Let f be defined as follows.
F(x)=y=x^2 -7
(a) Find the average rate of change of y with respect to x in the following intervals.
from x = 4 to x = 5
from x = 4 to x = 4.5
from x = 4 to x = 4.1
(b) Find the (instantaneous) rate of change of y at x = 4.
To find the average rate of change of y with respect to x, we need to calculate the difference in y-values divided by the difference in x-values.
(a) Average rate of change from x = 4 to x = 5:
To calculate the average rate of change from x = 4 to x = 5, we need to evaluate the function at both x-values and find the difference in y-values.
F(4) = 4^2 - 7 = 16 - 7 = 9
F(5) = 5^2 - 7 = 25 - 7 = 18
Now, we divide the difference in y-values by the difference in x-values:
Average rate of change = (F(5) - F(4)) / (5 - 4) = (18 - 9) / (5 - 4) = 9 / 1 = 9
The average rate of change of y with respect to x from x = 4 to x = 5 is 9.
(b) Instantaneous rate of change at x = 4:
To find the instantaneous rate of change at a specific point, we need to take the derivative of the function. In this case, we'll take the derivative of F(x) = x^2 - 7.
F'(x) = 2x
Now, we can substitute x = 4 into the derivative to find the slope at that point:
F'(4) = 2 * 4 = 8
The instantaneous rate of change of y at x = 4 is 8.