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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.

  1. The error in the of this formula approximation is $\mathcal{O}(x^3)$. Find the order of magnitude of $x^3$ in our problem.

  2. Compute the exact error term in the quadratic approximation.

  3. 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.

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    $\begingroup$ Note that $(1+c)^x \approx 1+cx +c^2x(x-1)/2+c^3x(x-1)(x-2)/6+...$. $\endgroup$ Jul 11, 2015 at 23:00
  • $\begingroup$ You have a $T$ and an $x$ ... do you know the relationship between them? $\endgroup$ Jul 13, 2015 at 17:55
  • $\begingroup$ they are the same. $T$ is the $x$ in equation. $\endgroup$
    – Sarah
    Jul 13, 2015 at 18:14

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