Question

Approximate Cos(3.1) Using A Quadratic Approximation: First Note That Cos(3.1) COS(T). Let F(X) = Cos(X). Then, F'(X) =_________

Answer 1

The first derivative of the cosine function, denoted as 'F'(X)', is '-sin(X)'. The second derivative for the cosine function is also '-sin(X)'. Quadratic approximation requires the function value and its first two derivatives. So, to build the quadratic approximation of ‘cos(3.1)’, we would evaluate ‘F(3.1)’, ‘F'(3.1)’, and ‘F''(3.1)’ and use these to construct a polynomial that approximates ‘cos(X)’ near ‘X = 3.1’.

Step-by-step explanation:

To approximate ‘cos(3.1)’ using a quadratic approximation, we start by noting that ‘cos(3.1)’ is essentially ‘cos(T)’ where ‘T’ is the angle in question and ‘F(X)’ represents the cosine function, so ‘F(X) = cos(X)’. The first derivative of the cosine function ‘F’ with respect to ‘X’, denoted as ‘F'(X)’, is equal to ‘-sin(X)’. Therefore, ‘F'(X) = -sin(X)’.

For the quadratic approximation, we also need the value of the second derivative of the cosine function at the point ‘X’. The second derivative of ‘F(X) = cos(X)’, denoted as ‘F''(X)’, is ‘-sin(X)’. So, to build the quadratic approximation of ‘cos(3.1)’, we would evaluate ‘F(3.1)’, ‘F'(3.1)’, and ‘F''(3.1)’ and use these to construct a polynomial that approximates ‘cos(X)’ near ‘X = 3.1’.