FP2 (MEI) Calculus (part 1) Using trigonometric identities in integration, the inverse trigonometric functions, differentiation of functions involving inverse trigonometric functions.

1
(No Transcript)
2
FP2 (MEI)Calculus (part 1) Using trigonometric
identities in integration, the inverse
trigonometric functions, differentiation of
functions involving inverse trigonometric
functions.

• Let Maths take you Further

3
Using trigonometric identities in integration,
the inverse trigonometric functions,
differentiation of functions involving inverse
trigonometric functions.

• Before you start
• You need to be familiar with the laws of indices
  (Core 1 chapter 5) and logarithms (Core 2 chapter
  11).
• You need to have covered all of the work on
  functions in Core 3 chapter 3. In particular, the
  section on inverse trigonometrical functions on
  pages 45 - 46 is a useful introduction.
• You need to be confident with all the techniques
  of differentiation and integration in C2 and C3,
  in particular differentiation using the chain
  rule, differentiation of trigonometric functions,
  implicit differentiation (C3 chapter 4),
  integration by substitution and integration of
  trigonometric functions (C3 chapter 5).
• You must also be confident with all the work on
  Trigonometry covered so far (C2 chapter 10 and C4
  chapter 8). In particular, the enrichment work on
  pages 218 222 of the A2 Pure Mathematics
  textbook covers some of the work in this section.

4
Using trigonometric identities in integration,
the inverse trigonometric functions,
differentiation of functions involving inverse
trigonometric functions.

• When you have finishedYou should
• Be able to use trigonometric identities to
  integrate functions such as sin2 x, sin3 x, sin 4
  x, tan x.
• Understand the definitions of inverse
  trigonometric functions.
• Be able to differentiate inverse trigonometric
  functions.

5
Calculus - Reminder
6
Calculus - Reminder
7
Integration of powers of sine and cosine
We can use this result to integrate odd powers of
sine for example
8
Try
9
Even powers of sine and cosine
10
(No Transcript)
11
Inverse trigonometric functions
It is useful to look at the graph of a function
together with its inverse (use of autograph)
12
arcsin
13
arccos
14
arctan
15
Look at yarcsecx on autograph and consider its
domain and range (if time permits)
Example show that
16
Differentiating inverse trigonometric functions
Use autograph to draw the gradient function of
yarcsinx
17
(No Transcript)
18
(No Transcript)
19
Summary of results (these are given in the exam
formula book)
Now that we have these results we can use the
chain rule to differentiate composite functions
that include inverse trigonometric functions
20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
Using trigonometric identities in integration,
The inverse trigonometric functions,
Differentiation of functions involving inverse
trigonometric functions.

• When you have finishedYou should
• Be able to use trigonometric identities to
  integrate functions such as sin2 x, sin3 x, sin 4
  x, tan x.
• Understand the definitions of inverse
  trigonometric functions.
• Be able to differentiate inverse trigonometric
  functions.

24
Independent study

• Using the MEI online resources complete the study
  plan for Calculus 1
• Do the online multiple choice test for this and
  submit your answers online.