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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.5.93d
Chapter 8, Problem 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?

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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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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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Finding Limits by Direct Substitution
Related Practice
Textbook Question

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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Textbook Question

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.

Textbook Question

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.

Textbook Question

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

69. Let f(x) = sin(eˣ).

d. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1.

Textbook Question

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

67. Let f(x) = √(x³ + 1).

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

Textbook Question

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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