Lab: 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. Let \(g(x) = \ln(x)\).

    1. Find a formula for the Taylor series for \(g(x)\) centered at \(a = 1\). Graph \(g(x)\) together with its 3\(^\text{rd}\) degree Taylor polynomial, \(P_3(x)\).

    2. Find the derivative \(g'(x)\) and its Taylor series centered at \(a = 1\) (meaning you should write the infinite series using \(\Sigma\)-notation).

      Graph \(g'(x)\) together with its 2\(^\text{nd}\) degree Taylor polynomial, \(P_2(x)\).

    3. Find an antiderivative \(G(x)\) and its Taylor series centered at \(a = 1\). Graph \(G(x)\) and its 4\(^\text{th}\) degree Taylor polynomial, \(P_4(x)\).

  2. Let \(p(x) = e^{-x^2}\).

    1. Substitute \(-x^2\) into the Taylor series for \(e^x\) centered at \(a = 0\) to find a Taylor series for \(p(x)\). Graph \(p(x)\) together with its 4\(^\text{th}\) degree Taylor polynomial, \(P_4(x)\).

    2. Find the derivative \(p'(x)\) and its Taylor series centered at \(a = 0\). Graph \(p'(x)\) together with its 3\(^\text{rd}\) degree Taylor polynomial, \(P_3(x)\).

    3. Can you find an antiderivative \(P(x)\)? Find the Taylor series for an antiderivative \(P(x)\) centered at \(x = 0\). Draw a graph of the 5\(^\text{th}\) degree Taylor polynomial, \(P_5(x)\). Even though you can't write a formula for \(P(x)\), qualitatively add an appropriate sketch of \(P(x)\) to your graph.

  3. Let \(r(x) = \arctan(x)\).

    1. Let \(r(x) = \arctan(x)\). Substitute \(-x^2\) into a geometric series to find a Taylor series for \(r'(x) = \frac{1}{1 + x^2}\) centered at \(a = 0\). What is the radius of convergence?

    2. Use Part a to find a Taylor series for \(r(x) = \arctan(x)\) centered at \(a = 0\). Graph \(r(x)\) together with its 9\(^\text{th}\) degree Taylor polynomial, \(P_9(x)\).

    3. Find an antiderivative \(R(x)\). Find the Taylor series for \(R(x)\) centered at \(a = 0\). Graph \(P(x)\) and its 10\(^\text{th}\) degree Taylor polynomial, \(P_{10}(x)\).