Find the average rate of change of f\left(x\right)=9x^2-3 on the interval [1,t]
. Your answer will be an expression involving t

To find the average rate of change of the function \( f(x) = 9x^2 - 3 \) over the interval \([1, t]\), we use the formula for the average rate of change:

\[
\text{Average rate of change} = \frac{f(t) - f(1)}{t - 1}
\]

1. **Evaluate \( f(t) \)**:
   \[
   f(t) = 9t^2 - 3
   \]

2. **Evaluate \( f(1) \)**:
   \[
   f(1) = 9(1)^2 - 3 = 9 - 3 = 6
   \]

3. **Substitute \( f(t) \) and \( f(1) \) into the formula**:
   \[
   \text{Average rate of change} = \frac{(9t^2 - 3) - 6}{t - 1}
   \]

4. **Simplify the expression**:
   \[
   \text{Average rate of change} = \frac{9t^2 - 9}{t - 1}
   \]
   \[
   = \frac{9(t^2 - 1)}{t - 1}
   \]
   \[
   = \frac{9(t-1)(t+1)}{t-1}
   \]

5. **Cancel the common factor \( t-1 \) (assuming \( t \neq 1 \))**:
   \[
   \text{Average rate of change} = 9(t + 1)
   \]

Thus, the average rate of change of \( f(x) \) over the interval \([1, t]\) is:

\[
9(t + 1)
\]

Would you like more details or have any questions about this solution?

Here are five related questions to deepen your understanding:
1. How would the solution change if the function were \( f(x) = 9x^2 + 3 \)?
2. What is the significance of the average rate of change in the context of this function?
3. Can you find the instantaneous rate of change at \( x = 1 \)?
4. How would the average rate of change be interpreted if \( t \to 1 \)?
5. What is the average rate of change for a linear function \( f(x) = mx + b \)?

**Tip:** When calculating the average rate of change, it's essentially finding the slope of the secant line connecting two points on the graph of the function.

Find the average rate of change of f(x) = 9x^2 - 3 on the interval [1, t]. Your answer will be an expression involving t.

Learn how to find the average rate of change for the quadratic function f(x) = 9x^2 - 3 over the interval [1, t]. This step-by-step guide explains the formula and provides the solution. Suitable for high school students (Grades 10-12).

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