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Let's solve each part step by step.

### (a) Find the tangent line approximation to \( \cos x \) at \( x = \frac{\pi}{4} \):

The tangent line approximation of a function \( f(x) \) at a point \( x = a \) is given by:
\[
f(x) \approx f(a) + f'(a)(x - a).
\]
Here, \( f(x) = \cos x \) and \( a = \frac{\pi}{4} \).

1. **Evaluate \( f(a) = \cos\left(\frac{\pi}{4}\right) \):**
   \[
   \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}.
   \]

2. **Find the derivative \( f'(x) = -\sin x \) and evaluate \( f'(a) = -\sin\left(\frac{\pi}{4}\right) \):**
   \[
   \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \text{so} \quad f'(a) = -\frac{\sqrt{2}}{2}.
   \]

3. **Form the tangent line approximation:**
   \[
   \cos x \approx \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}).
   \]
   For rounding to three decimal places:
   \[
   \cos x \approx 0.707 - 0.707(x - 0.785).
   \]

### (b) Estimate the error in the approximation for \( 0 \leq x \leq \frac{\pi}{2} \):

To estimate the error, we use the fact that the error for a Taylor approximation of the first degree (linear approximation) is given by:
\[
\left| R_1(x) \right| = \frac{M}{2} |x - a|^2,
\]
where \( M \) is an upper bound on the absolute value of the second derivative \( f''(x) \) over the interval.

1. **Find \( f''(x) = -\cos x \):**
   The maximum value of \( |\cos x| \) on \( [0, \frac{\pi}{2}] \) is 1, so \( M = 1 \).

2. **Estimate the error for \( x = \frac{\pi}{2} \):**
   \[
   \left| R_1\left(\frac{\pi}{2}\right) \right| \approx \frac{1}{2} \left| \frac{\pi}{2} - \frac{\pi}{4} \right|^2 = \frac{1}{2} \left(\frac{\pi}{4}\right)^2 = \frac{\pi^2}{32}.
   \]
   Calculating \( \frac{\pi^2}{32} \approx 0.308 \).

Thus, the error is less than 0.3 (to one decimal place).

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Would you like more details, or have any questions? Here are five related questions to explore further:

1. What is the tangent line approximation for other trigonometric functions?
2. How does the error change with higher-degree Taylor approximations?
3. What is the geometric interpretation of a tangent line approximation?
4. How can we determine a tighter bound on the error?
5. What happens if we approximate \( \cos x \) at points other than \( \frac{\pi}{4} \)?

**Tip:** The closer \( x \) is to the point of approximation, the more accurate the tangent line approximation will be.

Find the tangent line approximation to cos(x) at x = π/4 and estimate the error in the approximation for 0 ≤ x ≤ π/2.

Find the tangent line approximation of cos(x) at x = π/4 and estimate the error for 0 ≤ x ≤ π/2. This problem uses key calculus concepts like Taylor series and error estimation techniques. Suitable for Calculus I/II students.

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