Taylor series about π/4

10. Given that \(y = \tan x\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
2. Find the Taylor series expansion of \(\tan x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).
3. Hence show that \(\tan \frac { 3 \pi } { 10 } \approx 1 + \frac { \pi } { 10 } + \frac { \pi ^ { 2 } } { 200 } + \frac { \pi ^ { 3 } } { 3000 }\).

5. (a) Find the Taylor expansion of \(\cos 2 x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 5 }\).
(b) Use your answer to (a) to obtain an estimate of \(\cos 2\), giving your answer to 6 decimal places.
(3)(Total 8 marks)

5. $$y = \sec ^ { 2 } x$$

1. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x\).
2. Find a Taylor series expansion of \(\sec ^ { 2 } x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).

7. $$y = \tan ^ { 2 } x , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$

1. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x\)
2. Hence show that \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = 8 \sec ^ { 2 } x \tan x \left( A \sec ^ { 2 } x + B \right)\), where \(A\) and \(B\) are constants to be found.
3. Find the Taylor series expansion of \(\tan ^ { 2 } x\), in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\)

6. (a) Find the Taylor series expansion about \(\frac { \pi } { 4 }\) of \(\tan x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).
(b) Deduce that an approximation for \(\tan \frac { 5 \pi } { 12 }\) is \(1 + \frac { \pi } { 3 } + \frac { \pi ^ { 2 } } { 18 } + \frac { \pi ^ { 3 } } { 81 }\)

5. $$y = \sec ^ { 2 } x$$

1. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 \sec ^ { 4 } x - 4 \sec ^ { 2 } x\).
2. Find a Taylor series expansion of \(\sec ^ { 2 } x\) in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\).