Question

Consider the position function s(t)=−$$16t^2+100t. $$Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=3.

Accepted Answer

Step 1: Understand the problem.We are given a position function $$ s(t) = -16t^2 + 100t $$ and need to find average velocities over certain intervals and make a conjecture about the instantaneous velocity at $$ t = 3 . $$Step 2: Calculate the average velocity over an interval $$[a, b].$$The average velocity $$ v_{avg} $$ over an interval $$[a, b] is $$given by the formula:$$ v_{avg} = \frac{s(b) - s(a)}{b - a} $$Apply this formula to the intervals given in the table. Step 3: Evaluate the position function at the endpoints of each interval.For example, if the interval is $$[3, 3.1]$$, calculate $$ s(3) $$ and $$ s(3.1) $$.$$ s(3) = -16(3)^2 + 100(3) $ s(3.1) = -16(3.1)^2 + 100(3.1) $$ Step 4: Substitute the values into the average velocity formula.Using the example interval $$[3, 3.1]$$:$$ v_{avg} = \frac{s(3.1) - s(3)}{3.1 - 3} $$Repeat this process for each interval in the table. Step 5: Make a conjecture about the instantaneous velocity at $$ t = 3 .As $$the intervals get smaller and approach $$ t = 3 $$, observe the trend in the average velocities. This trend will help you conjecture the instantaneous velocity at $$ t = 3 $$, which is the derivative $$ s'(t) $$ evaluated at $$ t = 3 .$$

Consider the position function s(t)=−16t^2+100t. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=3. <IMAGE>

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Key Concepts

Position Function

Average Velocity

Instantaneous Velocity