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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.7.104
Chapter 4, Problem 4.7.104

Applications


Stopping a motorcycle The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from 30 mph (44 ft/sec) to 0 in 45 ft. What constant deceleration does it take to do that?

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1
Identify the known values: initial velocity \(v_0 = 44\) ft/sec, final velocity \(v = 0\) ft/sec, and stopping distance \(d = 45\) ft.
Recall the kinematic equation that relates velocity, acceleration, and distance without time: \(v^2 = v_0^2 + 2 a d\), where \(a\) is the acceleration (or deceleration in this case).
Substitute the known values into the equation: \(0 = (44)^2 + 2 \times a \times 45\).
Solve the equation for \(a\) to find the constant deceleration required: rearrange to \(a = -\frac{(44)^2}{2 \times 45}\).
Interpret the negative sign of \(a\) as indicating deceleration (slowing down), which matches the physical situation of braking.

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Key Concepts

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

Kinematic Equations for Constant Acceleration

Kinematic equations describe the motion of objects under constant acceleration. They relate velocity, acceleration, displacement, and time, allowing calculation of unknown variables when others are known. In this problem, the equation v² = v₀² + 2aΔx is useful to find acceleration given initial velocity, final velocity, and displacement.
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Using The Acceleration Function

Unit Conversion

Unit conversion ensures all quantities are expressed in compatible units before calculations. Here, speed is given in miles per hour and feet per second, so converting 30 mph to feet per second or vice versa is essential to maintain consistency and accuracy in solving the problem.
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Evaluate Composite Functions - Values on Unit Circle

Deceleration as Negative Acceleration

Deceleration refers to acceleration that reduces the speed of an object, represented as a negative acceleration value. Understanding that braking involves a negative acceleration helps correctly interpret the sign and magnitude of the acceleration calculated from the kinematic equations.
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Derivatives Applied To Acceleration
Related Practice
Textbook Question

Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = √(x(1 − x)), [0, 1]

Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(−2cost) dt

Textbook Question

Finding Extrema from Graphs


In Exercises 11–14, match the table with a graph.


1
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Textbook Question

Checking the Mean Value Theorem


Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.


g(x) = {x³, −2 ≤ x ≤ 0

x², 0 < x ≤ 2

Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(√x + ³√x) dx

Textbook Question

37. What value of a makes f(x) = x^2 +(a/x) have

a. a local minimum at x = 2?

b. a point of inflection at x = 1?