For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time.
a. s(t)=−16t^2+80t+60 at t=3
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Step 1: Understand the problem. We are given a position function s(t) = -16t^2 + 80t + 60, and we need to find the average velocities over intervals approaching t = 3.
Step 2: Calculate the average velocity over an interval [3, 3+h]. The average velocity is given by the formula: .
Step 3: Substitute s(t) into the average velocity formula. Calculate s(3) and s(3+h) using the position function.
Step 4: Simplify the expression for average velocity. This involves expanding s(3+h) and simplifying the difference quotient.
Step 5: Make a conjecture about the instantaneous velocity at t = 3 by observing the behavior of the average velocity as h approaches 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Average Velocity
Average velocity is defined as the change in position over the change in time. Mathematically, it is calculated using the formula (s(t2) - s(t1)) / (t2 - t1), where s(t) represents the position function. This concept is crucial for understanding how position changes over intervals and is foundational for analyzing motion.
Instantaneous velocity refers to the velocity of an object at a specific moment in time. It is determined by taking the limit of the average velocity as the time interval approaches zero. This concept is essential for understanding how an object's speed and direction change at any given point, and it is often represented as the derivative of the position function.
A position function describes the location of an object as a function of time, typically denoted as s(t). In this case, s(t) = -16t^2 + 80t + 60 represents a quadratic function that models the motion of an object under the influence of gravity. Understanding the position function is vital for calculating both average and instantaneous velocities.