 16E Ch. 10.2 - Prob. 17E Ch. 10.2 - Prob. 18E Ch. 10.2 - Prob. 19E Ch. 10.2 - Prob. 20E Ch. 10.2 - Prob. 21E Ch. 10.2 - Prob. 22E Ch. 10.2 - Prob. 23E Ch. 10.2 - Prob. 24E Ch. 10.2 - Prob. 25E Ch. 10.2 - Prob. 26E Ch. 10.2 - Prob. 27E Ch. 10.2 - Prob. 28E Ch. 10.2 - Prob. 29E Ch. 10.2 - Prob. 30E Ch. 10.2 - Prob. 31E Ch. 10.2 - (A) Graph the nth-degree Taylor polynomials at 0...Ch. 10.2 - Prob. 33E Ch. 10.2 - Prob. 34E Ch. 10.2 - In Problems 3338, find the nth-degree Taylor...Ch. 10.2 - Prob. 36E Ch. 10.2 - Prob. 37E Ch. 10.2 - Prob. 38E Ch. 10.2 - Prob. 39E Ch. 10.2 - Prob. 40E Ch. 10.2 - Prob. 41E Ch. 10.2 - Prob. 42E Ch. 10.2 - (A) Find the interval of convergence of the Taylor...Ch. 10.2 - Prob. 44E Ch. 10.2 - Prob. 45E Ch. 10.2 - Prob. 46E Ch. 10.2 - Prob. 47E Ch. 10.2 - Prob. 48E Ch. 10.2 - Prob. 49E Ch. 10.2 - Problems 4750 require a basic knowledge of the...Ch. 10.3 - Prob. 1MP Ch. 10.3 - Find the Taylor series at 0 for f(x) = 3x3 ln(1 ...Ch. 10.3 - Prob. 3MP Ch. 10.3 - Prob. 4MP Ch. 10.3 - Prob. 5MP Ch. 10.3 - Prob. 6MP Ch. 10.3 - Prob. 7MP Ch. 10.3 - Prob. 8MP Ch. 10.3 - Prob. 1ED Ch. 10.3 - Prob. 2ED Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - Prob. 3E Ch. 10.3 - Prob. 4E Ch. 10.3 - Prob. 5E Ch. 10.3 - Prob. 6E Ch. 10.3 - Prob. 7E Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - Prob. 10E Ch. 10.3 - Prob. 11E Ch. 10.3 - Prob. 12E Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Prob. 15E Ch. 10.3 - Prob. 16E Ch. 10.3 - Solve the problems by performing operations on the...Ch. 10.3 - Prob. 18E Ch. 10.3 - Prob. 19E Ch. 10.3 - Prob. 20E Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Prob. 24E Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Prob. 27E Ch. 10.3 - Prob. 28E Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Prob. 31E Ch. 10.3 - Prob. 32E Ch. 10.3 - Prob. 33E Ch. 10.3 - Find the Taylor series at 0 for (A) f(x)=x1x2 (B)...Ch. 10.3 - Prob. 35E Ch. 10.3 - If f(x) satisfies f(x) = ln (1 + x2) and f(0) = 1,...Ch. 10.3 - Prob. 37E Ch. 10.3 - Prob. 38E Ch. 10.3 - Prob. 39E Ch. 10.3 - Prob. 40E Ch. 10.3 - Prob. 41E Ch. 10.3 - Prob. 42E Ch. 10.3 - Prob. 43E Ch. 10.3 - Prob. 44E Ch. 10.3 - Prob. 45E Ch. 10.3 - Prob. 46E Ch. 10.3 - Prob. 47E Ch. 10.3 - Prob. 48E Ch. 10.3 - Prob. 49E Ch. 10.3 - Prob. 50E Ch. 10.3 - Prob. 51E Ch. 10.3 - Prob. 52E Ch. 10.3 - Prob. 53E Ch. 10.3 - Prob. 54E Ch. 10.3 - Prob. 55E Ch. 10.3 - Prob. 56E Ch. 10.3 - Prob. 57E Ch. 10.3 - Prob. 58E Ch. 10.3 - Prob. 59E Ch. 10.3 - Prob. 60E Ch. 10.3 - Prob. 61E Ch. 10.3 - Prob. 62E Ch. 10.3 - Prob. 63E Ch. 10.3 - Prob. 64E Ch. 10.3 - Prob. 65E Ch. 10.3 - Prob. 66E Ch. 10.4 - Prob. 1MP Ch. 10.4 - Prob. 2MP Ch. 10.4 - Prob. 3MP Ch. 10.4 - Prob. 4MP Ch. 10.4 - Prob. 1ED Ch. 10.4 - Suppose you wish to use a Taylor series for...Ch. 10.4 - Prob. 1E Ch. 10.4 - Prob. 2E Ch. 10.4 - Prob. 3E Ch. 10.4 - Prob. 4E Ch. 10.4 - Prob. 5E Ch. 10.4 - Prob. 6E Ch. 10.4 - Prob. 7E Ch. 10.4 - Prob. 8E Ch. 10.4 - Prob. 9E Ch. 10.4 - Prob. 10E Ch. 10.4 - Prob. 11E Ch. 10.4 - Prob. 12E Ch. 10.4 - Prob. 