Given that f(x)=x^2 e^x, what is an appropriate value of f(1.1) if you use the equation of the tangent line to the graph of f at x=1. Use a calculator to answer this question 

To approximate \( f(1.1) \) using the tangent line to the graph of \( f(x) = x^2 e^x \) at \( x = 1 \), we need to follow these steps:

1. **Find \( f(1) \):**
   \[
   f(1) = (1)^2 e^1 = e
   \]

2. **Find \( f'(x) \):**
   Using the product rule, where \( f(x) = x^2 e^x \):
   \[
   f'(x) = \frac{d}{dx}(x^2) \cdot e^x + x^2 \cdot \frac{d}{dx}(e^x)
   \]
   \[
   f'(x) = 2x e^x + x^2 e^x = (2x + x^2) e^x
   \]

3. **Evaluate \( f'(1) \):**
   \[
   f'(1) = (2 \cdot 1 + 1^2) e^1 = 3e
   \]

4. **Write the equation of the tangent line at \( x = 1 \):**
   The tangent line equation at \( x = 1 \) is:
   \[
   y = f(1) + f'(1)(x - 1)
   \]
   Substituting \( f(1) = e \) and \( f'(1) = 3e \):
   \[
   y = e + 3e(x - 1)
   \]
   \[
   y = e(1 + 3(x - 1))
   \]
   \[
   y = e(3x - 2)
   \]

5. **Approximate \( f(1.1) \):**
   Substitute \( x = 1.1 \) into the tangent line equation:
   \[
   y \approx e(3(1.1) - 2) = e(3.3 - 2) = e \cdot 1.3
   \]

Using a calculator to approximate this final result:
\[
f(1.1) \approx e \cdot 1.3 \approx 3.532 \quad \text{(approximately)}
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How would you find the second derivative of \( f(x) \)?
2. How can you determine if the tangent line underestimates or overestimates \( f(1.1) \)?
3. What would happen to the approximation if we chose a point closer to \( x = 1 \)?
4. How do we interpret the slope of the tangent line in this context?
5. What other methods besides the tangent line can approximate values of \( f(x) \) near \( x = 1 \)?

**Tip:** When using tangent line approximations, values closer to the point of tangency yield more accurate results.

Given that f(x)=x^2 e^x, what is an appropriate value of f(1.1) if you use the equation of the tangent line to the graph of f at x=1. Use a calculator to answer this question

Learn how to approximate f(1.1) for f(x) = x^2 e^x using the tangent line at x = 1. This calculus problem guides you through finding derivatives with the product rule and using tangent line equations to estimate function values. Suitable for advanced high school students (Grades 11-12).

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