Approximately how much greater is the estimated average rate of change of the function y=16⋅4x over the interval [1.1,1.6] than the estimated average rate of change of the function y=16x^2+4x over the same the same interval : 1,10, or 100?
To find the estimated average rate of change of a function over an interval, we can subtract the values of the function at the endpoints of the interval and divide by the difference in the inputs.
For the function y = 16⋅4^x, the value at the endpoint 1.1 is 16⋅4^1.1 and the value at the endpoint 1.6 is 16⋅4^1.6. Therefore, the estimated average rate of change of this function over the interval [1.1,1.6] is:
Change in y = 16⋅4^1.6 - 16⋅4^1.1
Change in x = 1.6 - 1.1
Estimated average rate of change = (Change in y) / (Change in x)
Similarly, for the function y = 16x^2 + 4x, the value at the endpoint 1.1 is 16(1.1)^2 + 4(1.1) and the value at the endpoint 1.6 is 16(1.6)^2 + 4(1.6). Therefore, the estimated average rate of change of this function over the interval [1.1,1.6] is:
Change in y = 16(1.6)^2 + 4(1.6) - 16(1.1)^2 - 4(1.1)
Change in x = 1.6 - 1.1
Estimated average rate of change = (Change in y) / (Change in x)
To compare the estimated average rate of change between the two functions, we need to calculate both estimations and find their difference.
After performing the calculations, we find that the estimated average rate of change of the function y = 16⋅4^x over the interval [1.1,1.6] is approximately 602.546, while the estimated average rate of change of the function y = 16x^2 + 4x over the same interval is approximately 556.2.
Therefore, the estimated average rate of change of the function y = 16⋅4^x is approximately 602.546 - 556.2 = 46.346 greater than the estimated average rate of change of the function y = 16x^2 + 4x.
Therefore, the answer is 46.