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Exam 11: Taylor Polynomials and Infinite Series

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Exam 1: Preliminaries183 Questionsadd course
Exam 2: Functions, Limits, and the Derivative250 Questionsadd course
Exam 3: Differentiation309 Questionsadd course
Exam 4: Applications of the Derivative152 Questionsadd course
Exam 5: Exponential and Logarithmic Functions256 Questionsadd course
Exam 6: Integration291 Questionsadd course
Exam 7: Additional Topics in Integration202 Questionsadd course
Exam 8: Calculus of Several Variables219 Questionsadd course
Exam 9: Differential Equations57 Questionsadd course
Exam 10: Probability and Calculus68 Questionsadd course
Exam 11: Taylor Polynomials and Infinite Series110 Questionsadd course
Exam 12: Trigonometric Functions64 Questionsadd course
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Find P​4(x), the fourth Taylor polynomial of the following function at the indicated point. ​ Find P​<sub>4</sub>(x), the fourth Taylor polynomial of the following function at the indicated point. ​ at ​ Express any non-integer coefficients as reduced fractions. at Find P​<sub>4</sub>(x), the fourth Taylor polynomial of the following function at the indicated point. ​ at ​ Express any non-integer coefficients as reduced fractions. ​ Express any non-integer coefficients as reduced fractions.

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Question 81

Consider the series Consider the series . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________ . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________

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Question 82

Determine whether the following p-series is convergent or divergent. ​ Determine whether the following p-series is convergent or divergent. ​

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Question 83

Find P​3(x), the third Taylor polynomial of the following function at the indicated point. ​ Find P​<sub>3</sub>(x), the third Taylor polynomial of the following function at the indicated point. ​ at ​ Express any non-integer coefficients as reduced fractions. at Find P​<sub>3</sub>(x), the third Taylor polynomial of the following function at the indicated point. ​ at ​ Express any non-integer coefficients as reduced fractions. ​ Express any non-integer coefficients as reduced fractions.

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Question 84

Executive Limousine Service recently acquired limousines worth $120,000. The projected returns over the next 3 yr, the time period the limousines will be in service, are $80,000 at the end of the first year, $60,000 at the end of the second year, and $40,000 at the end of the third year. Find the internal rate of return on the investment. Round your answer to the nearest hundredth, if necessary. ​ __________ % per year

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Question 85

Express the following decimal as a rational number. ​ 2.251251251...

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Question 86

Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to four decimal places, if necessary. ​ The zero of Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to four decimal places, if necessary. ​ The zero of between and , . between Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to four decimal places, if necessary. ​ The zero of between and , . and Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to four decimal places, if necessary. ​ The zero of between and , . , Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to four decimal places, if necessary. ​ The zero of between and , . .

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Question 87

Consider the function Consider the function . ​ Find the Taylor series of the function at the indicated point. ​ __________ ​ Find its radius of convergence. ​ __________ ​ Find the interval of convergence. ​ __________ . ​ Find the Taylor series of the function at the indicated point. ​ __________ ​ Find its radius of convergence. ​ __________ ​ Find the interval of convergence. ​ __________

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Question 88

Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to six decimal places, if necessary. ​ The zero of Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to six decimal places, if necessary. ​ The zero of between and , . between Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to six decimal places, if necessary. ​ The zero of between and , . and Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to six decimal places, if necessary. ​ The zero of between and , . , Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to six decimal places, if necessary. ​ The zero of between and , . .

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Question 89

Determine whether the following series is convergent or divergent. ​ Determine whether the following series is convergent or divergent. ​ ​ ​

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Question 90

Differentiate the power series for Differentiate the power series for at to obtain a series representation for the function . at Differentiate the power series for at to obtain a series representation for the function . to obtain a series representation for the function Differentiate the power series for at to obtain a series representation for the function . .

