Deck 11: Taylor Polynomials and Infinite Series

Question 1

Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to six decimal places, if necessary.
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Answer 1

2.659210

Question 2

Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to six decimal places, if necessary.
​ ; ;

Answer 2

2.410167

Question 3

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.
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The zero of between and , .

Answer 3

1.91515

Question 4

A division manufactures the "Futura" model microwave oven. Given that the daily cost of producing these microwave ovens (in dollars) obeys the rule
​ ​
where x stands for the number of units produced, find the level of production that minimizes the daily average cost per unit. Round your answer to the nearest integer.
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__________ units per day

Question 5

Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to five decimal places, if necessary.
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Question 6

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.
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The zero of between and , .

Question 7

Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to six decimal places, if necessary.
​

Question 8

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.
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The zero of between and ,

Question 9

Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to six decimal places, if necessary.
​

Question 10

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.
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The zero of between and , .

Question 11

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.
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The zero of between and , .

Question 12

The altitude (in feet) of a rocket t sec into flight is given by
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Find the time T when the rocket hits Earth. Round your answer to the nearest second.
​ __________ sec

Question 13

Let ​
Use the Newton method to find the zero of in the interval .
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Round your answer to five decimal places, if necessary.

Question 14

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.
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The zero of between and , .

Question 15

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.
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The zero of between and , .

Question 16

Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to six decimal places, if necessary.
​
​

Question 17

The temperature at 5 A.M. on a certain December day was measured at 18°F. In the next t hr, the temperature was given by the function
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where T is measured in degrees Fahrenheit. At what time was the temperature 0 °F?
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_____ : _____ P.M.

Question 18

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.
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The zero of between and , .

Question 19

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.
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__________ % per year

Question 20

The proprietor of Qwik Film Lab recently purchased $12,400 of new film-processing equipment. She expects that this investment, which has a useful life of 4 yr, will yield returns of $4,400 at the end of the first year, $5,200 at the end of the second year, $4,300 at the end of the third year, and $3,100 at the end of the fourth year. Find the internal rate of return on the investment.
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Round your answer to the nearest hundredth, if necessary.
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__________ % per yr

Question 21

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series.
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Question 22

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series.
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Question 23

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

Question 24

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

Question 25

Use the eighth-degree Taylor polynomial to approximate
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Round the answer to four decimal places, if necessary.

Question 26

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series.
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Question 27

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

Question 28

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series.
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Question 29

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

Question 30

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

Question 31

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series.
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Question 32

Use the sixth-degree Taylor polynomial to approximate
​ ​

Question 33

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.
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Round the answer to the nearest hundredth, if necessary.
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__________%

Question 34

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

Question 35

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series.
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Question 36

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series.
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Question 37

Integrate the power series for to obtain a power series representation for the function .

Question 38

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

Question 39

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series.
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Question 40

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series.
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Question 41

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

Question 42

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

Question 43

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

Question 44

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

Question 45

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

Question 46

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

Question 47

Use the integral test to determine whether the following series is convergent or divergent.
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Question 48

Use the integral test to determine whether the following series is convergent or divergent.
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Question 49

Use the integral test to determine whether the following series is convergent or divergent.
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Question 50

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

Question 51

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

Question 52

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

Question 53

Use the integral test to determine whether the following series is convergent or divergent.
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Question 54

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

Question 55

Use the integral test to determine whether the following series is convergent or divergent.
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Question 56

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

Question 57

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

Question 58

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

Question 59

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

Question 60

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

Question 61

Determine whether the following series is convergent or divergent.
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Question 62

Use the comparison test to determine whether the following series is convergent or divergent.
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Question 63

Consider the series
​ .
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Determine whether the geometric series converges or diverges.
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If it converges, find its sum.

Question 64

Show that the series is divergent.

Question 65

Determine whether the following series is convergent or divergent.
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Question 66

Determine whether the following p-series is convergent or divergent.
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Question 67

Determine whether the following series is convergent or divergent.
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Question 68

Use the comparison test to determine whether the following series is convergent or divergent.
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Question 69

Determine whether the following p-series is convergent or divergent.
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Question 70

Find a formula for the capital value of a perpetuity involving payments of P dollars paid at the end of each investment period into a fund that earns interest at the annual rate of k% compounded continuously.
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Question 71

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

Question 72

Determine whether the following series is convergent or divergent.
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Question 73

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

Question 74

Determine whether the following p-series is convergent or divergent.
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Question 75

Suppose the average wage earner saves 15% of her take-home pay and spends the other 85%. Estimate the impact that a proposed $25 billion tax cut will have on the economy over the long run due to the additional spending generated. Round the answer to the nearest billion dollars.
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$ __________ billion

Question 76

Use the comparison test to determine whether the following series is convergent or divergent.
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Question 77

Determine whether the following series is convergent or divergent.
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Question 78

Consider the series
​ .
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Determine whether the geometric series converges or diverges.
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If it converges, find its sum.

Question 79

Consider the series
​ .
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Determine whether the geometric series converges or diverges.
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If it converges, find its sum.

Question 80

Use the comparison test to determine whether the series is convergent or divergent.
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