Transcript FP2 MEI Lesson 8 Calculus part 1_trig in integration_inverse trig

the Further Mathematics network
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the Further Mathematics network
www.fmnetwork.org.uk
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…
Using trigonometric identities in integration, the inverse
trigonometric functions, differentiation of functions
involving inverse trigonometric functions.
Before you start:
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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.
Using trigonometric identities in integration, the
inverse trigonometric functions, differentiation of
functions involving inverse trigonometric
functions.
When you have finished…
You 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.
Calculus - Reminder
Calculus - Reminder
Integration of powers of sine and cosine
We can use this result to integrate odd powers of sine for example:
Try:
5
cos
 xdx
Even powers of sine and cosine
Inverse trigonometric functions
It is useful to look at the graph of a function together with its inverse
(use of autograph)
arcsin
arccos
arctan
Look at y=arcsecx on autograph and consider its domain and range
(if time permits)
Example: show that
1
arc sec x  arccos 
 x
Differentiating inverse trigonometric
functions
Use autograph to draw the gradient function of y=arcsinx
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
Using trigonometric identities in integration, The
inverse trigonometric functions, Differentiation
of functions involving inverse trigonometric
functions.
When you have finished…
You 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.
Independent study:
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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.