Determine the average rate of change form x = 1 to x = 4 for each function.
a) y= x
b) y=x^2
c) y=x^3
d) y=7
To determine the average rate of change for each function from x = 1 to x = 4, we need to calculate the slope of the line connecting the points (1, f(1)) and (4, f(4)) where the function is represented as f(x). The formula for calculating the slope between two points is:
slope = (change in y) / (change in x)
a) y = x:
For this linear function, the change in y is equal to the change in x since the coefficient is 1. Therefore, the average rate of change is:
slope = (4 - 1) / (4 - 1) = 3 / 3 = 1
b) y = x^2:
For this quadratic function, we substitute the given values of x into the equation and calculate the corresponding y values:
f(1) = 1^2 = 1
f(4) = 4^2 = 16
Now we can calculate the slope using the same formula:
slope = (16 - 1) / (4 - 1) = 15 / 3 = 5
c) y = x^3:
Similarly, we substitute the given values of x into the equation and calculate the corresponding y values:
f(1) = 1^3 = 1
f(4) = 4^3 = 64
Using the slope formula:
slope = (64 - 1) / (4 - 1) = 63 / 3 = 21
d) y = 7:
This is a constant function, meaning that the value of y is 7 for any x. Since the y-values do not change, the slope is zero:
slope = (7 - 7) / (4 - 1) = 0 / 3 = 0
Therefore, the average rate of change for each function from x = 1 to x = 4 is:
a) 1
b) 5
c) 21
d) 0