Rate of change How do you do this ????????? the online lecture is no help really

Question

Find the average rate of change of(x)=x^3−6x+5:
(a) From−8 to −5
(b) From −2 to 4
(c) From 4 to 5

Answer 1

The average rate of change is the slope of the line (called the secant line) that connects the end-points of the function on that interval.  I'll do the first interval [-8,-5]:

 

f(-8) = (-8)³ - 6(-8) + 5 = -512 + 48 + 5 = -459

f(-5) = (-5)3 - 6(-5) + 5 = -125 + 30 + 5 = -90

 

The endpoints of the function on the interval are (-8, -490) and (-5, -90).  Use the slope formula to find the slope of the line connecting the two points:

 

m = (y₂- y₁)/(x₂ - x₁) where (x₁,y₁) = (-8, -459) and (x₂, y₂) = (-5, -90)

 

m = (-90 - (-459))/(-5 - (-8)) = 123

 

The average rate of change of f(x) on the interval [-8,-5] is 123.  Do the same for the other two intervals

Answer 2

The average rate of change of f(x) from a to b is just

 

<Δf(x)/Δx> = [f(b) - f(a)] / (b - a)

 

For example, problem (a):

 

a = -8

b = -5

 

f(a) = (-8)³ - 6(-8) + 5

f(b) = (-5)³ - 6(-5) + 5

b - a = -5 - (-8)

 

Just substitute the numbers and evaluate [f(b) - f(a)] / (b - a).