Category: Mathematics

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The Ambiguous Case

Can you solve the ambiguous case?

Use GeoGebra to see if you can find the missing angle discussed in class.

In addition to this, complete the following set of questions from the textbook for our next lesson. (For a couple of these questions, you’ll need the definitions of angle of elevation and angle of depression.)
Pages 358–360 questions 31, 36, 40

Pages 367–369 questions 2, 11 c, 14 b, 16 + one of 20, 21, 22, or 23 (you choose)

Pages 380 questions 9, 21, 24

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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.
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Series and Convergence

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.

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The Integral Test

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.