Graphics Reference
In-Depth Information
8 Show that thetwo limits
f
(
a
h
)
f
(
a
)
f
(
x
)
f
(
a
)
x
a
lim
, and lim
,
h
are equivalent.
x
, what is
f
9 If
f
(
x
)
(
a
)?
Sums of functions
10 If two real functions
f
and
g
are both differentiable at
a
, prove
using qn 6.93, on thealgebra of limits, that thefunction
f
g
defined by
f
g
:
x
f
(
x
)
g
(
x
) is differentiable at
a
, and that
(
f
g
)
(
a
)
f
(
a
)
g
(
a
).
Explain how this result may be generalised to show that the sum of
n
real functions, each differentiable at
a
, is also differentiable at
a
.
11 If
f
is differentiable at
a
, and
g
(
x
)
k
·
f
(
x
), provethat
g
(
a
)
k
·
f
(
a
).
Questions 10 and 11 give linearity in determining derivatives, in the
sense that if the functions
f
and
g
are differentiable at
a
, then the
function given by
x
l
·
f
(
x
)
m
·
g
(
x
) has derivative
l
·
f
(
a
)
m
·
g
(
a
)at
a
.
The product rule
12 By considering the product
f
(
x
)
f
(
a
)
x
a
f
(
a
)
· (
x
a
), when
x
a
,
f
(
x
)
and, using qn 6.93, on the algebra of limits, show that if the real
function
f
is differentiable at
a
, then
f
is continuous at
a
.
13 If two real functions
f
and
g
are both differentiable at
a
, prove
using qns 6.93 and 12 that thefunction
f
·
g
defined by
f
·
g
:
x
f
(
x
)·
g
(
x
) is differentiable at
a
, and that
(
f
·
g
)
(
a
)
f
(
a
)·
g
(
a
)
f
(
a
) ·
g
(
a
).
14 Proveby induction that, if
f
(
x
)
x
, then
f
(
a
)
n
·
a
, for any
positive integer
n
.
Obtain the same result by considering the equation
x
a
(
x
a
)(
x
x
a
x
a
...
a
).
