problem 1\nthe temperature was recorded at several times in a\n24 - hour period. function ( f(n) ) gives…

problem 1\nthe temperature was recorded at several times in a\n24 - hour period. function ( f(n) ) gives the\ntemperature in degrees fahrenheit ( n ) hours after\nmidnight.\nuse the graph to determine if the average rate of\nchange for each interval is positive, negative, or\nzero.\npositive\nnegative\nzero\n( n = 1 ) to ( n = 5 )\npositive\nnegative\nzero\n( n = 5 ) to ( n = 7 )\npositive\nnegative\nzero\n( n = 10 ) to ( n = 20 )

problem 1\nthe temperature was recorded at several times in a\n24 - hour period. function ( f(n) ) gives the\ntemperature in degrees fahrenheit ( n ) hours after\nmidnight.\nuse the graph to determine if the average rate of\nchange for each interval is positive, negative, or\nzero.\npositive\nnegative\nzero\n( n = 1 ) to ( n = 5 )\npositive\nnegative\nzero\n( n = 5 ) to ( n = 7 )\npositive\nnegative\nzero\n( n = 10 ) to ( n = 20 )

Answer

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y = f(x)) over the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}). In the context of the function (t(n)) (where (y=t(n)) and (x = n)), the average rate of change over the interval ([n_1,n_2]) is (\frac{t(n_2)-t(n_1)}{n_2 - n_1}). Geometrically, if the function is increasing (going up from left - to - right) over the interval, the average rate of change is positive. If it is decreasing (going down from left - to - right), the average rate of change is negative.

Step2: Analyze the interval (n = 1) to (n=5)

Looking at the graph, as (n) (hours after midnight) increases from (n = 1) to (n = 5), the temperature (t(n)) (in degrees Fahrenheit) is increasing. For example, if we assume two points ((n_1,t(n_1))) and ((n_2,t(n_2))) with (n_1 = 1) and (n_2=5), since (t(n_2)>t(n_1)) and (n_2 - n_1=5 - 1>0), the average rate of change (\frac{t(5)-t(1)}{5 - 1}>0)

Step3: Analyze the interval (n = 5) to (n = 7)

As (n) increases from (n = 5) to (n = 7), the temperature (t(n)) is decreasing. If we take two points ((n_1,t(n_1))) with (n_1 = 5) and ((n_2,t(n_2))) with (n_2 = 7), since (t(n_2)<t(n_1)) and (n_2 - n_1=7 - 5>0), the average rate of change (\frac{t(7)-t(5)}{7 - 5}<0)

Step4: Analyze the interval (n = 10) to (n = 20)

As (n) increases from (n = 10) to (n = 20), the temperature (t(n)) is decreasing. If we take two points ((n_1,t(n_1))) with (n_1 = 10) and ((n_2,t(n_2))) with (n_2 = 20), since (t(n_2)<t(n_1)) and (n_2 - n_1=20 - 10>0), the average rate of change (\frac{t(20)-t(10)}{20 - 10}<0)

Answer:

  • For (n = 1) to (n = 5): Positive
  • For (n = 5) to (n = 7): Negative
  • For (n = 10) to (n = 20): Negative