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Past exam of the mathematics course of the University of Oxford
/
2023
/
Prelims
/
Mathematics III
/
3
/
a
/
Solution
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Past exam of the mathematics course of the University of Oxford
2023
Prelims
Mathematics III
3
a
2026-10-25
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The
Taylor expansions
sin
t
=
t
+
O
(
t
3
)
and
e
t
=
1
+
t
+
t
2
/2
+
O
(
t
3
)
give
f
(
x
,
y
)
=
−
3
+
6
x
+
3
y
+
2
1
(
x
+
2
y
)
2
+
O
((
∣
x
∣
+
∣
y
∣
)
3
)
.
(1)
Hence the
second
-order
Taylor polynomial
is
T
2
(
x
,
y
)
=
−
3
+
6
x
+
3
y
+
2
1
x
2
+
2
x
y
+
2
y
2
.
(2)
The
sine
contributes no quadratic term at zero.
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Mathematics III
Prelims
2023
Past exam of the mathematics course of the University of Oxford
Mathematics course of the University of Oxford
Undergraduate course of the University of Oxford
Course of the University of Oxford
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