Domanda del libro di testo
Does ƒ(x) = (x⁶/2) + (5x⁴/4) - 15x² have any inflection points? If so, identify them.
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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 24
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ ∞ (4x³ - 2x² + 6) / (πx³ + 4)
Guida verificata passo dopo passo1
First, identify the form of the limit as x approaches infinity. The given expression is (4x³ - 2x² + 6) / (πx³ + 4). As x approaches infinity, both the numerator and the denominator approach infinity, which is an indeterminate form ∞/∞.
Since the limit is in the indeterminate form ∞/∞, we can apply l'Hôpital's Rule. This rule states that if the limit of f(x)/g(x) as x approaches a value is in the form 0/0 or ∞/∞, then it can be evaluated as the limit of f'(x)/g'(x), provided this new limit exists.
Differentiate the numerator and the denominator separately. The derivative of the numerator 4x³ - 2x² + 6 is 12x² - 4x. The derivative of the denominator πx³ + 4 is 3πx².
Now, apply l'Hôpital's Rule by taking the limit of the new expression: lim_x→∞ (12x² - 4x) / (3πx²).
Simplify the expression by dividing each term by x², the highest power of x in the denominator. This gives lim_x→∞ (12 - 4/x) / (3π). As x approaches infinity, the term 4/x approaches 0, simplifying the limit to 12 / 3π.
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Limits
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior at points where it may not be explicitly defined, such as at infinity or at points of discontinuity. Evaluating limits is crucial for determining the continuity and differentiability of functions.
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One-Sided Limits
l'Hôpital's Rule
l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This process can be repeated if the result remains indeterminate.
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Polynomial Functions
Polynomial functions are expressions that consist of variables raised to whole number powers and their coefficients. In the context of limits, the degree of the polynomial in the numerator and denominator plays a crucial role in determining the limit as x approaches infinity. The leading terms of these polynomials dominate the behavior of the function at extreme values, simplifying the limit evaluation.
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6:04Introduction to Polynomial Functions
Pratica correlata
Domanda del libro di testo
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (3x⁴ - x²) / (6x⁴ + 12)
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Domanda del libro di testo
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 1 ln x / (4x - x² - 3)
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Domanda del libro di testo
Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = (x⁴/2) - 3x² + 4x + 1
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Domanda del libro di testo
Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = 3x/(x² + 3)
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Domanda del libro di testo
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = 3x/(x² - 1)
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