Average rate of change

Question

Definition: The average rate of change of a function f(x)f from x=a to x=b is the slope of the line which passes through (a,f(a) and (b,f(b)).

Consider the function f(x)=x^2+6 and find the following:

a) The average rate of change of f(x) from x=−1 to x=4.


b) The average rate of change of f(x) from x=a to x=b.


c) The average rate of change of f(x)f between the points (x,f(x)) and (x+h,f(x+h)). Assume h>0

Answer 1

f(x)=x^2+6

 

a)  from x=−1 to x=4.

f(-1) = (-1)² + 6 = 7 .. (-1, 7)

f(4) = 4² + 6 = 22 ... (4, 22)

 

slope:  m = (change in y)/(change in x)

= (22-7)/(4-(-1)) = 15/5 = 3

 

b)  from x=a to x=b. 

f(a) = a²+6  ... (a, a²+6)

f(b) = b²+6 ... (b, b²+6)

 

slope m = [(b²+6)-(a²+6)]/(b-a) = (b²-a²)/(b-a)

 

            = [(b-a)(b+a)]/(b-a) = b+a

 

c)  (x,f(x)) and (x+h,f(x+h))

 

    (x, x²+6) and (x+h, (x+h)²+6))

    (x, x²+6) and (x+h, x²+2xh + h²+6)

 

   m = (change in y)/(change in x)

 

  m = [(x²+2xh + h² + 6)-(x²+6)]/(x+h-x)

 

  = (2xh + h²)/h = h(2x+h)/h = 2x+h

Answer 2

a)

f(-1) = 7, f(4) = 22

m = (22 - 7) / (4 - 1)

b)

f(a) = a² + 6, f(b) = b² + 6

m = (b² + 6 - (a² + 6)) / (b - a)

c)

f(x) = x² + 6, f(x + h) = x² + 2hx + h² + 6

m = (x² + 2hx + h² + 6 - (x² + 6)) / (x + h - x)