The value of a family's home, in Camrose AB, is given by the following exponential function f(x), where x is the number of years after the family purchases the house for $130,000. What is the best estimate for the instantaneous rate of change in the value of the home when the family has owned it for 5 years?
Of course we see no "following exponential function".
let's assume it looks something like this
f(x) = 130000(e)^(.06x) , in this case the rate of increase would be 6%
f ' (x) = 130000(.06)(e^.06x)
so if x = 5 we get f ' (x) = $10,429
Change my solution to reflect your actual case
the base could be something like 1.06^x
To find the instantaneous rate of change, we need to compute the derivative of the exponential function.
The given exponential function f(x) represents the value of the home as a function of the number of years x. Let's denote the value of the home at time x as V(x).
We are given f(x) = V(x) = $130,000 * e^(kx), where k is a constant.
To find the instantaneous rate of change, we need to compute the derivative of f(x) with respect to x, or dV/dx.
Taking the derivative of f(x) = $130,000 * e^(kx) with respect to x, we get:
dV/dx = k * $130,000 * e^(kx)
Now, we can substitute x = 5 into dV/dx to find the best estimate for the instantaneous rate of change when the family has owned the home for 5 years.
dV/dx = k * $130,000 * e^(k * 5)
However, without knowing the value of k, we cannot determine the exact value of the instantaneous rate of change. We can only estimate its value for a given k.
If the value of k is known, you can substitute it into the equation above and evaluate the expression to find the best estimate for the instantaneous rate of change in the value of the home when the family has owned it for 5 years.
To find the instantaneous rate of change of the value of the home after 5 years, we need to differentiate the given exponential function f(x).
The given exponential function for the value of the home is: f(x) = $130,000 * e^kx, where x is the number of years after purchasing the house.
To find the instantaneous rate of change, we differentiate f(x) with respect to x, and evaluate it at x=5.
Differentiating f(x) = $130,000 * e^kx:
f'(x) = 130,000 * k * e^kx
Now we evaluate f'(5) to find the instantaneous rate of change at 5 years:
f'(5) = 130,000 * k * e^(k*5)
At this point, we need more information to find the best estimate for the instantaneous rate of change. The value of k in the exponential function is not given, which affects the rate of change.