Month: April 2017

Now that we’ve looked at the Sine and Cosine Rules (as well as the ambiguous case of the sine rule), we can apply these to solve some 3D problems.

Have a look at the following questions tonight.

Pages 381–382 questions 24 and 25

Pages 391–393 questions 12, 17, 19

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

Complete the following questions for our lesson tomorrow.

Pages 344–345 questions 3, 6, 9, 10, 11, 13, 17, 21, 28

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.

Complete the following set of questions for tomorrow’s lesson.

Page 321 questions 1, 3, 5, 10, 11, 14, 18, 20, 26