Consider the functionf(x)=3/x.

Question

a. Determine an expression, in terms of a and h, for the average rate of change between the points (a,f(a)) and ( a + h, f(a+h) ) for the function . Show all steps needed to find a simplified algebraic expression.

b. Using your expression from (a), determine the average rate of change from x = -1 to x = 2. (no decimal values)

Answer 1

y(x+h) = 3/(x+h)
y(x) = 3/x

y(x+h)- y(x) = [3x -3(x+h) ] / [x(x+h)]
= -h/(x^2 + xh)
divide by h for average
-1/(x^2+h)

for part b
x = -1
x+h = 2
h = 2 - -1 = 3
so
we want change in y divided by h
= -1/[-1^2 + -1*3]
= -1/-2 = 1/2
=============================
check
y(2) = 1/2
y(-1) = -1
change in y = 3/2
change in x = 3
change in y/change in x = 1/2 whew ! Caramba

Answer 2

By the way, about the next chapter
the derivative is the limit of [ y(x+h) -y(x) ]/ h as h--->0
here that is
dy/dx = -1/(x^2+xh) as h---->0
= -1/x^2

Answer 3

so
if y = 1/x
then dy/dx = -1/x^2

Answer 4

isnt it -3/(x^2+x)?????

Answer 5

Oh yes, forgot the 3

Answer 6

-3/(x^2+x h)