Consider the position function s(t)=−16t^2+128t (Exercise 13). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=1. <IMAGE>

Verified step by step guidance

1. Step 1: Understand the problem. The position function s(t) = -16t^2 + 128t represents the position of an object at time t. We need to find average velocities over certain intervals and make a conjecture about the instantaneous velocity at t = 1.
2. Step 2: Recall the formula for average velocity over an interval [a, b], which is given by (s(b) - s(a)) / (b - a).
3. Step 3: Calculate the average velocity over the interval [1, 1+h] for small values of h. This involves computing (s(1+h) - s(1)) / h.
4. Step 4: Substitute s(t) = -16t^2 + 128t into the average velocity formula: (s(1+h) - s(1)) / h = ((-16(1+h)^2 + 128(1+h)) - (-16(1)^2 + 128(1))) / h.
5. Step 5: Simplify the expression from Step 4 and evaluate the limit as h approaches 0 to conjecture the instantaneous velocity at t = 1.

Key Concepts

Position Function

A position function describes the location of an object at a given time, typically represented as s(t). In this case, s(t) = -16t^2 + 128t models the vertical position of an object under the influence of gravity, where t is time in seconds. Understanding this function is crucial for analyzing motion and calculating velocities.

Average Velocity

Average velocity is defined as the change in position over the change in time, calculated as (s(t2) - s(t1)) / (t2 - t1). It provides a measure of how fast an object is moving over a specific interval. In the context of the given position function, calculating average velocities at different intervals helps in understanding the object's overall motion.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific moment in time, represented mathematically as the derivative of the position function, v(t) = s'(t). It provides a precise measure of how fast the object is moving at that exact time. Making a conjecture about the instantaneous velocity at t=1 involves evaluating the derivative of the position function at that point.