Find the average rate of change of the function over the given interval. \[ f(t)=4 t^{2}-3, \quad[3,3.1] \] \( \square \) Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval: left endpoint \( \square \) right endpoint \( \square \) Show My Work (Optional)

To calculate the average rate of change of the function \( f(t) = 4t^2 - 3 \) over the interval \([3, 3.1]\), we use the formula: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] Here, \(a = 3\) and \(b = 3.1\). First, calculate \(f(3)\) and \(f(3.1)\): 1. \(f(3) = 4(3)^2 - 3 = 4 \cdot 9 - 3 = 36 - 3 = 33\) 2. \(f(3.1) = 4(3.1)^2 - 3 = 4 \cdot 9.61 - 3 = 38.44 - 3 = 35.44\) Now, plug these values into the average rate of change formula: \[ \text{Average Rate of Change} = \frac{35.44 - 33}{3.1 - 3} = \frac{2.44}{0.1} = 24.4 \] Next, we find the instantaneous rates of change at the endpoints by calculating the derivative of \(f(t) = 4t^2 - 3\): \[ f'(t) = \frac{d}{dt}(4t^2 - 3) = 8t \] Now evaluate \(f'\) at the endpoints: - Left endpoint \(f'(3) = 8(3) = 24\) - Right endpoint \(f'(3.1) = 8(3.1) = 24.8\) In summary: - The average rate of change over the interval \([3, 3.1]\) is \(24.4\). - The instantaneous rate of change at the left endpoint is \(24\), while at the right endpoint, it is \(24.8\). Now let’s have some additional fun facts! The thrill of calculus can be traced back to the Renaissance when mathematicians like Newton and Leibniz were making groundbreaking advances. They explored the relationships between functions and their rates of change, laying the groundwork for modern mathematics! Understanding these concepts not only enhances your problem-solving skills but they are also super useful in real-life scenarios, like predicting population growth, calculating speed, or even in optimizing business costs. Calculus helps you make smart decisions by understanding how things change over time!