Revision Notes For Core

ADVERTISEMENT

Revision Notes for Core 4
Differentiating Trigonometric Functions
d
d
sin x = cos x and
cos x = - sin x. All the rest can be found by expressing the
dx
dx
function in terms or sin or cos then using the product, chain or quotient rule.
Use formula booklet to help and look at both differentiating and integrating
columns as one is the reverse of the other.
2
2
Use 2sinAcosA = sin 2A, 2cos
A = 1 + cos2A and 2sin
A = 1 – cos2A to change into
1
1
2
expressions you can differentiate or integrate. e.g. from cos
x into
+
cos2A
2
2
d
1
Remember
sin 2x = 2 cos 2x (chain rule) and
sin
2
x
= -
cos 2x + c
dx
2
Integration
Either by parts or substitution
dv
du
dv
By parts:
u
dx = uv -
v dx which means call one bit u and the other
so
dx
dx
dx
dv
du
integrate the
to give v and differentiate the u to give
then sub in to the formula.
dx
dx
Remember: Diff. one and Int. the other and choose to diff. the one that will
disappear.
If still have a product in the second integral bit, swap around which bit to integrate and
do again then if still horrid, do substitution.
Substitution
Always given the substitution so differentiate this then substitute in for dx and then can
cancel OR rearrange and express x in terms of u then differentiate and substitute in.
1
1/2
Example
dx where u = 1 – x
1 x
2 /
1
dx
2
x = (1 – u)
and
= -2(1 – u) = 2u – 2 so dx = (2u – 2)du
du
1
2
(2u – 2) du =
2 (
)
du = 2u – 2 ln |u| + c
u
u
1/2
1/2
Substitute back in for u therefore = 2(1 – x
) – 2 ln |1 – x
| + c
Parametric Equations
Uses an equation for x and y separately but with an extra variable t to link the two
together. To change to a cartesian equation, solve simultaneously to find the value of t
then write as y=mx + c.
If the parametric equations are for curves, then use chain rule to find the gradient at a
dy
dt
dy
particular point, e.g.
=
x
dx
dx
dt
dy
dx
dx
3
e.g. if y = t
and x = 2t then find
and
then invert
and multiply the two together.
dt
dt
dt
Remember to use the 2 equations to find the x and y coordinates then use y = mx +c if
finding the equation of a tangent or normal.
Binomial Expansion
n −
n
n (
) 1
n
n (
1
)(
n
) 2
n
2
3
(1 + x)
= 1 +
x +
x
+
x
+ …
THIS IS IN THE FORMULA BOOKLET
1
x 1
2
x 1
2
x
3
n
2
2
If (1 + mx)
then remember that it is (mx)
not just mx
and you may need to take out a
common factor to get it in the binomial form. Also used when expressing partial fractions
and expanding them.

ADVERTISEMENT

00 votes

Related Articles

Related forms

Related Categories

Parent category: Education
Go
Page of 4