Category: 12 HL Homework

Complete the following two questions for next class (Friday).

1. Find \(\int x \sin x \; dx\), and verify that your answer is correct.
2. Find \(\int x^2 \sin x \; dx\), and verify that your answer is correct.

[spoiler title=’Hint for question 2′ style=’blue’ collapse_link=’true’]You may find it helpful to use integration by parts twice.[/spoiler]

Complete the following questions for our next class (exercises are from the Cambridge textbook).

17H questions 1bc, 2–5
17J 1a, 2–5, 6, 8, 9

Complete the following questions, along with the questions that were emailed to you on June 18th, before our next lesson.

20C questions 10, 11, 20, 33
20D questions 4, 9, 17
21E.1 questions 1, 2, 4, 5, 7, 9–11

Complete pages 751–752 questions 1–6, 8, 14 from Chapter 15 in the Pearson textbook.

Now that we’ve looked at operations involving Taylor series, complete the questions below for our next lesson.

1. Find the Taylor polynomial of degree 3, centred at \(0\), for \(e^x\sin 2x\).
2. Find the Taylor series, centred at \(0\), for \(\sin x+\cos x\).
3. Us the series
   \[\frac{1}{1-x}=1+x+x^2+\cdots+x^n+\cdots\]
   to find the Taylor series for
   1. \(\displaystyle{\frac{1}{1-2x}}\)
   2. \(\displaystyle{\frac{1}{1+x}}\)
   3. \(\displaystyle{\frac{1}{1+x^2}}\)
4. Determine the interval of convergence for each series in questions 1 to 3.
5. Use the Taylor series for \(\displaystyle{\frac{1}{1+x^2}}\) to find the Taylor series centred at \(0\) for \(\arctan x\). Determine the interval of convergence for the Taylor series at \(0\) for \(\arctan x\).

Taylor’s Theorem gives us a bound on the error that would result from using a Taylor polynomial \(P_n(x)\) to calculate the approximate value of a function \(f(x)\) at a given value.

Use this result to answer the following questions for our next lesson.

1. Consider the Taylor polynomials for \(e^x\), centred at \(a=0\).
   1. Using the fact that \(e^x\) is an increasing function, and \(e<3\), find a value of \(n\) such that \(|R_n(1)|<10^{-5}\).
   2. Hence, determine the value of \(e\) accurate to 4 decimal places.
2. 1. Generate the Taylor polynomial of degree \(3\) at \(x=0\) for the function \(f(x)=\ln(x+1)\).
   2. Hence, calculate an approximate value for \(\ln(1.1)\). Give a bound on the error of your approximation.

From the Cambridge materials (Chapter 29), complete the following questions over the break.

Page 28 questions 1 and 3

Pages 32–34 questions 2, 3, 7, 9, 12, 13

Have a look at the following questions from option topic (Series and Differential Equations) past papers shown below.

November 2011, questions 2 and 4.

November 2012, question 3.

We will discuss these in our next lesson.

One nice feature of the integral test is that—when it can be applied to show that a given function converges—we also get an easy way to find a bound on the size of the error term.

In other words, we can use a partial sum to find an approximation of the actual value of a series while at the same time knowing how far off we are (at worst) from the true value of the sum of that series.

Complete the question below for tomorrow’s lesson.

Consider the \(p\)-series with \(p=3\).

• Find a general expression for a bound on the error term when \(S_k\), for some \(k \in \mathbb{Z}^+\), is used to approximate the sum.
• Find the number of terms required to approximate the value of this series to within 0.0005.
• Find the value of \(\displaystyle{\sum_{n=1}^\infty \frac{1}{n^3}}\), accurate to three decimal places.