   Chapter 10.4, Problem 34E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Science Research in Europe The following table shows the number of science research articles authored by researchers in the European Union during the period 1980-2010: year  t (  year since 1980 ) 0 5 10 15 20 25 30 Articales   N ( t ) ( thousands  ) 140 170 190 260 300 340 430 a. Find the interval(s) over which the average rate of change of N was the least positive. What was that rate of change? Interpret your answer.b. The percentage change of N over the interval [ a , b ] is defined to bePercentage change of N =  Change in  N First value = N ( b ) − N ( a ) N ( a ) Compute the percentage change of N over the interval [ 10 , 30 ] and also the average rate of change. Interpret the answers.

(a)

To determine

To calculate: The interval at which the average rate of change of N is least positive, where N is the number of science research article. Also calculate the rate of change of N and interpreted the answer. The number of science research article with respect to years is shown below.

 Year t (year since 1980) 0 5 10 15 20 25 30 Article N(t)(thousand) 140 170 190 260 300 340 430
Explanation

Given Information:

The number of science research article with respect to years is shown below.

 Year t (year since 1980) 0 5 10 15 20 25 30 Article N(t)(thousand) 140 170 190 260 300 340 430

Formula used:

Average rate of change of f(x) over the interval [a,b] is:

Average rate of change of f=Change in fChange in x=ΔfΔx=f(b)f(a)ba

Calculation:

Consider the table shown below:

 Year t (year since 1980) 0 5 10 15 20 25 30 Article N(t)(thousand) 140 170 190 260 300 340 430

The possible intervals are [0,5], [5,10], [10,15], [15,20], [20,25], [25,30].

Compute the average rate of change for every interval:

Apply the average rate of change formula for the interval [0,5]:

Average rate of change of N=N(5)N(0)50

From the table, the values are N(5)=170 and N(0)=140:

Now, substitute the values in the formula:

N(5)N(0)50=1701405=305=6

Apply the average rate of change formula for the interval [5,10]:

Average rate of change of N=N(10)N(5)105

From the table, the values are N(5)=170 and N(10)=190:

Now, substitute the values in the formula:

N(10)N(5)105=1901705=205=4

Apply the average rate of change formula for the interval [10,15]:

Average rate of changex

(b)

To determine

To calculate: The percentage change and average rate of change of the N over the interval [10,30] and also to interpret the solution where the number of science research article with respect to years is shown below.

 Year t (year since 1980) 0 5 10 15 20 25 30 Article N(t)(thousand) 140 170 190 260 300 340 430

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