Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.7.26
Capitolo 4, Problema 4.7.26

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ 1 (4 tan⁻¹ x- π) / (x-1)

Guida verificata passo dopo passo
1
First, identify the form of the limit as x approaches 1. Substitute x = 1 into the expression: (4 tan⁻¹(1) - π) / (1 - 1). This results 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 results in an indeterminate form like 0/0, you can take the derivative of the numerator and the derivative of the denominator separately and then evaluate the limit again.
Differentiate the numerator: The derivative of 4 tan⁻¹(x) with respect to x is 4/(1 + x²). The derivative of the constant π is 0.
Differentiate the denominator: The derivative of x - 1 with respect to x is 1.
Re-evaluate the limit using the derivatives: lim x→1 (4/(1 + x²)) / 1. Substitute x = 1 into the new expression to find the limit.

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:
4m

Concetti 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 where the function may not be defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
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) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, and then re-evaluating the limit.
Video consigliato:
5:50
Power Rules

Inverse Tangent Function

The inverse tangent function, denoted as tan⁻¹(x) or arctan(x), is the function that returns the angle whose tangent is x. It is important in calculus for evaluating limits and derivatives involving trigonometric functions. Understanding its behavior, particularly near specific values like 1, is essential for solving limit problems involving arctan.
Video consigliato:
3:17
Inverse Tangent
Pratica correlata
Domanda del libro di testo

Suppose f is differentiable on (-∞,∞), f(5.99) = 7, and f(6) = 7.002. Use linear approximation to estimate the value of f'(6).

253
views
Domanda del libro di testo

{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = 2x cos x on [0,2]

211
views
Domanda del libro di testo

Use linear approximation to estimate f (3.85) given that f(4) = 3 and f'(4) = 2.

265
views
Domanda del libro di testo

Graphing functions Use the guidelines of this section to make a complete graph of f.


f(x) = x³ - 6x² - 135x

248
views
Domanda del libro di testo

Graphing functions Use the guidelines of this section to make a complete graph of f.


f(x) = ln (x² + 1)

254
views
Domanda del libro di testo

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (1/2y)dy

54
views