Category: 12 Mathematics HL

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12 HL Integration by Parts (Part 2)

On Tuesday we’ll be talking about the Exploration, but we can also discuss solutions to the questions below.

From the Cambridge book, Chapter 19, complete Exercise 19F.

Have a look at these, and also start thinking about an application of mathematics, or a particular topic in mathematics, that you might consider for your Exploration. You can also have a look at the resources posted here.

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12 HL Integration by Parts

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]
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Calculus Mock Examination

On Wednesday this week during our normal lesson time we’ll have a mock examination for the Calculus option topic. Since this will be a mock examination, it will include all material from the option topic (and not just the material we’ve covered recently). We can then discuss solutions during Thursday’s lesson.

In order to prepare for this, have a look at the past papers that have been made available to you.

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Manipulating Taylor Series

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 \(\arctan x\).
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Taylor’s Theorem and Taylor Polynomials

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.
    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.