Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 15e

Chapter 3, Problem 15e

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
On what intervals is the speed increasing?
f(t) = t² - 4t; 0 ≤ t ≤ 5

Verified step by step guidance

Step 1: Understand that speed is the absolute value of velocity. To find when speed is increasing, we need to determine when the derivative of the speed function is positive.

Step 2: Find the velocity function by taking the derivative of the position function f(t). The velocity v(t) is given by v(t) = f'(t) = \(\frac{d}{dt}\)(t^2 - 4t).

Step 3: Calculate the derivative: f'(t) = 2t - 4. This is the velocity function v(t).

Step 4: Determine when the velocity is increasing by finding the derivative of the velocity function, which is the acceleration a(t). So, a(t) = v'(t) = \(\frac{d}{dt}\)(2t - 4).

Step 5: Calculate the acceleration: a(t) = 2. Since the acceleration is constant and positive, the velocity is increasing for all t in the interval [0, 5]. Therefore, the speed is increasing wherever the velocity is positive.

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Key Concepts

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

Position Function

The position function, denoted as s = f(t), describes the location of an object at any given time t. In this case, f(t) = t² - 4t represents a quadratic function that models the object's position in feet. Understanding this function is crucial for analyzing the object's motion and determining its speed and acceleration.

Velocity and Speed

Velocity is the rate of change of position with respect to time, calculated as the derivative of the position function, v(t) = f'(t). Speed, being the absolute value of velocity, indicates how fast the object is moving regardless of direction. To determine when speed is increasing, one must analyze the behavior of the velocity function over the specified interval.

Acceleration

Acceleration is the rate of change of velocity with respect to time, represented as the derivative of the velocity function, a(t) = v'(t). When acceleration is positive, the speed of the object is increasing. Thus, to find the intervals where speed is increasing, one must identify where the acceleration function is greater than zero within the given time frame.

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Derivatives Applied To Acceleration

Related Practice

Textbook Question

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = x² - 5; P(3,4)

Textbook Question

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

Determine the velocity and acceleration of the object at t = 1.

f(t) = t² − 4t; 0 ≤ t ≤ 5

Textbook Question

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = sin⁵x

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Textbook Question

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = 1/x; P(-1,-1)

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Textbook Question

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = (5x²+11x)^4/3

Textbook Question

Determine the acceleration of the object when its velocity is zero.

f(t) = t² - 4t; 0 ≤ t ≤ 5

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.On what intervals is the speed increasing?f(t) = t2 - 4t; 0 ≤ t ≤ 5

Verified video answer for a similar problem:

Key Concepts

Position Function

Velocity and Speed

Acceleration

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
On what intervals is the speed increasing?
f(t) = t² - 4t; 0 ≤ t ≤ 5