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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.31b
Chapter 2, Problem 2.31b

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.


b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t=a.

Verified step by step guidance
1
insert step 1: Understand that the instantaneous velocity of the projectile is given by the derivative of the position function s(t).
insert step 2: Differentiate the position function s(t) = -16t^2 + 128t + 192 with respect to t to find the velocity function v(t).
insert step 3: Set the velocity function v(t) equal to zero to find the time when the projectile has an instantaneous velocity of zero.
insert step 4: Solve the equation from step 3 for t to find the value of t = a.
insert step 5: Verify that the value of t = a is within the given interval 0 ≤ t ≤ 9.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Position Function

The position function describes the location of an object at any given time. In this case, s(t) = -16t^2 + 128t + 192 represents the height of a projectile over time, where 't' is the time in seconds. Understanding this function is crucial for analyzing the projectile's motion and determining its behavior at different time intervals.
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Relations and Functions

Instantaneous Velocity

Instantaneous velocity is the rate of change of the position function with respect to time, represented mathematically as the derivative of the position function, s'(t). For the given position function, finding the time when the instantaneous velocity is zero involves calculating the derivative and solving for t when s'(t) = 0, indicating the moment the projectile reaches its peak height.
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Derivatives Applied To Velocity

Critical Points

Critical points occur where the derivative of a function is zero or undefined, indicating potential maxima, minima, or points of inflection. In the context of the projectile's motion, identifying the critical points of the velocity function helps determine when the projectile stops ascending and starts descending, which is essential for understanding its overall trajectory.
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Related Practice
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Determine the following limits.


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Complete the following steps for the given functions. 


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Determine the following limits.


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