Answer:
The correct answers are:
1) [tex]f(x)=-5^x[/tex]
2) [tex]f(x)=7-12x[/tex]
3) [tex]f(x)=-3(3^x)[/tex]
Step-by-step explanation:
The rate of change of a function in the interval [a,b] is calculated by:
[tex]Rate\ of\ change=\dfrac{f(b)-f(a)}{b-a}[/tex]
We have: [a,b]=[0,2]
1)
[tex]f(x)=-5^x[/tex]
Hence, we have:
[tex]f(2)=-5^2\\\\\\f(2)=-25[/tex]
and
[tex]f(0)=-5^0\\\\\\f(0)=-1[/tex]
Hence, the rate of change in the interval [0,2] is:
[tex]=\dfrac{-25-(-1)}{2-0}\\\\\\=\dfrac{-24}{2}\\\\\\=-12[/tex]
Hence, option: (1) is correct.
2)
[tex]f(x)=7-12x[/tex]
We know that for any linear function of the type y=mx+c
The rate of change in any interval =m
Here we have: m= -12
Hence, Rate of change= -12
Option: 2 is correct.
3)
[tex]f(x)=-3(3^x)[/tex]
[tex]f(2)=-3\times 3^2\\\\\\f(2)=-27[/tex]
and
[tex]f(0)=-3\times 3^0\\\\\\f(0)=-3[/tex]
Hence, Rate of change is:
[tex]=\dfrac{-27-(-3)}{2-0}\\\\\\=\dfrac{-24}{2}\\\\\\=-12[/tex]
Hence, option: 3 is correct.
4)
[tex]f(x)=-12^x[/tex]
[tex]f(2)=-12^2=-144[/tex]
[tex]f(0)=-12^0=-1[/tex]
Hence, rate of change is:
[tex]=\dfrac{-144-(-1)}{2-0}\\\\\\=\dfrac{-143}{2}[/tex]
Option: 4 is incorrect.
5)
[tex]f(x)=-\dfrac{1}{12}x+9[/tex]
It is linear function.
[tex]Hence\ ,\ rate\ of\ change=\dfrac{-1}{12}[/tex]
Hence, option: 5 is incorrect.
6)
[tex]f(x)=12x-\dfrac{2}{3}[/tex]
Again it is a linear function with rate of change: 12
Hence, option: 6 is incorrect.
