Let f(x)=sec(x) Find the 2nd order polynomial of f at c=pi/4. Please show the ca

Question

Let f(x)=sec(x)

Find the 2nd order polynomial of f at c=pi/4.   Please show the calculation of the 2 derivatives using hand rules.

Using the polynomial obtained above, approximate sec(.7). Then use the calculator to calculate sec(.7), and using these answers to calculate E_2(.7). If the absolute value of your error is more than .01, then find your error and correct it.

Then use the polynomial above to approximate sec(3.0) and compute the error E_2(3.0).

Then find the exact value of x in the interval [pi/4,3.0] at which all derivatives of f fail to exist.

THANK YOU!!!

Explanation / Answer

f(x) = sec x
f '(x) = sec x tan x
f ''(x) = (sec x tan x) tan x + sec x (sec^2(x))
.......= sec x (sec^2(x) - 1) + sec^3(x)
.......= 2 sec^3(x) - sec x

at pi/4

1.41 + 1.41 x +(9.64-1.41) x^2

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