Category: 12 Mathematics HL

Here’s the question we considered on Thursday. Now that you’ve got a solution for parts 1 and 2, complete part 3 as a homework assignment due on Tuesday, the 25th of October.

The function \(f\)is defined by \(f(x) = e^x \sin x\).

1. Show that \(f”(x) = 2e^x \sin \left(x+\frac{\pi}{2}\right)\).
2. Obtain a similar expression for \(f^{(4)}(x)\).
3. Suggest an expression for \(f^{(2n)}(x), n \in \mathbb{Z}^+\), and prove your conjecture using mathematical induction.

Today we saw how to use a definite integral to calculate the volume of a solid of revolution. (Some textbooks will refer to this as finding a “volume of revolution.”)

The solid we studied today is shown below, and the equation used to generate this solid was \(y=\cos x +2\), with \(x\) running from 0 to 5. Can you use your knowledge of solids of revolution to derive the formulas for

• the volume of a cone with height \(h\) and a base of radius \(r\), \(V=\frac{1}{3}\pi r^2h\)
• the volume of a sphere of radius \(r\), \(V=\frac{4}{3}\pi r^3\)

As discussed on Thursday, try the following questions this weekend. Some may not be as difficult as they first appear, and others…

Complete pages 780–781 questions 26, 28, 33, 45, and 49.
If you’re interested in (what may be) a challenge, also try questions 43, 48, and 50.

Below is a question that you should complete for tomorrow.

Find the area enclosed between the graphs of the functions \(f(x)=x^3-3x^2+2x+2\) and \(g(x)=-x^3+4x^2-4x+3\).

Complete the following questions for Sunday, the 18 of September.

Pages 751–752 questions 1–6, 8, 14

Complete the following questions on indefinite integrals for tomorrow’s lesson.

Page 780 questions 1, 3, 7, 8, 9, 10, 12, 13, 14