Complete the following questions for next class.
Exercise 11A questions 2, 3, 7, 9, 10, 11
Exercise 11C.1 questions 1, 2
11 HL Trigonometric Identities and Equations Continued
Below is the list of questions we began in class today. Aim to finish up to the end of 13E before our lesson on Thursday.
Exercise 13A.2 questions 3cd
Exercise 13B question 5
Exercise 13C.2 questions 2h, 3c
Exercise 13D question 12
Exercise 13E questions 3ab, 5ab, 26, 27
Exercise 13F questions 3ac, 4ab
Exercise 13G question 5
12 SL Derivatives, Tangents and Normals
After today’s review of techniques for differentiation, try the questions listed below before our next class.
We’ve already seen how to find the equation of the tangent line to a curve that passes through a given point, and finding a normal through a given point on a curve is essentially the same. A normal is a line that is perpendicular to a tangent, and you will recall that if one line has slope \(m\), any line perpendicular to it will have slope \(-\frac{1}{m}\). For example, the equation of the line tangent to the function \(f(x)=x^2\) that passes through the point \((1,1)\) has slope \(2\) (found by taking the derivative), and so has equation \(y=2x-1\). Similarly, the normal to the function \(f(x)=x^2\) that passes through the point \((1,1)\) has equation \(y=-\frac{1}{2}x-1\).
Complete 16A questions 1acf, 2cd, and 4
11 SL Modelling with Exponential Functions
Complete the following questions for our class tomorrow. We’ll have a short (1 question) quiz on this material tomorrow. Remember, if you get stuck, just post a comment below!
Exercise Set 3G.1 questions 1–3
Exercise Set 3G.2 questions 1, 4, 5
11 HL Inverse Trigonometric Functions and Equations
Complete the following exercises before our next class (after the break).
Exercise Set 13A.3 questions 6bdij and 12
Also, have a look at the question below (which was discussed in class). It was mentioned in class that a solution can be found (to both parts!) without a calculator. Can you figure it out over the break?
\[\arctan\left(\frac{1}{2}\right)-\arctan\left(\frac{1}{3}\right) = \arctan(a), a \in \mathbb{Q}^+\]
- Find the value of \(a\).
- Hence, or otherwise, solve the equation \(\arcsin (x)=\arctan(a)\).
I’ll post a hint in the comments below [update: one error in that comment has been corrected, and the hint has been extended]—you may find it useful.
11 SL Exponential Functions
In class we looked at functions of the form \(f(x)=a\cdot b^x+d\), and we had a brief discussion of the effect of the values of \(a, b,\) and \(c\) on the graph of the function. Below, you’ll extend this treatment to understand the effect of the values of \(a, b, c,\) and \(d\) on the graph of functions of the form \(f(x)=a\cdot b^{x-c}+d\).
Read through Investigation 1 on page 95 of the textbook, and use graphing software (I suggest either GeoGebra or Desmos) to explore the effects of various values on the graph of each function. As discussed in class, using sliders can make it easier to see the effect of each, and you can see an example of this below. (But it’s more fun to try to make your own version of this!)
Once you’ve completed your investigation, complete questions 4 and 5 from Exercise 3F. (Record your answers in your notebook, and then bring your notebook to class!)