Explanation Results used: Basic Taylor series: Function f ( x ) Taylor series at 0 Interval of convergence ln ( 1 + x ) x − 1 2 x 2 + 1 3 x 3 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 n x n + ⋅ ⋅ ⋅ − 1 < x < 1 Table 1 Properties used: Integration : If f ( x ) = a 0 + a 1 x + a 2 x 2 + ⋅ ⋅ ⋅ + a n x n + ⋅ ⋅ ⋅ be a Taylor series at 0, then ∫ f ( x ) d x = C + a 0 x + 1 2 a 1 x 2 + 1 3 a 2 x 3 + ⋅ ⋅ ⋅ + 1 n + 1 a n x n + 1 + ⋅ ⋅ ⋅ , where C is the constant of integration. The Taylor series for ∫ f ( x ) d x has the same interval of convergence as the Taylor series for f . Calculation: Consider derivative function f ″ ( x ) = ln ( 1 + x ) . From Table 1, the Taylor series at 0 for ln ( 1 + x ) is x − 1 2 x 2 + 1 3 x 3 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 n x n + ⋅ ⋅ ⋅ . Since ∫ f ″ ( x ) d x = f ′ ( x ) , integrate on both sides of the equation f ″ ( x ) = x − 1 2 x 2 + 1 3 x 3 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 n x n + ⋅ ⋅ ⋅ and simplify as follows. ∫ f ″ ( x ) d x = ∫ ( x − 1 2 x 2 + 1 3 x 3 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 n x n + ⋅ ⋅ ⋅ ) d x = ∫ x d x − 1 2 ∫ x 2 d x + 1 3 ∫ x 3 d x − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 n ∫ x n d x + ⋅ ⋅ ⋅ = C + x 1 + 1 1 + 1 − 1 2 ( x 2 + 1 2 + 1 ) + 1 3 ( x 3 + 1 3 + 1 ) − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 n ( x n + 1 n + 1 ) + ⋅ ⋅ ⋅ = C + x 2 2 − x 3 6 + x 4 12 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 x n + 1 n ( n + 1 ) + ⋅ ⋅ ⋅ So, f ′ ( x ) = C + x 2 2 − x 3 6 + x 4 12 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 x n + 1 n ( n + 1 ) + ⋅ ⋅ ⋅ . Substitute f ′ ( 0 ) = 3 in f ′ ( x ) = C + x 2 2 − x 3 6 + x 4 12 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 x n + 1 n ( n + 1 ) + ⋅ ⋅ ⋅ and obtain the value of C as follows. f ′ ( x ) = C + x 2 2 − x 3 6 + x 4 12 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 x n + 1 n ( n + 1 ) + ⋅ ⋅ ⋅ f ′ ( 0 ) = C + 0 2 2 − 0 3 6 + 0 4 12 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 ( 0 n + 1 ) n ( n + 1 ) + ⋅ ⋅ ⋅ 3 = C + 0 − 0 + 0 − ⋅ ⋅ ⋅ + 0 + ⋅ ⋅ ⋅ C = 3 Substitute the value of C in f ′ ( x ) = C + x 2 2 − x 3 6 + x 4 12 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 x n + 1 n ( n + 1 ) + ⋅ ⋅ ⋅ , f ′ ( x ) = 3 + x 2 2 − x 3 6 + x 4 12 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 x n + 1 n ( n + 1 ) + ⋅ ⋅ ⋅ . Since ∫ f ′ ( x ) d x = f ( x ) , integrate on both sides of the equation f ′ ( x ) = 3 + x 2 2 − x 3 6 + x 4 12 − ⋅ ⋅ ⋅ + ( − 1 ) n − 1 x n + 1 n ( n + 1 ) + ⋅ ⋅ ⋅ and simplify as follows

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Chapter 10.3, Problem 57E

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To find: The Taylor series at 0 for f if f″(x)=ln(1+x), f′(0)=3 and f(0)=−2.

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Chapter 10.3, Problem 56E

Chapter 10.3, Problem 58E

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You may need to use the appropriate appendix table or technology to answer this question. You are given the following information obtained from a random sample of 4 observations. 24 48 31 57 You want to determine whether or not the mean of the population from which this sample was taken is significantly different from 49. (Assume the population is normally distributed.) (a) State the null and the alternative hypotheses. (Enter != for ≠ as needed.) H0: Ha: (b) Determine the test statistic. (Round your answer to three decimal places.) (c) Determine the p-value, and at the 5% level of significance, test to determine whether or not the mean of the population is significantly different from 49. Find the p-value. (Round your answer to four decimal places.) p-value = State your conclusion. Reject H0. There is insufficient evidence to conclude that the mean of the population is different from 49.Do not reject H0. There is sufficient evidence to conclude that the…

Chapter 10 Solutions

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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. 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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 - 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The Taylor series at 0 for f if f ″ ( x ) = ln ( 1 + x ) , f ′ ( 0 ) = 3 and f ( 0 ) = − 2 .

Solution Summary: The author explains the Taylor series at 0 for f and its interval of convergence.

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To find: The Taylor series at 0 for f if f″(x)=ln(1+x), f′(0)=3 and f(0)=−2.

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Chapter 10 Solutions