13E Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Prob. 16E Ch. 10.4 - Prob. 17E Ch. 10.4 - Prob. 18E Ch. 10.4 - Prob. 19E Ch. 10.4 - Prob. 20E Ch. 10.4 - Prob. 21E Ch. 10.4 - Prob. 22E Ch. 10.4 - Prob. 23E Ch. 10.4 - Prob. 24E Ch. 10.4 - Prob. 25E Ch. 10.4 - Prob. 26E Ch. 10.4 - Prob. 27E Ch. 10.4 - Prob. 28E Ch. 10.4 - Prob. 29E Ch. 10.4 - Prob. 30E Ch. 10.4 - Prob. 31E Ch. 10.4 - Prob. 32E Ch. 10.4 - In Problems 938, use Theorem 1 to perform the...Ch. 10.4 - Prob. 34E Ch. 10.4 - Prob. 35E Ch. 10.4 - Prob. 36E Ch. 10.4 - Prob. 37E Ch. 10.4 - Prob. 38E Ch. 10.4 - Prob. 39E Ch. 10.4 - Prob. 40E Ch. 10.4 - Prob. 41E Ch. 10.4 - Prob. 42E Ch. 10.4 - Prob. 43E Ch. 10.4 - Prob. 44E Ch. 10.4 - In Problems 4548, use the second-degree Taylor...Ch. 10.4 - Prob. 46E Ch. 10.4 - In Problems 4548, use the second-degree Taylor...Ch. 10.4 - Prob. 48E Ch. 10.4 - Prob. 49E Ch. 10.4 - Prob. 50E Ch. 10.4 - Prob. 51E Ch. 10.4 - To estimate 01.511+x2dx a student takes the first...Ch. 10.4 - There are different ways to approximate a function...Ch. 10.4 - There are different ways to approximate a function...Ch. 10.4 - In Problems 5566, use Theorem 1 to perform the...Ch. 10.4 - Prob. 56E Ch. 10.4 - Prob. 57E Ch. 10.4 - Prob. 58E Ch. 10.4 - Useful life. A computer store rents time on...Ch. 10.4 - Average price. Given the demand equation...Ch. 10.4 - Temperature. The temperature (in degrees Celsius)...Ch. 10.4 - Temperature. Repeat Problem 61 for...Ch. 10.4 - Prob. 63E Ch. 10.4 - Prob. 64E Ch. 10.4 - Prob. 65E Ch. 10.4 - Prob. 66E Ch. 10 - Prob. 1RE Ch. 10 - Prob. 2RE Ch. 10 - Prob. 3RE Ch. 10 - Prob. 4RE Ch. 10 - Prob. 5RE Ch. 10 - Prob. 6RE Ch. 10 - Prob. 7RE Ch. 10 - Use Theorem 1 of Section 10.2 to find the interval...Ch. 10 - Prob. 9RE Ch. 10 - Prob. 10RE Ch. 10 - In Problems 10 and 11, use the formula an =...Ch. 10 - Prob. 12RE Ch. 10 - Prob. 13RE Ch. 10 - Prob. 14RE Ch. 10 - Prob. 15RE Ch. 10 - Prob. 16RE Ch. 10 - Prob. 17RE Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Prob. 20RE Ch. 10 - Prob. 21RE Ch. 10 - Prob. 22RE Ch. 10 - Prob. 23RE Ch. 10 - Prob. 24RE Ch. 10 - In Problems 25 and 26, use the second-degree...Ch. 10 - Prob. 26RE Ch. 10 - Prob. 27RE Ch. 10 - In Problems 27 and 28, use a Taylor polynomial at...Ch. 10 - Prob. 29RE Ch. 10 - Prob. 30RE Ch. 10 - Prob. 31RE Ch. 10 - Prob. 32RE Ch. 10 - Prob. 33RE Ch. 10 - Prob. 34RE Ch. 10 - Prob. 35RE Ch. 10 - Prob. 36RE Ch. 10 - Prob. 37RE Ch. 10 - Prob. 38RE Ch. 10 - Prob. 39RE Ch. 10 - Prob. 40RE Ch. 10 - Medicine. The rate of healing for a skin wound (in...Ch. 10 - Prob. 42RE Ch. 10 - Prob. 43RE

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The n th-degree Taylor polynomial and Taylor series of the of the function f ( x ) = 2 2 − x at zero and the values of x for which f ( x ) = lim n → ∞ p n ( x ) .

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To find: The nth-degree Taylor polynomial and Taylor series of the of the function f(x)=22−x at zero and the values of x for which f(x)=limn→∞pn(x).

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Chapter 10 Solutions