Lab 12: Taylor series - part 1

Instructions: Work with your group on the problem assigned to you. We encourage you to collaborate both in and out of class, but you must write up your responses individually.

  1. Write the \(n^\text{th}\) degree Taylor Polynomial for \(f(x) = e^x\) centered at \(a = 0\). Use the Lagrange remainder formula (or Taylor's inequality) to determine the smallest degree Taylor polynomial you can use to approximate \(e\) to within 0.02% of its actual value, that is \(\epsilon = 0.0002e\).

  2. In the sciences, the approximation \(\sin\theta \approx \theta\) is often used for small angles.

    1. Justify this using Taylor series. Use the Lagrange remainder to determine for what small angles this approximation is accurate to within an error bound of \(\epsilon = 0.1\).

    2. Approximate \(\sin 92^\circ\) to 5 decimals accuracy (\(\epsilon = 5\times 10^{-6}\)) using the Taylor series for \(\sin(x)\) centered at \(x = 0\). What degree Taylor polynomial did you need to use?

    3. Approximate \(\sin 92^\circ\) to 5 decimals using the Taylor series of \(\sin(x)\) at \(x = \frac{\pi}{2}\). What degree Taylor polynomial did you need to use? 

  3. Let \(f(x) = \cos x\).

    1. If we want to use a 7\(^\text{th}\) degree Taylor polynomial centered at \(a = 0\) to approximate \(f\), and we want to bound the error to be less than \(\epsilon = 0.1\), find the largest possible interval for \(x\).

    2. Now say we want the approximation to be accurate to within \(\epsilon = 0.1\) of the right answer on a larger interval, say \(-5 \leq x \leq 5\). How many terms do we need to use in the Taylor series?  (Hint: it's more than 10)

  4. The force due to gravity on object \(h\) meters above the surface of the earth is \(F(h) = \frac{mgR^2}{(R + h)^2}\) Newtons, where \(m\) is the mass of the object in kg, \(g = 9.8\) m/s\(^2\) is the acceleration due to gravity at sea level, and \(R \approx 6.4\times 10^6\) m is the radius of the earth.
    1. Show that the first term in the Taylor series of \(F(h)\) centered at \(h = 0\) is \(P_0(h) = mg\).

    2. Use the Lagrange remainder to determine how far from sea level you can travel before the error on \(F\) is more than 10% of the approximation \(P_0(h) = mg\)?

      (Hint: This means the error bound is \(\epsilon = .1 * mg\)).

    3. What is the linear approximation \(P_1(h)\)? What is the error bound, \(\epsilon\), for the linear approximation at the distance you determined in Part b (you may use \(mg\) in your bound)?

  5. Use a Taylor series centered at \(a = 1\) to approximate \(\ln(2)\) so that the error is less than \(\epsilon = 0.1\).