Domanda del libro di testo
{Use of Tech} Graph carefully Graph the function f(x) = 60x⁵ - 901x³ + 27x in the window [-4,4] x [-10,000, 10,000]. How many extreme values do you see? Locate all the extreme values by analyzing f'.
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 100
Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.
x² ln x; x³
Guida verificata passo dopo passo1
First, understand that to compare the growth rates of two functions, we can use the concept of limits. Specifically, we can evaluate the limit of the ratio of the two functions as x approaches infinity.
Set up the limit expression: \( \lim_{x \to \infty} \frac{x^2 \ln x}{x^3} \). This will help us determine which function grows faster.
Simplify the expression inside the limit: \( \frac{x^2 \ln x}{x^3} = \frac{\ln x}{x} \). This simplification is achieved by canceling \( x^2 \) from both the numerator and the denominator.
Evaluate the limit \( \lim_{x \to \infty} \frac{\ln x}{x} \). As x approaches infinity, \( \ln x \) grows slower than any polynomial function, and \( x \) grows faster, leading the limit to approach 0.
Conclude that since the limit is 0, \( x^3 \) grows faster than \( x^2 \ln x \) as x approaches infinity.
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Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for analyzing the behavior of functions, especially as they tend toward infinity or a specific value. In this context, limits help determine the growth rates of the functions x² ln x and x³ as x becomes very large.
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05:50
One-Sided Limits
Growth Rates
Growth rates describe how a function increases as its input increases. In calculus, we often compare functions to see which grows faster by examining their limits. This comparison can be made using techniques like L'Hôpital's Rule or by analyzing the leading terms of the functions involved, which is crucial for understanding the relative growth of x² ln x and x³.
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Percorso guidato
04:16Intro To Related Rates
L'Hôpital's Rule
L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in comparing the growth rates of functions like x² ln x and x³, allowing for a clearer analysis of their behavior as x approaches infinity.
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5:50Power Rules
Pratica correlata
Domanda del libro di testo
Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule.
lim_x→0 (e²ˣ + 4eˣ - 5) / (e²ˣ - 1)
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Domanda del libro di testo
More limits Evaluate the following limits.
lim_x→1 (x ln x - x + 1) / (xln²x)
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Domanda del libro di testo
Interpreting the derivative The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>
a. On what interval(s) is f increasing? Decreasing?
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Domanda del libro di testo
Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.
x²⁰ ; 1.0001ˣ
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Domanda del libro di testo
{Use of Tech} Optimal boxes Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 125 ft³.
b. Based on your graph in part (a), estimate the value of x that produces the box with a minimum surface area.
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