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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
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Capitolo 2, Problema 14d

The position of an object moving vertically along a line is given by the function s(t)=−4.9t2+30t+20s\(\left\)(t\(\right\))=-4.9t^2+30t+20. Find the average velocity of the object over the following intervals.
[0,h]\(\left\[\lbrack\)0,h\(\right\]\rbrack\), where h>0h\(\gt{0}\) is a real number

Guida verificata passo dopo passo
1
Identify the formula for average velocity over an interval [a, b], which is given by the change in position divided by the change in time: \( v_{avg} = \frac{s(b) - s(a)}{b - a} \).
In this problem, the interval is [0, h], so we need to find the average velocity over this interval. Set \( a = 0 \) and \( b = h \).
Substitute the values into the average velocity formula: \( v_{avg} = \frac{s(h) - s(0)}{h - 0} \).
Calculate \( s(h) \) by substituting \( t = h \) into the position function: \( s(h) = -4.9h^2 + 30h + 20 \).
Calculate \( s(0) \) by substituting \( t = 0 \) into the position function: \( s(0) = -4.9(0)^2 + 30(0) + 20 = 20 \).

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Position Function

The position function describes the location of an object at any given time, represented mathematically as s(t). In this case, the function s(t) = -4.9t² + 30t + 20 models the vertical motion of an object under the influence of gravity, where t is time in seconds. Understanding this function is crucial for analyzing the object's motion and calculating its velocity.
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Relations and Functions

Average Velocity

Average velocity is defined as the change in position over the change in time, calculated using the formula (s(b) - s(a)) / (b - a) for an interval [a, b]. In this context, to find the average velocity over the interval [0, h], one would evaluate the position function at the endpoints and apply this formula. This concept is essential for understanding how the object's speed changes over time.
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Average Value of a Function

Intervals and Limits

Intervals in calculus refer to the range of values over which a function is analyzed. In this question, the interval [0, h] indicates that we are examining the object's motion from time t = 0 to t = h, where h is a positive real number. Understanding how to work with intervals is important for evaluating functions and determining properties like average velocity over specific time frames.
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One-Sided Limits
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