Let's solve each part step by step: --- ### 1. **Maclaurin Series for \( \cos x \) and \( x^2 \cos^2 x \):** #### **Maclaurin Series for \( \cos x \):** \[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \] #### **Maclaurin Series for \( \cos^2 x \):** Recall, \[ \cos^2 x = \frac{1 + \cos 2x}{2} \] So, \[ \cos^2 x = \frac{1}{2} + \frac{1}{2}\cos 2x \] Now, \( \cos 2x \) series: \[ \cos 2x = 1 - \frac{(2x)^2}{2!} + \frac{(2x)^4}{4!} - \cdots = 1 - 2x^2 + \frac{4x^4}{3} - \cdots \] So, \[ \cos^2 x = \frac{1}{2} + \frac{1}{2} \left[1 - 2x^2 + 2x^4/3 - \cdots \right] = 1 - x^2 + \frac{x^4}{3} + \cdots \] #### **Multiply by \( x^2 \):** \[ x^2 \cos^2 x = x^2 \left[1 - x^2 + \frac{x^4}{3} + \cdots \right] = x^2 - x^4 + \frac{x^6}{3} + \cdots \] --- ### 2. **Integral as Power Series:** Evaluate: \[ \int_0^1 x^2 \cos^2 x \, dx \] Using the expansion above: \[ \int_0^1 x^2 \cos^2 x \, dx = \int_0^1 \left(x^2 - x^4 + \frac{x^6}{3} + \cdots \right) dx = \int_0^1 x^2 dx - \int_0^1 x^4 dx + \frac{1}{3} \int_0^1 x^6 dx + \cdots \] Each term: - \( \int_0^1 x^2 dx = \frac{1}{3} \) - \( \int_0^1 x^4 dx = \frac{1}{5} \) - \( \int_0^1 x^6 dx = \frac{1}{7} \) So, \[ = \frac{1}{3} - \frac{1}{5} + \frac{1}{3} \cdot \frac{1}{7} + \cdots = \frac{1}{3} - \frac{1}{5} + \frac{1}{21} + \cdots \] --- ### 3. **Approximation Using First Three Terms & Error Estimate** **Approximate Value:** \[ \frac{1}{3} - \frac{1}{5} + \frac{1}{21} \] **Error Estimate:** The error is less than the absolute value of the next term in the series, which is: \[ -\int_0^1 x^8 dx = -\frac{1}{9} \] But the coefficient for the \( x^8 \) term in the series \( x^2 \cos^2 x \) is \( -\frac{2}{45} \), so the error estimate: \[ \text{Error} < \left|\text{next term}\right| = \left|-\frac{2}{45} \int_0^1 x^8 dx\right| = \frac{2}{45} \cdot \frac{1}{9} = \frac{2}{405} \] --- ## **Final Answers (Boxed):** ### **1. Power Series for \( x^2\cos^2 x \):** \[ \boxed{x^2 \cos^2 x = x^2 - x^4 + \frac{x^6}{3} + \cdots} \] ### **2. Integral as a Power Series:** \[ \boxed{ \int_0^1 x^2 \cos^2 x \, dx = \frac{1}{3} - \frac{1}{5} + \frac{1}{21} + \cdots } \] ### **3. Approximation (First 3 Terms) and Error Estimate:** \[ \boxed{ \text{Approximation: } \frac{1}{3} - \frac{1}{5} + \frac{1}{21} } \] \[ \boxed{ \text{Error Estimate: } < \frac{2}{405} } \] Let me know if you want a numerical value!