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.