Part II: The Answers
230
Answers
201–300
285.
−
−
(
)
1
2
1
3 2
x
Use the derivative definition with f x
x
( ) =
−
1
1
and with f x h
x h
+
(
)= + −
1
1
:
lim
lim
lim
h
h
h
x h
x
h
x
x h
x
x h
h
x
x h
→
→
→
+ −
−
−
=
− −
+ −
−
+ −
=
− −
+
0
0
0
1
1
1
1
1
1
1
1
1
1
− −
−
+ −






− +
+ −
− +
+ −






=
−
(
)− +
→
1
1
1
1
1
1
1
1
0
h x
x h
x
x h
x
x h
x
x h
h
lim
− −
(
)
−
+ −
(
) − + + −
(
)
=
−
−
+ −
(
) − + + −
(
)
→
1
1
1
1
1
1
1
1
1
0
h x
x h
x
x h
h
h x
x h
x
x h
h
lim
= =
−
−
+ −
(
) − + + −
(
)
=
−
−
+ −
(
) − + + −
(
)
=
−
→
lim
h
x
x h
x
x h
x
x
x
x
0
1
1
1
1
1
1
1
0 1
1
0 1
1
x x
x
x
−
(
)
−
(
)
=
−
−
(
)
1 2
1
1
2
1
3 2
286.
3x
2
+ 3
Use the derivative definition with f
 
(x) = x
3
+ 3x and with f
 
(x + h) = (x + h)
3
+ 3(x + h) =
x
3
+ 3x
2
h + 3xh
2
+ h
3
+ 3x + 3h:
lim
lim
h
h
x
x h
xh h
x
h x
x
h
x h
xh h
h
→
→
+
+
+ +
+
−
+
(
)
=
+
+ +
0
3
2
2
3
3
0
2
2
3
3
3
3
3
3
3
3
3
h h
h x
xh h
h
x
xh h
x
h
h
=
+
+ +
(
)
=
+
+ +
(
)
=
+ + +
=
→
→
lim
lim
0
2
2
0
2
2
2
3
3
3
3
3
3
3
0 0 3
3x x
2
3
+