Question

1. 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.

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 \(f^\prime(5)=130000\times1.3382256\times0.0582689\).

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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.