Domanda del libro di testo
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
e. On what intervals (approximately) is f concave up?
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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.61
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_t→0 (1 - cos 6t) / 2t
Guida verificata passo dopo passo1
First, identify the form of the limit as t approaches 0. The expression (1 - cos(6t)) / 2t results in the indeterminate form 0/0, which is suitable for applying l'Hôpital's Rule.
Apply l'Hôpital's Rule, which states that if the limit of f(t)/g(t) as t 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 1 - cos(6t) with respect to t is 6sin(6t).
Differentiate the denominator: The derivative of 2t with respect to t is 2.
Re-evaluate the limit using the derivatives: The limit now becomes lim_t→0 (6sin(6t)) / 2. Simplify the expression and evaluate the limit as t approaches 0.
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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 of discontinuity or where the function is not 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 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) as x approaches a point yields an indeterminate form, then 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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5:50Power Rules
Trigonometric Limits
Trigonometric limits involve evaluating limits that include trigonometric functions, such as sine and cosine. A common limit is lim_x→0 (1 - cos(x))/x^2, which can be evaluated using Taylor series or L'Hôpital's Rule. Understanding the behavior of trigonometric functions near specific points is essential for solving problems involving limits in calculus.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ (1 - (3/x))ˣ
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Domanda del libro di testo
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
f. On what intervals (approximately) is f concave down?
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Domanda del libro di testo
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
x¹⸍² and x¹⸍³
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Domanda del libro di testo
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
c. Give the approximate coordinates of the inflection point(s) of f.
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Domanda del libro di testo
24–34. Curve sketching 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) = 10x² / (x² + 3)
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