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 = √7x-1
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 20e
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) = 6t3 + 36t2 - 54t; 0 ≤ t ≤ 4
Verified step by step guidance1
Step 1: Understand that speed is the absolute value of velocity. To find when speed is increasing, we need to analyze the derivative of the velocity function.
Step 2: First, find the velocity function v(t) by taking the derivative of the position function f(t). So, v(t) = f'(t) = \(\frac{d}{dt}\)(6t^3 + 36t^2 - 54t).
Step 3: Calculate the derivative: v(t) = 18t^2 + 72t - 54.
Step 4: To find when the speed is increasing, we need to find the derivative of the velocity function, which is the acceleration a(t). So, a(t) = v'(t) = \(\frac{d}{dt}\)(18t^2 + 72t - 54).
Step 5: Calculate the acceleration: a(t) = 36t + 72. Determine the intervals where a(t) > 0, as this indicates the velocity is increasing, and thus the speed is increasing.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity and Speed
Velocity is the rate of change of position with respect to time, represented mathematically as the derivative of the position function, f(t). Speed, being the absolute value of velocity, indicates how fast an object is moving regardless of direction. To determine when speed is increasing, one must analyze the behavior of the velocity function over the given interval.
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Derivatives Applied To Velocity
Acceleration
Acceleration is the rate of change of velocity with respect to time, which can be found by taking the derivative of the velocity function. If the acceleration is positive, the speed of the object is increasing. Therefore, to find intervals where speed is increasing, we need to examine where the derivative of the velocity function is positive.
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Critical Points and Intervals
Critical points occur where the first derivative of a function is zero or undefined, indicating potential maxima, minima, or points of inflection. By analyzing these points and the sign of the derivative in the intervals between them, one can determine where the function is increasing or decreasing. This is essential for identifying the intervals where speed is increasing in the context of the given position function.
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Related Practice
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 = e^4x²+1
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 = e^√x
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 = tan 5x²
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) = 2t3 - 21t2 + 60t; 0 ≤ t ≤ 6
Textbook Question
Position, velocity, and 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.
a. Graph the position function.
1
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