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Find the average rate of change of f ( x ) = 3 x 2 − 9 on the interval [ 2 , b ] . Your answer will be an expression involving b

Sagot :

MrRoyal

Answer:

[tex]m = 3b+6[/tex]

Step-by-step explanation:

Given

[tex]f(x)=3x^2 - 9[/tex]

Required

The average rate over [tex][2,b][/tex]

Average rate (m) is calculated using:

[tex]m = \frac{f(b) - f(a)}{b - a}[/tex]

Where

[tex][a,b] = [2,b][/tex]

So, we have:

[tex]m = \frac{f(b) - f(2)}{b - 2}[/tex]

Calculate f(b) and f(2)

[tex]f(x)=3x^2 - 9[/tex]

[tex]f(b)=3b^2 - 9[/tex]

[tex]f(2)=3*2^2 - 9 = 12 - 9 = 3[/tex]

So, we have:

[tex]m = \frac{f(b) - f(2)}{b - 2}[/tex]

[tex]m = \frac{3b^2 - 9 - 3}{b - 2}[/tex]

[tex]m = \frac{3b^2 - 12}{b - 2}[/tex]

Expand the numerator

[tex]m = \frac{3b^2 + 6b-6b-12}{b - 2}[/tex]

Factorize

[tex]m = \frac{b(3b + 6)-2(3b+6)}{b - 2}[/tex]

Factor out 3b + 6

[tex]m = \frac{(b -2)(3b+6)}{b - 2}[/tex]

Cancel out b - 2

[tex]m = 3b+6[/tex]

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