Domanda del libro di testo
Linear approximation Find the linear approximation to the following functions at the given point a.
f(x) = 4x² + x; a = 1
292
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 4.7.46
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (ln(3x + 5eˣ)) / (ln(7x + 3e²ˣ)
Guida verificata passo dopo passo1
Identify the form of the limit as x approaches infinity. Both the numerator and the denominator are logarithmic functions, which tend to infinity as x approaches infinity. This suggests an indeterminate form of type ∞/∞, making l'Hôpital's Rule applicable.
Apply l'Hôpital's Rule, which states that for limits of the form ∞/∞, the limit of the ratio of the derivatives of the numerator and the denominator can be taken. Differentiate the numerator: d/dx[ln(3x + 5e^x)] = (3 + 5e^x) / (3x + 5e^x).
Differentiate the denominator: d/dx[ln(7x + 3e^(2x))] = (7 + 6xe^(2x)) / (7x + 3e^(2x)).
Substitute the derivatives back into the limit expression: lim_x→∞ [(3 + 5e^x) / (3x + 5e^x)] / [(7 + 6xe^(2x)) / (7x + 3e^(2x))].
Simplify the expression by dividing the numerators and denominators, and evaluate the limit as x approaches infinity. Consider the dominant terms in the expressions to determine the behavior of 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:
8mConcetti 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 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.
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 ∞/∞. It states that if the limit of the ratio of two functions yields an indeterminate form, the limit of their derivatives can be taken instead. This rule simplifies the process of finding limits, especially when dealing with logarithmic or exponential functions.
Video consigliato:
5:50Power Rules
Natural Logarithm
The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a key function in calculus, particularly in growth and decay problems, and is often used in conjunction with limits and derivatives. Understanding the properties of logarithms, such as their behavior at infinity, is essential for evaluating limits involving logarithmic expressions.
Video consigliato:
05:18Derivative of the Natural Logarithmic Function
Pratica correlata
Domanda del libro di testo
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 (sin x - x) / 7x³
200
views
Domanda del libro di testo
Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.
v(t) = 2t + 4; s(0) = 0
74
views
Domanda del libro di testo
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (eˣ⁺²) dx
98
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) = tan x/2 on (-π,π)
179
views
Domanda del libro di testo
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
ƒ(x) = eˣ
79
views