12 HL: Definite Integrals

Complete as many of the following questions as you can before our next class (we’ll continue to work on these in our next class as well).

Exercise 22A questions 3–8, 12 (just complete the centre column for 3–8 and 12), 14, 16, 21

12 HL: Trigonometric Substitutions

In class we arrived at part of the solution to $$\int \sqrt{2-x^2}\;d x$$, using the substitution $$x = \sqrt{2}\sin \theta$$. Complete this question by showing that $\int \sqrt{2-x^2}\;d x=\left( \frac{x}{\sqrt{2}} \right) \sqrt{ 1-\frac{x^2}{2}} +\arcsin \left(\frac{x}{\sqrt{2}}\right) + C$

Hint
Note that, given our substitution choice, it follows that $$\sin \theta = \frac{x}{\sqrt{2}}$$. Can we do something similar for $$\cos \theta$$?

Also complete

Exercise 21E questions 4, 5d, 6
Exercise 21F questions 12ad, 17, 18bd

12 HL: Integration by Substitution

Complete the following questions before our next class.

Exercise 21F questions 3, 4bce, 5bf, 8dh, 9ae

12 HL: Integration by Parts

Complete the following questions before our next class.

Exercise 21G questions 1, 3, 4ab, 6ac
Exercise 21D questions 12, 15, 19, 20

12 HL: Rules for Integration

Complete the following questions before our next class.

Exercise 21B questions 1acgh, 2ac, 3def, 4dg, 7ae, 8aef, 9, 12a, 14
Exercise 21C questions 1ac, 2, 6, 7b

12 HL: Indefinite Integrals

Complete the following questions before our next class.

Exercise 20C questions 1, 2ai, ii, b, 3ai, ii, b, 4abc
Exercise 21A questions 2abd, 3c, 9

12 HL: Applications of Derivatives

Start working on the following problems tonight, and you’ll have time during our next class to complete these.

Exercise 19A questions 1, 2, 4, 9, 12, 14
Exercise 19B questions 5, 6, 12, 20, 27
Exercise 19C questions 2, 3, 5, 8, 9, 13

12 HL: Related Rates

The question we discussed in class today is included below. If you didn’t get the answer in class, try to see if you can complete it now.

12 HL: l’Hôpital’s Rule

Complete the following questions before our next class.

Exercise 18H questions 1a–d, 2, 4, 6a, 7, 10, 11 (the last two are challenge questions—give them a try!)

12 HL: Derivatives and Graphs of Functions

Complete the following questions before our next class.

Exercise 18F question 10
Exercise 18G questions 1–3