Textbook Question
112. True, or false? Give reasons for your answers.
a. 1/x⁴ = O(1/x² + 1/x⁴)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.P.118
118. A particle is traveling upward and to the right along the curve y=ln(x). Its x-coordinate is increasing at the rate (dx/dt)=√x m/sec. At what rate is the y-coordinate changing at the point (e², 2)?
Verified step by step guidance1
Identify the given information: The particle moves along the curve \(y = \ln(x)\), and the rate of change of the x-coordinate with respect to time is given by \(\frac{dx}{dt} = \sqrt{x}\) meters per second.
Recall that the rate of change of the y-coordinate with respect to time, \(\frac{dy}{dt}\), can be found using the chain rule: \(\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}\).
Calculate \(\frac{dy}{dx}\) by differentiating \(y = \ln(x)\) with respect to \(x\): \(\frac{dy}{dx} = \frac{1}{x}\).
Evaluate \(\frac{dy}{dx}\) and \(\frac{dx}{dt}\) at the point \((e^{2}, 2)\): Substitute \(x = e^{2}\) into both expressions to find their values at that point.
Multiply the values found in the previous step to get \(\frac{dy}{dt}\) at the point \((e^{2}, 2)\): \(\frac{dy}{dt} = \frac{1}{e^{2}} \times \sqrt{e^{2}}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Related Rates
Related rates problems involve finding the rate at which one quantity changes by relating it to the rate of change of another quantity through their functional relationship. Here, the rates of change of x and y with respect to time are connected via the curve equation.
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Intro To Related Rates
Chain Rule in Differentiation
The chain rule allows differentiation of composite functions. Since y = ln(x) depends on x, which in turn depends on time t, dy/dt is found by multiplying dy/dx by dx/dt, linking the rates of change through the chain rule.
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05:02Intro to the Chain Rule
Properties of the Natural Logarithm Function
The natural logarithm function y = ln(x) has a derivative dy/dx = 1/x. This derivative is essential to find how y changes with respect to x, which is then used to determine the rate of change of y with respect to time.
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06:21Properties of Functions
Related Practice
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
13. y = (x+2)^(x+2)
Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
29. y = (sin θ)^√θ
Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
25. y = 2(x² + 1)/√(cos 2x)
Textbook Question
111. True, or false? Give reasons for your answers.
c. x = o(x + ln(x))
Textbook Question
111. True, or false? Give reasons for your answers.
e. arctan x = O(1)