11 HL Working with Derivatives

Following on from our discussion from class, complete the following question.

Let \(f(x)=\frac{1}{3}x^3-2x^2\). Find the coordinates of the local maximum and the local minimum of \(f\).

Are you interested in seeing something else that’s sort of neat? Read on.

We’ll eventually be discussing something called the second derivative. Once you’ve found the derivative of a given function, you can then go on to find the derivative of that derivative. For example, if \(f(x)=2x^5\), then \(f'(x)=10x^4\). The second derivative is represented as \(f^{\prime \prime}(x)\); in this case, we have \(f^{\prime \prime}(x)=40x^3\).

Find the second derivative of the function \(f(x)=\frac{1}{3}x^3-2x^2\), then solve the equation \(f^{\prime \prime}(x)=0\). On the graph of \(f\), plot the point on \(f\) whose \(x\)-coordinate is the solution you found to \(f^{\prime \prime}(x)=0\). What do you notice about the location of this point?

11 SL Trigonometric Identities and Expressions

In order to make use of the trigonometric identities we discussed today, it is frequently also necessary to be able to simplify or factor expressions involving trigonometric functions.

The questions below give you some opportunity to practice simplifying and factoring, then you’ll use those skills when applying the double-angle formulae. Complete these questions before our next class (and these may be checked on Monday…).

Exercise 11C.1 questions 1aef, 2d, 3acd, 4ad
Exercise 11C.2 questions 1ace, 2af, 3af
Exercise 11D questions 1–5

11 SL Trigonometric Models Review

We’ve now finished the chapter on modelling with trigonometric functions, and the questions listed below will help you review this material. We’ll begin our next class with a discussion of the solutions, then we’ll begin looking at the next (and final) chapter on trigonometry.

Review Set 10A questions 3–6
Review Set 10B questions 2–5
Review Set 10C questions 3 and 6

11 SL Modelling with Trigonometric Functions Part 2

Now that we’ve built our London Eye model, we can use the same technique (applying our knowledge of transformations) to model other situations that involve periodic behaviour (and to answer more general questions about transformations involving trigonometric functions).

Read the (brief) section at the top of page 242 about the “general” sine function, then get a good start on the following questions before our next class (tomorrow).

Exercise 10B.1 questions 3 and 4
Exercise 10C questions 4 and 5 (for more of a challenge, try question 1)
Exercise 10F question 1

11 SL Modelling with Trigonometric Functions

Now that we’ve learned about the graphs of trigonometric functions, we can apply our understanding of transformations of functions to transform trigonometric functions (particularly sine and cosine) to model period behaviour, like the tides, temperature, etc.

Try to complete the following questions before our next lesson (I suggest leaving the horizontal stretch until the end).

The London Eye is 15 m off the ground, and has a diameter of 120 m. The London Eye completes one revolution every 30 minutes.

  1. Develop a function to model the height of a passenger on the London Eye t minutes after boarding a capsule.
  2. How far above ground would passengers in a capsule be after
    a. 10 minutes have passed?
    b. 25 minutes have passed?