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

Expert Answer to "Find the quadratic approximation of: s e c..."

Micaela Simon

2022-06-16

Find the quadratic approximation of:

\frac{s e c (t)}{\sqrt{1 - t^{2}}}

for

t \approx 0

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

\frac{1}{\sqrt{1 - t^{2}}}

, and multiply.
Or else (recycling is a good thing) we can note that

\cos t \approx 1 - t^{2} / 2

and

\sqrt{1 - t^{2}} \approx 1 - t^{2} / 2

. Thus

(\cos t) \sqrt{1 - t^{2}} \approx 1 - t^{2}

, and therefore its reciprocal is

\approx 1 + t^{2}

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