Find the quadratic approximation of: s e c ( t

Micaela Simon

Micaela Simon

Answered question

2022-06-16

Find the quadratic approximation of:
s e c ( t ) 1 t 2
for t 0

Answer & Explanation

zalitiaf

zalitiaf

Beginner2022-06-17Added 27 answers

If we go via the direct differentiation route, we need f ( 0 ), f ( 0 ), and f ( 0 ). Not too bad, but the second derivative is somewhat unpleasant to calculate.
We can alternately find the quadratic approximations of sec t and 1 1 t 2 , and multiply.
Or else (recycling is a good thing) we can note that cos t 1 t 2 / 2 and 1 t 2 1 t 2 / 2. Thus ( cos t ) 1 t 2 1 t 2 , and therefore its reciprocal is 1 + t 2 .

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