Domanda del libro di testo
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x³ - 3x² on [-1, 3]
214
views
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 4.7.55
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 csc 6x sin 7x
Guida verificata passo dopo passo1
Identify the form of the limit as x approaches 0. The expression csc(6x) is equivalent to 1/sin(6x), so the limit becomes lim_{x→0} (sin(7x) / sin(6x)). This is an indeterminate form 0/0, which suggests the use of l'Hôpital's Rule.
Apply l'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, then lim_{x→c} f(x)/g(x) = lim_{x→c} f'(x)/g'(x), provided the latter limit exists.
Differentiate the numerator and the denominator separately. The derivative of sin(7x) with respect to x is 7cos(7x), and the derivative of sin(6x) with respect to x is 6cos(6x).
Substitute these derivatives back into the limit: lim_{x→0} (7cos(7x) / 6cos(6x)).
Evaluate the new limit as x approaches 0. Since cos(0) = 1, the limit simplifies to 7/6.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
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 the continuity and differentiability of functions.
Video consigliato:
05:50
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) 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. This process can be repeated if the result remains indeterminate.
Video consigliato:
5:50Power Rules
Cosecant and Sine Functions
The cosecant function, csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). Understanding the behavior of these trigonometric functions, especially near critical points like zero, is essential for evaluating limits involving them. In the given limit, the interaction between csc(6x) and sin(7x) as x approaches zero is key to finding the limit's value.
Video consigliato:
Percorso guidato
6:22Graphs of Secant and Cosecant Functions
Pratica correlata
Domanda del libro di testo
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x² - 10 on [-2, 3]
246
views
Domanda del libro di testo
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x² - 2 ln x
235
views
Domanda del libro di testo
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = x √(x-a)
308
views
Domanda del libro di testo
{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.
f(x) = x² - 6; x₀ = 3
254
views
Domanda del libro di testo
Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.
f(x) = 6x² - x³
252
views