Estimate the instantaneous rate of change of \( P(t)=4 t^{2}-2 \) at the point: \( t=3 \) nother words, choose \( x \)-values that are getting closer and closer to 3 and compute the slope of the secant ines at each value. Then, use the trend/pattern you see to estimate the slope of the tangent line. Your answer should be accurate to at least 2 decimal places.

To estimate the instantaneous rate of change of the function \( P(t) = 4t^2 - 2 \) at the point \( t = 3 \), we can compute the slopes of secant lines using values of \( t \) that are close to 3. The slope of the secant line between two points \( (t_1, P(t_1)) \) and \( (t_2, P(t_2)) \) is given by: \[ \text{slope} = \frac{P(t_2) - P(t_1)}{t_2 - t_1} \] We will choose values of \( t \) that approach 3 from both sides. Let's calculate the slopes for the following pairs of \( t \) values: 1. **Using \( t_1 = 3 \) and \( t_2 = 3.1 \)**: \[ P(3) = 4(3^2) - 2 = 4(9) - 2 = 36 - 2 = 34 \] \[ P(3.1) = 4(3.1^2) - 2 = 4(9.61) - 2 = 38.44 - 2 = 36.44 \] \[ \text{slope} = \frac{36.44 - 34}{3.1 - 3} = \frac{2.44}{0.1} = 24.4 \] 2. **Using \( t_1 = 3 \) and \( t_2 = 2.9 \)**: \[ P(2.9) = 4(2.9^2) - 2 = 4(8.41) - 2 = 33.64 - 2 = 31.64 \] \[ \text{slope} = \frac{34 - 31.64}{3 - 2.9} = \frac{2.36}{0.1} = 23.6 \] 3. **Using \( t_1 = 3 \) and \( t_2 = 3.01 \)**: \[ P(3.01) = 4(3.01^2) - 2 = 4(9.0601) - 2 = 36.2404 - 2 = 34.2404 \] \[ \text{slope} = \frac{34.2404 - 34}{3.01 - 3} = \frac{0.2404}{0.01} = 24.04 \] 4. **Using \( t_1 = 3 \) and \( t_2 = 2.99 \)**: \[ P(2.99) = 4(2.99^2) - 2 = 4(8.9401) - 2 = 35.7604 - 2 = 33.7604 \] \[ \text{slope} = \frac{34 - 33.7604}{3 - 2.99} = \frac{0.2396}{0.01} = 23.96 \] Now, we can summarize the slopes we calculated: - From \( t = 3 \) to \( t = 3.1 \): \( 24.4 \) - From \( t = 3 \) to \( t = 2.9 \): \( 23.6 \) - From \( t = 3 \) to \( t = 3.01 \): \( 24.04 \) - From \( t = 3 \) to \( t = 2.99 \): \( 23.96 \) As \( t \) approaches 3, the slopes are converging around \( 24 \). Thus, we can estimate the instantaneous rate of change of \( P(t) \) at \( t = 3 \) to be approximately: \[ \boxed{24.00} \]