Complete the following questions for our lesson tomorrow.
Page 642 questions 1ace, 2bc, 7a, 9, 12, 16, 19, 22
Working with Vectors
Complete the following questions for our lesson tomorrow.
Pages 417–419 questions 10, 12, 14, 15, 18, 20, 24
You’ll find it useful to read the textbook pages 412–416 (especially the examples) when answering some of these questions. The textbook also mentions something they call a direction angle for a vector, and although we won’t have much use for that angle right now, it may be useful later in the course, so I suggest you look at that material as well (in particular, look at the example on page 411).
Introduction to Vectors
Complete the following questions (in addition to those left on the board) for our next lesson.
Pages 407–409 questions 1, 3, 6, 10, 12, 19, 20, 22
Trigonometry Test [Updated]
On Tuesday, the 2nd of May, we’ll have a test on trigonometry (corresponding to the material in Chapters 7 and 8 in our textbook).
Complete the following questions in preparation for the test, and we’ll discuss solutions in our next lesson.
Page 393 questions 18 and 20 (20 is challenging!)
Pages 394–397 questions 5–9, 11–13, 18
Pages 348–349 questions 17–23
Update: The date of the test was corrected to Tuesday.
Applications of Trigonometry
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
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
Inverse Trigonometric Functions
Complete the following questions for our lesson tomorrow.
Pages 344–345 questions 3, 6, 9, 10, 11, 13, 17, 21, 28
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.
Manipulating Taylor Series
Now that we’ve looked at operations involving Taylor series, complete the questions below for our next lesson.
- Find the Taylor polynomial of degree 3, centred at \(0\), for \(e^x\sin 2x\).
- Find the Taylor series, centred at \(0\), for \(\sin x+\cos x\).
- Us the series
\[\frac{1}{1-x}=1+x+x^2+\cdots+x^n+\cdots\] to find the Taylor series for- \(\displaystyle{\frac{1}{1-2x}}\)
- \(\displaystyle{\frac{1}{1+x}}\)
- \(\displaystyle{\frac{1}{1+x^2}}\)
- Determine the interval of convergence for each series in questions 1 to 3.
- 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 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.
- Consider the Taylor polynomials for \(e^x\), centred at \(a=0\).
- Using the fact that \(e^x\) is an increasing function, and \(e<3\), find a value of \(n\) such that \(|R_n(1)|<10^{-5}\).
- Hence, determine the value of \(e\) accurate to 4 decimal places.
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- Generate the Taylor polynomial of degree \(3\) at \(x=0\) for the function \(f(x)=\ln(x+1)\).
- Hence, calculate an approximate value for \(\ln(1.1)\). Give a bound on the error of your approximation.