Question

If dy/dx=1/x, then the average rate of change of y with respect to x on the closed interval [1, 4] is

a) -1/4
b) (1/2)*ln(2)
c) (2/3)*ln(2)
d) 2/5
e) 2

Answer 1

The first thing to do here is to integrate the ordinary differential equation dy/dx = 1/x. This is equal to  y = ln x + C. Assuming C is zero. y = lnx. substituting the two interval, 
y1 = 0 y2 = ln 4
the rate of change then is equal to (ln4 - 0) / (4-1) equal to 2/3 ln 2. Answer is C