The estimated average rate of change of the function y = 16 x 4^x is approximately ___ greater than the estimated average rate of change of the function y = 16x^2 + 4x over the interval [1.1, 1.6].
To find the average rate of change of a function over an interval, we need to use the formula:
Average rate of change = (f(b) - f(a))/(b-a)
For the function y = 16x^2 + 4x, let's find the average rate of change over the interval [1.1, 1.6].
f(b) = 16(1.6)^2 + 4(1.6) = 16(2.56) + 6.4 = 41.6 + 6.4 = 48
f(a) = 16(1.1)^2 + 4(1.1) = 16(1.21) + 4.4 = 19.36 + 4.4 = 23.76
b - a = 1.6 - 1.1 = 0.5
Average rate of change of y = 16x^2 + 4x over the interval [1.1, 1.6] = (f(b) - f(a))/(b-a) = (48 - 23.76)/0.5 = 24.24/0.5 = 48.48
Now let's find the average rate of change of y = 16 x 4^x over the same interval.
f(b) = 16 x 4^(1.6)
f(a) = 16 x 4^(1.1)
Average rate of change of y = 16 x 4^x over the interval [1.1, 1.6] = (f(b) - f(a))/(b-a) = (16 x 4^(1.6) - 16 x 4^(1.1))/0.5
To find this value, we need to evaluate the expressions for f(b) and f(a):
f(b) = 16 x 4^1.6 ➝ Evaluate this as (16 x 4) x (4^(0.6))
= 64 x (4^(0.6)) = 64 x 2.639
= 169.216
f(a) = 16 x 4^1.1 ➝ Evaluate this as (16 x 4) x (4^(0.1))
= 64 x (1.584) = 101.376
Average rate of change of y = 16 x 4^x over the interval [1.1, 1.6] = (f(b) - f(a))/(b-a) = (169.216 - 101.376)/0.5 = 67.84/0.5 = 135.68
Therefore, the estimated average rate of change of the function y = 16 x 4^x is approximately 135.68 - 48.48 = 87.2 greater than the estimated average rate of change of the function y = 16x^2 + 4x over the interval [1.1, 1.6].