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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.1.16b
Capitolo 6, Problema 6.1.16b

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


b. Find the displacement over the given interval. 


v(t) = 50e^−2t on [0, 4]

Guida verificata passo dopo passo
1
Identify the displacement as the definite integral of the velocity function over the given time interval. Displacement is given by \(\int_{a}^{b} v(t) \, dt\), where \(a=0\) and \(b=4\) in this problem.
Write down the integral to find displacement: \(\int_{0}^{4} 50 e^{-2t} \, dt\).
Recall the integral formula for an exponential function: \(\int e^{kt} \, dt = \frac{1}{k} e^{kt} + C\). Here, \(k = -2\).
Apply the integral formula to \(50 e^{-2t}\): the antiderivative is \(50 \times \frac{1}{-2} e^{-2t} = -25 e^{-2t}\).
Evaluate the definite integral by substituting the limits: calculate \([-25 e^{-2t}]\) from \(t=0\) to \(t=4\), which means computing \(-25 e^{-8} - (-25 e^{0})\).

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Displacement and Velocity Relationship

Displacement represents the change in position of an object and is found by integrating the velocity function over a given time interval. Since velocity is the rate of change of position, integrating velocity with respect to time gives the net displacement.
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Derivatives Applied To Velocity

Definite Integral

A definite integral calculates the accumulated quantity, such as displacement, over a specific interval. For velocity functions, the definite integral from time a to b gives the total displacement between those times.
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Definition of the Definite Integral

Exponential Decay Function

The velocity function v(t) = 50e^(-2t) is an exponential decay, meaning velocity decreases rapidly over time. Understanding how to integrate exponential functions is essential to find displacement accurately.
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Exponential Growth & Decay
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