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Turning a corner with a pole
What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is a ft wide and a corridor that is b ft wide meet at right angles?
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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 89
More limits Evaluate the following limits.
lim_x→1 (x ln x - x + 1) / (xln²x)
Guida verificata passo dopo passo1
First, identify the form of the limit as x approaches 1. Substitute x = 1 into the expression (x ln x - x + 1) / (x ln²x) to check if it results in an indeterminate form like 0/0.
Since substituting x = 1 results in an indeterminate form, apply L'Hôpital's Rule, which is used to evaluate limits of indeterminate forms. This rule states that if the limit of f(x)/g(x) as x approaches a value results in 0/0 or ∞/∞, then it can be evaluated as the limit of f'(x)/g'(x).
Differentiate the numerator f(x) = x ln x - x + 1 with respect to x. Use the product rule for the term x ln x, which gives the derivative as ln x + 1. The derivative of -x is -1, and the derivative of the constant 1 is 0.
Differentiate the denominator g(x) = x ln²x with respect to x. Again, use the product rule. The derivative of x is 1, and the derivative of ln²x is 2 ln x * (1/x) by the chain rule. Combine these to get the derivative of the denominator.
Substitute the derivatives back into the limit expression and evaluate the new limit as x approaches 1. If necessary, apply L'Hôpital's Rule again if the result is still an indeterminate form.
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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 understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches 1 involves analyzing the behavior of the function near that point, which may require techniques like substitution or L'Hôpital's rule if the limit results in an indeterminate form.
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One-Sided Limits
Indeterminate Forms
Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is crucial for applying methods such as L'Hôpital's rule, which allows for differentiation of the numerator and denominator to resolve the limit. In the given limit, substituting x = 1 results in an indeterminate form, necessitating further analysis.
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L'Hôpital's Rule
L'Hôpital's Rule is a powerful tool in calculus used to evaluate limits that result in indeterminate forms. It states that if the limit of f(x)/g(x) as x approaches a point yields 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the evaluation process, making it easier to find the limit of complex functions like the one presented in the question.
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{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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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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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
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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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{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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