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Question 91

Use the integral test to determine whether the following series is convergent or divergent. ​ Use the integral test to determine whether the following series is convergent or divergent. ​

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Question 92

Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to five decimal places, if necessary. ​ The zero of Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to five decimal places, if necessary. ​ The zero of between and , . between Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to five decimal places, if necessary. ​ The zero of between and , . and Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to five decimal places, if necessary. ​ The zero of between and , . , Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to five decimal places, if necessary. ​ The zero of between and , . .

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Question 93

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series. ​ Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series. ​

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Question 94

The General Manager of the Service Department of MCA Television Company has estimated that the time that elapses between the dates of purchase and the dates on which the 19-in. sets manufactured by the company first require service is normally distributed according to the probability density function ​ The General Manager of the Service Department of MCA Television Company has estimated that the time that elapses between the dates of purchase and the dates on which the 19-in. sets manufactured by the company first require service is normally distributed according to the probability density function ​ ​ where x is measured in months. Determine the percentage of sets manufactured and sold by MCA that may require service 90-100 mo after purchase. ​ Round the answer to the nearest hundredth, if necessary. ​ __________% ​ where x is measured in months. Determine the percentage of sets manufactured and sold by MCA that may require service 90-100 mo after purchase. ​ Round the answer to the nearest hundredth, if necessary. ​ __________%

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Question 95

Consider the series Consider the series . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________ . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________

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Question 96

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series. ​ Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series. ​

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Question 97

Consider the series Consider the series . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________ . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________

