Suppose that you take out a $\$15,000$ loan to purchase a car. The loan has an interest rate of $3\%$ per year, compounded monthly.
The formula for the amount of interest accrued over a time period $T$ measured in number of months is
$$f(x) =15,000\left(1+\frac{0.03}{12}\right)^T-15,000$$ Use a quadratic approximation to approximate the amount of interest accrued over 4 years.
the approximation for $T=48$
$$f(48) \approx -15,000+15,000(1+0.12+0.007)+ \mathcal{O}(x^3)$$
Then I need to find.
The error in the of this formula approximation is $\mathcal{O}(x^3)$. Find the order of magnitude of $x^3$ in our problem.
Compute the exact error term in the quadratic approximation.
Use the error to identify the order of magnitude of $k$ since the error is known to be $kx^3$.
Any help will be appreciated.