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Ch. 8 - Integration Techniques
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. 8 - Integration TechniquesProblema 8.R.110
Capitolo 8, Problema 8.R.110

110. Comparing distances Suppose two cars started at the same time and place (t = 0 and s = 0). The velocity of car A (in mi/hr) is given by
u(t) = 40 / (t + 1) and the velocity of car B (in mi/hr) is given by v(t) = 40 * e^(-t/2).
b. After t = 3 hr, which car has traveled farther?

Guida verificata passo dopo passo
1
Recall that the distance traveled by a car from time \(t=0\) to \(t=3\) is the integral of its velocity function over that interval. So, for each car, we need to compute \(\int_0^3 u(t) \, dt\) and \(\int_0^3 v(t) \, dt\) respectively.
Set up the integral for car A's distance: \(\int_0^3 \frac{40}{t+1} \, dt\). This integral involves a rational function and can be solved using the natural logarithm function.
Set up the integral for car B's distance: \(\int_0^3 40 e^{-t/2} \, dt\). This integral involves an exponential function and can be solved using the formula for integrating exponentials.
Evaluate both integrals separately by applying the appropriate integration techniques: for car A, use the substitution \(u = t+1\); for car B, use the standard integral \(\int e^{kt} dt = \frac{1}{k} e^{kt} + C\).
After finding the expressions for the distances traveled by both cars at \(t=3\), compare the two values to determine which car has traveled farther.

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Velocity and Displacement Relationship

Velocity is the rate of change of displacement with respect to time. To find the total distance traveled by an object over a time interval, you integrate its velocity function over that interval. This integral gives the displacement, which in this context represents the distance traveled by each car.
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Derivatives Applied To Velocity

Definite Integration

Definite integration calculates the accumulated quantity, such as distance, over a specific interval. Here, integrating the velocity functions from t = 0 to t = 3 hours will yield the total distance each car has traveled. Understanding how to set up and evaluate these integrals is essential.
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Definition of the Definite Integral

Exponential and Rational Functions

The velocity functions involve different types of functions: car A's velocity is a rational function, and car B's velocity is an exponential decay function. Recognizing their forms helps in choosing appropriate integration techniques and understanding how their speeds change over time.
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Graphs of Exponential Functions
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