Graphics Reference
In-Depth Information
Sketch the graphs of (i)
f
, (ii)
f
, (iii) the derived function of
f
.
, and
the derived function
f
:
B
R has a derived function (
f
)
:
C
R, with
C
When the real function
f
:
A
R
has a derived function
f
:
B
R
B
A
, wewrite(
f
)
f
and call
f
the
second derivative
of
f
.
When
f
(
a
) exists, we say that
f
is
twice differentiable
at
a
.
36 How do you know that, when
f
(
a
) exists,
f
is continuous at
a
?
37 A function
f
:
R
R
is defined by
x
x
when
x
0 and
f
(
x
)
when
x
0.
Determine whether
(i)
f
is continuous at 0,
(ii)
f
(0) exists,
(iii)
f
is continuous at 0,
(iv)
f
(0) exists.
38 Define the third derivative of
f
, and inductively, the
n
th derivative of
f
,
which is written
f
.
Inverse functions
39 Find thedrivativeof thefunction
f
given by
f
(
x
)
x
at some
point
a
0, from first principles. The function
f
is continuous at
a
from qn 6.27.
If
g
(
x
)
x
, attempt to relate
f
(
a
)to
g
(
f
(
a
)).
40 If thereal function
f
:
A
B
is a bijection, and
g
:
B
A
is its
inverse, so that
f
(
g
(
b
))
b
for any
b
B
, and
g
(
f
(
a
))
a
for any
