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104–107. Functions from derivatives Find the function f with the following properties.
h'(x) = (x⁴ -2) /(1 + x²) ; h (1) = -(2/3)
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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 4.R.63
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 (3 sin² 2Θ) / Θ²
Guida verificata passo dopo passo1
First, identify the form of the limit as Θ approaches 0. The expression (3 sin² 2Θ) / Θ² is in the indeterminate form 0/0, which suggests that l'Hôpital's Rule can be applied.
Apply l'Hôpital's Rule, which states that if the limit of f(Θ)/g(Θ) as Θ approaches a value results in an indeterminate form, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately.
Differentiate the numerator: The derivative of 3 sin² 2Θ with respect to Θ involves using the chain rule. First, differentiate sin² 2Θ to get 2 sin 2Θ * cos 2Θ, and then multiply by the derivative of 2Θ, which is 2. Therefore, the derivative of the numerator is 12 sin 2Θ * cos 2Θ.
Differentiate the denominator: The derivative of Θ² with respect to Θ is straightforward, which is 2Θ.
Now, evaluate the limit of the new expression (12 sin 2Θ * cos 2Θ) / (2Θ) as Θ approaches 0. Simplify the expression and check if further application of l'Hôpital's Rule is needed or if the limit can be directly evaluated.
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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 near points of interest, including points where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
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One-Sided Limits
L'Hôpital's Rule
L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It 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 rule simplifies the process of finding limits in complex functions.
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5:50Power Rules
Trigonometric Functions
Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In calculus, they are often involved in limits and derivatives due to their unique properties, such as periodicity and boundedness. Understanding their behavior, especially near critical points like 0, is essential for evaluating limits involving trigonometric expressions.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Pratica correlata
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Change in elevation The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 - 16t², where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 ≤ t ≤ 5.7 (recall that Δh ≈ h' (a) Δt).
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Domanda del libro di testo
90–103. Indefinite integrals Determine the following indefinite integrals.
∫(1 + 3 cosΘ) dΘ
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Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x sin⁻¹ x on [-1, 1]
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60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 2Θ cot 3Θ
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Domanda del libro di testo
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (12/x)dx
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