2. the value of a familys home is given by $f(n)=130000(1.06)^{n}$, where $n$ is the number of years after…

2. the value of a familys home is given by $f(n)=130000(1.06)^{n}$, where $n$ is the number of years after the family purchases the house for $130000. what is the instantaneous rate of change in the value of the home when the family has owned it for 5 years? \\\\ \\\\ \\\\ note: use the tools taught in this course. we do not use calculus.

2. the value of a familys home is given by $f(n)=130000(1.06)^{n}$, where $n$ is the number of years after the family purchases the house for $130000. what is the instantaneous rate of change in the value of the home when the family has owned it for 5 years? \\\\ \\\\ \\\\ note: use the tools taught in this course. we do not use calculus.

Answer

Explanation:

Step1: Recall the formula for the instantaneous rate of change of an exponential function

For a function (y = a\cdot b^{x}), the instantaneous rate of change is given by (y^\prime=a\cdot b^{x}\cdot\ln(b)). In our case, (a = 130000), (b = 1.06), and (x=n). So the formula for the instantaneous rate of change of (f(n)=130000(1.06)^{n}) is (f^\prime(n)=130000(1.06)^{n}\ln(1.06)).

Step2: Substitute (n = 5) into the derivative formula

We know that (\ln(1.06)\approx0.0582689). When (n = 5), (f(5)=130000(1.06)^{5}) and (f^\prime(5)=130000(1.06)^{5}\ln(1.06)). First, calculate ((1.06)^{5}=1.06\times1.06\times1.06\times1.06\times1.06\approx1.3382256). Then (f^\prime(5)=130000\times1.3382256\times0.0582689). [ \begin{align*} f^\prime(5)&=130000\times1.3382256\times0.0582689\ &=(130000\times1.3382256)\times0.0582689\ &=173969.328\times0.0582689\ &\approx10145.97 \end{align*} ]

Answer:

The instantaneous rate of change of the value of the home when the family has owned it for (5) years is approximately ($10146) per year.