Author: Dr. McDonald

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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

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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

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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?

\[\arctan\left(\frac{1}{2}\right)-\arctan\left(\frac{1}{3}\right) = \arctan(a), a \in \mathbb{Q}^+\]

  1. Find the value of \(a\).
  2. 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.

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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!)

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11 SL Exponents

Remember to complete the following questions for our lesson tomorrow.

Exercise Set 3A questions 3 and 4
Exercise Set 3B questions 3–5
Exercise Set 3C questions 3, 5

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11 HL Modelling with Trigonometric Functions

In our last class we looked at how we could transform the sine and cosine functions to model periodic behaviour. The questions posted during class are listed below. Try to get through to the end of Section 12D, and we’ll continue with these questions during our next lesson.

Exercise 12C questions 1, 3
Exercise 12D questions 2–4
Exercise 12E question 1
Exercise 12F questions 1, 3, 5, 6def, 8

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12 SL Explorations

At this point you should have created (and shared with me) a Google Doc that indicates your exploration’s

  1. Title
  2. Aim
  3. Rationale
  4. Area of Mathematics Involved

Your completed draft should be in by the end of the day on Friday next week (October 20th).

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11 SL Functions Test

We’ll have our functions test on Thursday, the 19th of October.

In order to prepare for this test, have a look at the questions below. We’ll be spending part of Tuesday’s lesson looking at inverse functions and working on these questions, but it’s a good idea to start these early so you can ask questions if you get stuck.

Review Set 2A questions 1–3, 7, 10, 11
Review Set 2B questions 4, 6, 7
Review Set 2C questions 4, 10, 12