If f(x) = x² + 2x + 3, what is the average rate of change of f(x) over the interval [-4, 6]?

Solution:

Given: Function f(x) = x² + 2x + 3,

a = -4 and b = 6

The average rate of change of g(x) over an interval between 2 points (a ,f(a)) and (b ,f(b)) is the slope of the secant line connecting the 2 points.

The condition to calculate the average rate of change between the 2 points is [f(b) - f(a)]/(b - a)

To find the secant line,

Consider f(b) = f(6) = 6² + 2× 6 + 3

= 36 + 12 + 3

f(b) = 51

f(a) = f(-4) = (-4)² + 2× (-4) + 3

= 16 - 8 + 3

f(a) = 11

The average rate of change = (51 - 11)/[6 - (-4)]

The average rate of change = 40/10

Therefore, the average rate of change = 4

This means that the average of all the slopes of lines tangent to the graph of f(x) at (-4, 6) is 4.

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 If f(x) = x² + 2x + 3, what is the average rate of change of f(x) over the interval [-4, 6]?

Summary:

The average rate of all the slopes of lines tangent to the graph of f(x) = x² + 2x + 3 at (-4, 6) is 4.