Suppose that the function f has the same average rate of cha

Suppose that the function f has the same average rate of change c between any two points, Find the average rate of change of f between the points a and x to show that c= f(x) - f(a)/x - a. The rate of change between any two points is Rearrange the equation in part (a) to show that f(x) = cx + (f(a) - ca). Multiplying the equation in part(a) by x - a, we obtain f(x) - f(a) = c Rearranging and adding f(a) to both sides, we have f(x) = cx + (f(a) - ,as desired. How does this show that f is a linear function? What is the slope, and what is the y-intercept? Because this equation is of form f(x) = Ax + B with constants and B = f(a) -, it represents a linear function with slope and y-intercept f(a)s

Solution

a) Average rate of change between any two points x1 and x2 is given by:

Avg rate of change = {f(x2) -f(x1)} /(x2-x1)

Let x1 =a and x2 =x:

Avg rate of change = ={ f(x) - f(c)/( x-a) = c

c ={ f(x) - f(c)/( x-a)

b) Multiply aboveequation by ( x-a):

c(x-a) ={ f(x) - f(a)/( x-a)}(x-a)

cx - ca = f(x) - f(c)

f(x) - f(a) = c( x-a)

f(x) = f(a) + cx -ca

f(x) = cx + ( f(a) - ca )

Std linear function : y = mx +c where m is slope and c is a constant term

So, comparing with this equation we have :: f(x) = cx + ( f(a) - ca )

Its a linear equation

B = f(a) - ca

slope = c ad y-intercept = f(a) -ca

where m = c (slope)