Consider the functionf(x)=3/x.
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)
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
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
so
if y = 1/x
then dy/dx = -1/x^2
isnt it -3/(x^2+x)?????
Oh yes, forgot the 3
-3/(x^2+x h)
a. To find the expression for the average rate of change, we need to calculate the difference in function values between the two points and divide it by the difference in x-values.
Given the points (a, f(a)) and (a + h, f(a + h)), the average rate of change is given by:
Average rate of change = (f(a + h) - f(a)) / [(a + h) - a]
Now let's substitute the function f(x) = 3/x into this expression:
Average rate of change = [3/(a + h) - 3/a] / h
To simplify further, we can combine the fractions:
Average rate of change = [(3a - 3(a + h)) / (a(a + h))] / h
Simplifying the numerator:
Average rate of change = [3a - 3a - 3h] / (a(a + h)h)
We can cancel out the -3h in the numerator:
Average rate of change = -3h / (a(a + h)h)
Now, cancelling out the h terms:
Average rate of change = -3 / (a(a + h))
So, the expression for the average rate of change between the two points is -3 / (a(a + h)).
b. To find the average rate of change from x = -1 to x = 2, we can substitute the values into the expression we derived in part (a):
Average rate of change = -3 / ((-1)(-1 + 2))
Simplifying further:
Average rate of change = -3 / (-1)(1)
Average rate of change = -3 / (-1)
Since dividing a negative by a negative gives a positive result, the average rate of change is 3.