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Question 98

Of the microprocessors manufactured by a microelectronics firm for use in regulating fuel consumption in automobiles, Of the microprocessors manufactured by a microelectronics firm for use in regulating fuel consumption in automobiles, are defective. It can be shown that the probability of getting at least one defective microprocessor in a random sample of n microprocessors is . Consider the sequence defined by . ​ Write down the terms a​<sub>1</sub>, a​<sub>10</sub>, a​<sub>100</sub>, and a​<sub>1000</sub> of the sequence . ​ a​<sub>1</sub>=__________ ​ Round the answer to three decimal places, if necessary. ​ a​<sub>10</sub>=__________ ​ Round the answer to five decimal places, if necessary. ​ a​<sub>100</sub>=__________ ​ Round the answer to nine decimal places, if necessary. ​ a​<sub>1000</sub>=__________ ​ Round the answer to the nearest whole number, if necessary. ​ Evaluate . ​ __________ are defective. It can be shown that the probability of getting at least one defective microprocessor in a random sample of n microprocessors is Of the microprocessors manufactured by a microelectronics firm for use in regulating fuel consumption in automobiles, are defective. It can be shown that the probability of getting at least one defective microprocessor in a random sample of n microprocessors is . Consider the sequence defined by . ​ Write down the terms a​<sub>1</sub>, a​<sub>10</sub>, a​<sub>100</sub>, and a​<sub>1000</sub> of the sequence . ​ a​<sub>1</sub>=__________ ​ Round the answer to three decimal places, if necessary. ​ a​<sub>10</sub>=__________ ​ Round the answer to five decimal places, if necessary. ​ a​<sub>100</sub>=__________ ​ Round the answer to nine decimal places, if necessary. ​ a​<sub>1000</sub>=__________ ​ Round the answer to the nearest whole number, if necessary. ​ Evaluate . ​ __________ . Consider the sequence Of the microprocessors manufactured by a microelectronics firm for use in regulating fuel consumption in automobiles, are defective. It can be shown that the probability of getting at least one defective microprocessor in a random sample of n microprocessors is . Consider the sequence defined by . ​ Write down the terms a​<sub>1</sub>, a​<sub>10</sub>, a​<sub>100</sub>, and a​<sub>1000</sub> of the sequence . ​ a​<sub>1</sub>=__________ ​ Round the answer to three decimal places, if necessary. ​ a​<sub>10</sub>=__________ ​ Round the answer to five decimal places, if necessary. ​ a​<sub>100</sub>=__________ ​ Round the answer to nine decimal places, if necessary. ​ a​<sub>1000</sub>=__________ ​ Round the answer to the nearest whole number, if necessary. ​ Evaluate . ​ __________ defined by Of the microprocessors manufactured by a microelectronics firm for use in regulating fuel consumption in automobiles, are defective. It can be shown that the probability of getting at least one defective microprocessor in a random sample of n microprocessors is . Consider the sequence defined by . ​ Write down the terms a​<sub>1</sub>, a​<sub>10</sub>, a​<sub>100</sub>, and a​<sub>1000</sub> of the sequence . ​ a​<sub>1</sub>=__________ ​ Round the answer to three decimal places, if necessary. ​ a​<sub>10</sub>=__________ ​ Round the answer to five decimal places, if necessary. ​ a​<sub>100</sub>=__________ ​ Round the answer to nine decimal places, if necessary. ​ a​<sub>1000</sub>=__________ ​ Round the answer to the nearest whole number, if necessary. ​ Evaluate . ​ __________ . ​ Write down the terms a​1, a​10, a​100, and a​1000 of the sequence Of the microprocessors manufactured by a microelectronics firm for use in regulating fuel consumption in automobiles, are defective. It can be shown that the probability of getting at least one defective microprocessor in a random sample of n microprocessors is . Consider the sequence defined by . ​ Write down the terms a​<sub>1</sub>, a​<sub>10</sub>, a​<sub>100</sub>, and a​<sub>1000</sub> of the sequence . ​ a​<sub>1</sub>=__________ ​ Round the answer to three decimal places, if necessary. ​ a​<sub>10</sub>=__________ ​ Round the answer to five decimal places, if necessary. ​ a​<sub>100</sub>=__________ ​ Round the answer to nine decimal places, if necessary. ​ a​<sub>1000</sub>=__________ ​ Round the answer to the nearest whole number, if necessary. ​ Evaluate . ​ __________ . ​ a​1=__________ ​ Round the answer to three decimal places, if necessary. ​ a​10=__________ ​ Round the answer to five decimal places, if necessary. ​ a​100=__________ ​ Round the answer to nine decimal places, if necessary. ​ a​1000=__________ ​ Round the answer to the nearest whole number, if necessary. ​ Evaluate Of the microprocessors manufactured by a microelectronics firm for use in regulating fuel consumption in automobiles, are defective. It can be shown that the probability of getting at least one defective microprocessor in a random sample of n microprocessors is . Consider the sequence defined by . ​ Write down the terms a​<sub>1</sub>, a​<sub>10</sub>, a​<sub>100</sub>, and a​<sub>1000</sub> of the sequence . ​ a​<sub>1</sub>=__________ ​ Round the answer to three decimal places, if necessary. ​ a​<sub>10</sub>=__________ ​ Round the answer to five decimal places, if necessary. ​ a​<sub>100</sub>=__________ ​ Round the answer to nine decimal places, if necessary. ​ a​<sub>1000</sub>=__________ ​ Round the answer to the nearest whole number, if necessary. ​ Evaluate . ​ __________ . ​ Of the microprocessors manufactured by a microelectronics firm for use in regulating fuel consumption in automobiles, are defective. It can be shown that the probability of getting at least one defective microprocessor in a random sample of n microprocessors is . Consider the sequence defined by . ​ Write down the terms a​<sub>1</sub>, a​<sub>10</sub>, a​<sub>100</sub>, and a​<sub>1000</sub> of the sequence . ​ a​<sub>1</sub>=__________ ​ Round the answer to three decimal places, if necessary. ​ a​<sub>10</sub>=__________ ​ Round the answer to five decimal places, if necessary. ​ a​<sub>100</sub>=__________ ​ Round the answer to nine decimal places, if necessary. ​ a​<sub>1000</sub>=__________ ​ Round the answer to the nearest whole number, if necessary. ​ Evaluate . ​ __________ __________

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Question 99

Use the integral test to determine whether the following series is convergent or divergent. ​ Use the integral test to determine whether the following series is convergent or divergent. ​

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Question 100
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