For the function :

i) Find the linear and quadratic approximations.

ii) Derive the Maclaurin series expansion.
b) Let  be defined as
 

Find the Jacobian  at the point .

Answer:

### i) Linear and Quadratic Approximations for \( f(x) = \cos(x) \)
The linear and quadratic approximations of a function can be derived using Taylor series expansion around a point, typically \( x = 0 \). The general Taylor series expansion is:

\[
f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots
\]

#### Linear Approximation:
For \( f(x) = \cos(x) \), we compute:
- \( f(0) = \cos(0) = 1 \)
- \( f'(x) = -\sin(x), \, f'(0) = -\sin(0) = 0 \)

Thus, the linear approximation is:
\[
f(x) \approx 1
\]

#### Quadratic Approximation:
Next, compute the second derivative:
- \( f''(x) = -\cos(x), \, f''(0) = -\cos(0) = -1 \)

The quadratic approximation is:
\[
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