To find information about the function \( y = 7t^2 + 6e^t \), we could perform several analyses, such as finding its derivative, second derivative, or critical points. Here are a few possible directions:

1. **First Derivative (dy/dt)**:
   To determine the rate of change of \( y \) with respect to \( t \), we take the derivative:
   \[
   \frac{dy}{dt} = \frac{d}{dt}(7t^2) + \frac{d}{dt}(6e^t)
   \]
   Applying the power rule and exponential rule:
   \[
   \frac{dy}{dt} = 14t + 6e^t
   \]

2. **Second Derivative (d²y/dt²)**:
   To analyze the concavity or the acceleration of change, we find the second derivative:
   \[
   \frac{d^2y}{dt^2} = \frac{d}{dt}(14t) + \frac{d}{dt}(6e^t)
   \]
   \[
   \frac{d^2y}{dt^2} = 14 + 6e^t
   \]

3. **Critical Points**:
   Set the first derivative equal to zero to find critical points:
   \[
   14t + 6e^t = 0
   \]
   This equation could be solved numerically for specific values of \( t \).

4. **Behavior Analysis**:
   - As \( t \to \infty \), the term \( 6e^t \) dominates, so \( y \to \infty \).
   - As \( t \to -\infty \), \( 6e^t \) approaches zero, so \( y \approx 7t^2 \), which still tends to infinity.

5. **Graphical Analysis**:
   Plotting the function \( y = 7t^2 + 6e^t \) can give insight into the behavior, concavity, and growth rate of the function.

Would you like to explore any of these points in more detail?

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### Expanding Knowledge

1. What are the critical points of the function?
2. How does the function behave as \( t \) approaches zero?
3. What is the significance of the second derivative in understanding concavity?
4. How would the function change if we used a different coefficient for \( t^2 \) or \( e^t \)?
5. How does the exponential term \( e^t \) affect the long-term behavior of the function compared to polynomial growth?

**Tip:** Analyzing the first and second derivatives of a function gives essential insights into its growth and shape, especially when combining polynomial and exponential terms.

Explore the function y = 7t^2 + 6e^t, including its derivatives, critical points, and concavity analysis. Learn how to calculate the first and second derivatives, find critical points, and understand the long-term behavior of polynomial-exponential functions. This content is suitable for students in Grades 11-12 studying calculus.

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