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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 8.5.93d
Capitolo 8, Problema 8.5.93d

93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:
vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).
d. Which car ultimately gains the lead and remains in front?

Guida verificata passo dopo passo
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Understand that the position of each car at time \( t \) is found by integrating its velocity function from 0 to \( t \). This gives the displacement \( s(t) \) for each car.
Write the position functions as integrals of the velocity functions: \( s_A(t) = \int_0^t \frac{88u}{u + 1} \, du \), \( s_B(t) = \int_0^t \frac{88u^2}{(u + 1)^2} \, du \), and \( s_C(t) = \int_0^t \frac{88u^2}{u^2 + 1} \, du \).
Evaluate or analyze the behavior of each position function as \( t \to \infty \) to determine which car travels the farthest distance over a long time, since the car with the greatest displacement will be in the lead ultimately.
Compare the limits or growth rates of \( s_A(t) \), \( s_B(t) \), and \( s_C(t) \) as \( t \to \infty \) to see which one dominates.
Conclude which car remains in front based on which position function grows fastest or has the greatest limit as time goes to infinity.

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Velocity and Position Relationship

Velocity is the rate of change of position with respect to time. To determine which car is ahead, we need to find each car's position by integrating its velocity function over time. The position function gives the displacement from the start, allowing comparison of their locations at any time.
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Percorso guidato
06:29
Derivatives Applied To Velocity

Definite Integration for Displacement

Displacement over a time interval is found by integrating the velocity function from the start time to the given time. This process accumulates the total distance traveled, accounting for changes in velocity, and is essential to compare the cars' positions as time progresses.
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Percorso guidato
05:43
Definition of the Definite Integral

Limits and Long-Term Behavior

Analyzing the limit of the position functions as time approaches infinity helps determine which car ultimately leads. Understanding how velocity and position behave for large time values reveals which car maintains the lead indefinitely.
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05:21
Finding Limits by Direct Substitution
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Domanda del libro di testo

87. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. If ∫(from 1 to ∞) x^(-p) dx exists, then ∫(from 1 to ∞) x^(-q) dx exists (where q > p).

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81. Possible and impossible integrals

Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.

d. Show that, in general, if n is odd, then Iₙ = -½ e⁻ˣ² pₙ₋₁(x), where pₙ₋₁ is a polynomial of degree n - 1.

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Domanda del libro di testo

57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The integral ∫ dx/(x² + 4x + 9) cannot be evaluated using a trigonometric substitution.

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Domanda del libro di testo

2. Give an example of each of the following.

d. A repeated irreducible quadratic factor

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66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

68. Let f(x) = e^(x²).

d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).

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Domanda del libro di testo

77. Tabular integration Consider the integral ∫ f(x)g(x) dx, where f can be differentiated repeatedly and g can be integrated repeatedly

Let Gₖ represent the result of calculating k indefinite integrals of g (omitting constants of integration).

d. The tabular integration table from part (c) is easily extended to allow for as many steps as necessary in the process of integration by parts.

Evaluate ∫ x² e^(x/2) dx by constructing an appropriate table, and explain why the process terminates after four rows of the table have been filled in.

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