Time allowed: 15 minutes.
1. (2 marks)
Use logarithmic differentiation to find
d
dx
x
sin
x
for
x >
0.
2. (2 marks)
By expanding (
x
−
y
)
2
prove that
x
2
+
y
2
≥
2
xy
for all real numbers
x, y
.
Hence, or otherwise, determine the minimum value of
y
=
x
6
+
1
x
6
.
3. (1 mark)
Find the exact value of sin
−
1
sin
200
π
3
.
4. (5 marks)
(a) Given that cosh(
A
+
B
) = sinh
A
sinh
B
+ cosh
A
cosh
B
, find a formula for cosh 2
x
.
By differentiation or otherwise, find a formula for sinh 2
x
.
(b) Using the results of part (i), express cosh 3
x
as a cubic polynomial in cosh
x
.
Hence, or otherwise, find
cosh
3
x
.
Week 3
Friday 11-12 Class

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