Core 3 Algebra Revision Notes
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3Revision Notes for Core 3
Functions
Domain is the set of inputs for a function. Can be all real numbers or only a particular
few.
Range is the set of output numbers obtained from the domain. Use completing the
square to find the lowest and highest values for the range if a quadratic.
Composite functions
gf(x) = g(f(x)) means substitute in x into function f then substitute in that result into
function g. It is a function of a function.
Inverse function means just doing the function in reverse
-1
f
(x) when f(x) = (2x+3)/5 becomes (5x-3)/2. Think of as a flow diagram followed in
reverse. The range and domain swap around when doing the inverse and it is a
reflection in the line y=x if they are one-to-one functions (one value for the range for one
value from the domain)
n
Differentiating (ax+b)
Use the chain rule
dy/dx = dy/du x du/dx where u=ax+b Use whenever you have a function of a function.
Diff one function then diff the other and multiply the results together. Use for rates of
change as well to find dy/dt for instance.
Product Rule
Quotient Rule
dy
du
dv
u
dy
du
dv
If y=uv then
= v
+ u
If y=
then
= v
-u
dx
dx
dx
v
dx
dx
dx
dy
2
= ( diff leave + leave diff )
v
dx
Make sure u is the numerator and v the
denominator.
d
1
2
2
e.g
x
lnx = 2x lnx + x
x
dx
n
Integrating (ax+b)
1
1
∫
+ ) b
n
n+1
(
ax
dx =
x
(ax+b)
+ c
This is because if you differentiate
+
n
1
a
n+1
n
(ax+b)
you get: a x (n+1) (ax+b)
so
1
1
the
x
counteracts this.
+
n
1
a
x
Differentiating and integrating e
and ln x
d
x
x
e
= e
as by definition the gradient is the same as the function and therefore
dx
d
1
x
x
3x
3x
3x
3x
∫ e
and ∫ e
dx = e
+ c
e
use chain rule so = 3e
dx =
e
+ c for the same
dx
3
reason as the reverse of the chain rule.
Log Laws
a
x
ln a + ln b = ln ab
ln a – ln b = ln
ln 1 = 0
ln a
= x ln a
b
x
Remember that if y = e
then if I take the natural log (ln) of both sides, then x = ln y
x
x
(e
and ln x are the inverse of each other).
LEARN: y = e
so x = ln y
d
1
1
and ∫
ln x =
dx = ln x + c
dx
x
x
1
1
1
1
If you can take out a coefficient then do ∫
∫
dx =
dx =
ln x + c
x
3
x
3
3
1
1
1
If of the form ∫
dx then =
ln (ax+b) as
counteracts the differentiation of ax+b.
ax + + + +
b
a
a
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