Consider the function f(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)
as always, the average rate of change is just the slope between the two points:
∆y/∆x = (f(a+h)-f(a))/h
Now just plug in a=-1, h=3
oh, yeah. with f(x) = 3/x,
∆y = 3/(a+h) - 3/a = -3h/(a(a+h))
how did you get h=3? and isnt it (-3h/(a(a+h)))/h
h is x2-x1, the change in x
a. To find the average rate of change between two points, we need to calculate the slope of the line connecting those two points. In this case, we are given two points: (a, f(a)) and (a + h, f(a + h)).
The slope of a line passing through two points (x1, y1) and (x2, y2) can be found using the formula:
slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1)
In our case, we have:
x1 = a
y1 = f(a) = 3/a
x2 = a + h
y2 = f(a + h) = 3/(a + h)
Plugging these values into the slope formula:
slope = (3/(a + h) - 3/a) / ((a + h) - a)
Simplifying the expression:
slope = (3a - 3(a + h)) / (a(a + h))
Expanding and collecting like terms:
slope = (3a - 3a - 3h) / (a^2 + ah)
Canceling out the common terms:
slope = -3h / (a^2 + ah)
So, the expression for the average rate of change between the points (a, f(a)) and (a + h, f(a + h)) for the function f(x) = 3/x is:
slope = -3h / (a^2 + ah)
b. To determine the average rate of change from x = -1 to x = 2, we need to substitute the values into the expression we derived in part (a).
In this case:
a = -1
h = 2 - (-1) = 3
Substituting these values into the expression:
slope = -3(3) / ((-1)^2 + (-1)(3))
Simplifying the expression:
slope = -9 / (1 - 3)
Further simplification:
slope = -9 / (-2)
Finally:
slope = 9/2
Therefore, the average rate of change from x = -1 to x = 2 for the function f(x) = 3/x is 9/2.