Question

Type the correct answer in each box. Round your answers to two decimal places. Find the average rate of change of f(x) = log2(3x − 6) on [3, 4], [4, 5], and [5, 6].

Answer 1

Answer:CORRECT ANSWER ON PLATO

[3,4]= 1

[4,5]=0.59

[5,6]= 0.41

Answer 2

Answer:

on [3, 4] = 0.30

on [4, 5] = 0.18

on [5, 6] = 0.12

Step-by-step explanation:

The average rate of change f, of a function f(x) on an interval [a, b] is given by;

$f = \frac{f(b) - f(a)}{b - a}$              -------------(i)

In our case,

f(x) = log 2(3x - 6)

Now let's get the average rate of change of f(x) on;

(i) [3, 4]

Here, a = 3 and b = 4

f(a) = f(3)        [This is f(x) at x = 3]

=> f(3) = log[2(3(3) - 6)]

=> f(3) = log[2(9 - 6)]

=> f(3) = log[2(3)]

=> f(3) = log[6]

Also,

f(b) = f(4)        [This is f(x) at x = 4]

=> f(4) = log[2(3(4) - 6)]

=> f(4) = log[2(12 - 6)]

=> f(4) = log[2(6)]

=> f(4) = log[12]

Now substitute the values of a, b, f(a) and f(b) into equation (i) as follows;

$f = \frac{log 12 - log 6}{4 - 3}$             [Remember that log m - log n = log (m / n)]

$f = \frac{log (12 / 6)}{4 - 3}$

$f = \frac{log (2)}{1}$

f = log 2 = 0.3010

f = 0.30          [to two decimal places]

∴ The average rate of change on [3, 4] = 0.30

(ii) [4, 5]

Here, a = 4 and b = 5

f(a) = f(4)        [This is f(x) at x = 4]

=> f(4) = log[2(3(4) - 6)]

=> f(4) = log[2(12 - 6)]

=> f(4) = log[2(6)]

=> f(4) = log[12]

Also,

f(b) = f(5)        [This is f(x) at x = 5]

=> f(5) = log[2(3(5) - 6)]

=> f(5) = log[2(15 - 6)]

=> f(5) = log[2(9)]

=> f(5) = log[18]

Now substitute the values of a, b, f(a) and f(b) into equation (i) as follows;

$f = \frac{log 18 - log 12}{5 - 4}$             [Remember that log m - log n = log (m / n)]

$f = \frac{log (18 / 12)}{5 - 4}$

$f = \frac{log (1.5)}{1}$

f = log 1.5 = 0.176

f = 0.18          [to two decimal places]

∴ The average rate of change on [4, 5] = 0.18

(iii) [5, 6]

Here, a = 5 and b = 6

f(a) = f(5)        [This is f(x) at x = 5]

=> f(5) = log[2(3(5) - 6)]

=> f(5) = log[2(15 - 6)]

=> f(5) = log[2(9)]

=> f(5) = log[18]

Also,

f(b) = f(6)        [This is f(x) at x = 6]

=> f(6) = log[2(3(6) - 6)]

=> f(6) = log[2(18 - 6)]

=> f(6) = log[2(12)]

=> f(6) = log[24]

Now substitute the values of a, b, f(a) and f(b) into equation (i) as follows;

$f = \frac{log 24 - log 18}{6 - 5}$             [Remember that log m - log n = log (m / n)]

$f = \frac{log (24 / 18)}{6 - 5}$

$f = \frac{log (1.33)}{1}$

f = log 1.33 = 0.124

f = 0.12         [to two decimal places]

∴ The average rate of change on [5, 6] = 0.12