Compare the estimated average rates of change of the functions f(x)=3x−4−−−−−√ and g(x)=2x−43−−−−−√3 over the interval [2,3]. State the difference in the estimated average rates of change to the nearest tenth.
To find the average rate of change of a function over an interval, we can use the formula:
Average Rate of Change = (f(b) - f(a))/(b - a)
For f(x) = √(3x - 4), the average rate of change over the interval [2, 3] can be calculated as:
[√(3(3) - 4) - √(3(2) - 4)]/(3 - 2)
= (√(5) - √2)/(1)
For g(x) = √3(2x - 4), the average rate of change over the interval [2, 3] can be calculated as:
[√3(2(3) - 4) - √3(2(2) - 4)]/(3 - 2)
= [√6 - √(12 - 4)]/1
= [√6 - √8]/1
To compare the two estimated average rates of change, we need to find the difference between them:
Difference = (√6 - √8) - (√5 - √2)
To approximate the difference to the nearest tenth, we can use a calculator to find the decimal values of each square root and subtract:
Difference ≈ (2.449 - 2.828) - (2.236 - 1.414)
≈ -0.379
So, the difference in the estimated average rates of change to the nearest tenth is approximately -0.4.