Graphics Reference
In-Depth Information
Answers
1 One. No.
2 Two. Yes.
3 Three if
m
0. Oneif
m
0. Only
m
0. Slope
0.
4
y
f
(
a
)
m
(
x
a
).
f
(
a
h
)
f
(
a
)
y
f
(
a
)
(
x
a
). With thegiven
m
this is a tangent.
h
5 There may still be a tangent when the slope of the chord
.
6
f
(
a
)
0.
7
f
(
a
)
m
.
f
(
a
h
)
f
(
a
)
8
If, given
0, 0
h
l
h
f
(
x
)
f
(
a
)
then 0
x
a
a
l
; and conversely.
x
9 2
a
.
10 Generalise by induction as in qn 6.25.
11
g
(
x
)
g
(
a
)
k
·
f
(
x
)
f
(
a
)
a
and useqn 6.93, thealgebra of limits.
x
x
a
12
By qn 6.93, thealgebra of limits, lim
f
(
x
)
f
(
a
)
f
(
a
) · 0.
So lim
f
(
x
)
f
(
a
) and
f
is continuous at
a
by qn 6.89.
13
f
(
x
) ·
g
(
x
)
f
(
a
)·
g
(
a
)
f
(
x
) ·
g
(
x
)
g
(
a
)
g
(
a
)·
f
(
x
)
f
(
a
)
a
.
x
a
x
x
a
This tends to
f
(
a
) ·
g
(
a
)
g
(
a
)·
f
(
a
) by qn 6.93, thealgebra of limits,
and thecontinuity of
f
.
14 Trivial for
n
1. If
f
(
x
)
x
f
(
a
)
na
, then for
g
(
x
)
x
x
,
g
(
a
)
a
· 1
na
·
a
(
n
1)
a
by qn 13.
lim
x
x
a
...
a
a
a
...
a
na
by
qn 6.93, thealgebra of limits.
15
f
(
a
)
b
2
b
a
3
b
a
...
nb
a
.
16
f
(
x
)
f
(
a
)
1/
x
1/
a
a
x
a
)
na
a
a
, using qn 14.
x
x
a
x
(
x
a
a
