BC Calculus: AP Essays on Sequences and Series Worksheet 2

• 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.

1. Find \( f (4) \) and \( f'''(4) \).
2. Write the second-degree Taylor polynomial for \( f' \) about 4 and use it to approximate \( f'(4.3) \) .
3. Write the fourth-degree Taylor polynomial for \( g(x) = \displaystyle \int_4^x f(t) \, dt\) about 4.
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 \( f '''(0) = 4\).

1. Write the third-degree Taylor polynomial for \( f \) about \(x = 0\) and use it to approximate \( f(0.2) \) .
2. Write the fourth-degree Taylor polynomial for \(g\), where \(g(x) = f \left( x^2 \right) \), about \(x = 0\).
3. Write the third-degree Taylor polynomial for \(h\), where \(h(x) = \displaystyle \int_0^x f(t) \, dt\) about \(x = 0\).
4. 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 \( f'''(2) = -8\).

1. Write the third-degree Taylor polynomial for \( f \) about \(x = 2\) and use it to approximate \( f(1.5) \) .
2. 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 \( f (1.5) \neq -5\).
3. 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 \( f (5) = \frac{1}{2}\) .

1. Write the third-degree Taylor polynomial for \( f \) about \(x = 5\).
2. Find the radius of convergence of the Taylor series for \( f \) about \(x = 5\).
3. 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.

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