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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 2.21a
Chapter 2, Problem 2.21a

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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1
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: \( v_{avg} = \frac{s(3+h) - s(3)}{h} \).
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.
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Derivatives Applied To Velocity

Instantaneous Velocity

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.
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Derivatives Applied To Velocity

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.
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Relations and Functions
Related Practice
Textbook Question

a. Estimate lim x→π/4 cos 2x / cos x − sin x by making a table of values of cos 2x / cos x − sin x for values of x approaching π/4. Round your estimate to four digits.

Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=x2−2x+53x−2f\(\left\)(x\(\right\))=\(\frac{x^2-2x+5}{3x-2}\)

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Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)

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Textbook Question

a. Use a graphing utility to estimate lim x→0 tan 2x / sin x, lim x→0 tan 3x / sin x, and lim x→0 tan 4x / sin x.

Textbook Question

Given the graph of f in the following figures, find the slope of the secant line that passes through (0,0) and (h,f(h))in terms of h, for h>0 and h<0.


f(x)=x1/3 <IMAGE>

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Textbook Question

A rock is dropped off the edge of a cliff, and its distance s (in feet) from the top of the cliff after t seconds is s(t)=16t^2. Assume the distance from the top of the cliff to the ground is 96 ft.


a. When will the rock strike the ground?