Question

The function $\cos(x)$ is approximated using the Taylor series around $x = 0$ as \( \cos(x) \approx 1 + a x + b x^2 + c x^3 + d x^4. \) The values of $a, b, c$ and $d$ are: 
 

• A. $a = 1,\; b = -0.5,\; c = -1,\; d = -0.25$
• B. $a = 0,\; b = -0.5,\; c = 0,\; d = 0.042$
• C. $a = 0,\; b = 0.5,\; c = 0,\; d = 0.042$
• D. $a = -0.5,\; b = 0,\; c = 0.042,\; d = 0$

Hint:

When matching Taylor expansions, always compare term-by-term and remember that odd powers vanish for even functions like $\cos(x)$.

Answer 1

The Correct Option is B

To find the coefficients \(a, b, c, d\), we use the Taylor expansion of \(\cos(x)\) about \(x = 0\): \[ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \]

Step 1: Match Taylor expansion terms.
The standard Taylor series is: \[ \cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \]

Step 2: Write the given polynomial form.
\[ \cos(x) \approx 1 + a x + b x^2 + c x^3 + d x^4 \]

Step 3: Compare coefficients.
- There is no \(x\) term in cosine ⇒ \(a = 0\).
- Coefficient of \(x^2\): \(b = -\frac{1}{2} = -0.5\).
- There is no \(x^3\) term ⇒ \(c = 0\).
- Coefficient of \(x^4\): \(d = \frac{1}{24} \approx 0.041666 \approx 0.042\).

Step 4: Conclusion.
\[ (a,\; b,\; c,\; d) = (0,\; -0.5,\; 0,\; 0.042) \] which matches Option (B).