- Class: 5H: BC Calculus
- Author: Peter Atlas
- Text:
__Calculus__Finney, Demana, Waits, Kennedy

**Directions: Do these essay problems one-per-page. Make sure your name is on each page. Be ready to switch them in class so we can correct each other's essays. **

Questions came from Advanced Placement Calculus tests as indicated.

Let \(P(x) = 7 - 3(x - 4) + 5(x - 4)^2 - 2(x - 4)^3 + 6(x - 4)^4\) be the fourth-degree Taylor polynomial for the function \( f \) about 4. Assume \( f \) has derivatives of all orders for all real numbers.

- Find \( f (4) \) and \( f'''(4) \).
- Write the second-degree Taylor polynomial for \( f' \) about 4 and use it to approximate \( f'(4.3) \) .
- Write the fourth-degree Taylor polynomial for \( g(x) = \displaystyle \int_4^x f(t) \, dt\) about 4.
- Can \( f (3)\) be determined from the information given? Justify your answer.

Let \( f \) be a function that has derivatives of all orders for all real numbers. Assume \( f (0) = 5, f '(0) = -3, f ''(0) = 1\), and \( f '''(0) = 4\).

- Write the third-degree Taylor polynomial for \( f \) about \(x = 0\) and use it to approximate \( f(0.2) \) .
- Write the fourth-degree Taylor polynomial for \(g\), where \(g(x) = f \left( x^2 \right) \), about \(x = 0\).
- Write the third-degree Taylor polynomial for \(h\), where \(h(x) = \displaystyle \int_0^x f(t) \, dt\) about \(x = 0\).
- Let \(h\) be defined as in part (c). Given that \( f (1) = 3\), either find the exact value of \(h(1)\) or explain why it cannot be determined.

The function \( f \) has derivatives of all orders for all real numbers \(x\). Assume \( f (2) = -3, f'(2) = 5, f''(2) = 3\), and \( f'''(2) = -8\).

- Write the third-degree Taylor polynomial for \( f \) about \(x = 2\) and use it to approximate \( f(1.5) \) .
- The fourth derivative of \( f \) satisfies the inequality \( \left| f^{(4)} (x) \right| \leq 3\) for all \(x\) in the closed interval [1.5, 2]. Use the Lagrange error bound on the approximation to \( f(1.5) \) found in part (a) to explain why \( f (1.5) \neq -5\).
- Write the fourth-degree Taylor polynomial \(P(x)\), for \(g(x) = f(x^2 + 2)\) about \(x = 0\). Use \(P\) to explain why \(g\) must have a relative minimum at \(x = 0\).

The Taylor series about \(x = 5\) for a certain function \( f \) convrges to \( f(x) \) for all \(x\) in the interval of convergence. The n^{th} derivative of \( f \) at \(x = 5\) is given by \( f^n (5) = (-1)^n \frac{n!}{2^n (n + 2)}\), and \( f (5) = \frac{1}{2}\) .

- Write the third-degree Taylor polynomial for \( f \) about \(x = 5\).
- Find the radius of convergence of the Taylor series for \( f \) about \(x = 5\).
- Show that the sixth-degree Taylor polynomial for \( f \) about \(x = 5\) approximates \( f(6) \) with error less than \(\frac{1}{1000}\).

The Maclaurin series for the function \( f \) is given by \( \displaystyle f (x) = \sum_{n = 0}^{\infty} \frac{(2x)^{n + 1}}{n + 1} = 2x + \frac{4x^2}{2} + \frac{8x^3}{3 }+ \frac{16x^4}{4} + ... + \frac{(2x)^{n + 1}}{n + 1} + ...\) on its interval of convergence.

- Find the interval of convergence of the Maclaurin series for \( f \) . Justify your answer.
- Find the first four terms and the general term for the Maclaurin series for \( f'(x) \) .
- Use the Maclaurin series you found in part (b) to find the value of \( f' \left( -\frac{1}{3} \right) \).