Domanda del libro di testo
Evaluate the integrals in Exercises 97–110.
103. ∫₁⁴ (ln 2 · log₂x / x) dx
16
views
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.1.27
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x³ + 1
Guida verificata passo dopo passo1
Start by writing the function given: \(f(x) = x^3 + 1\).
To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = x^3 + 1\).
Swap the roles of \(x\) and \(y\) to find the inverse: \(x = y^3 + 1\).
Solve this equation for \(y\) to express \(y\) in terms of \(x\): subtract 1 from both sides to get \(x - 1 = y^3\), then take the cube root to find \(y = \sqrt[3]{x - 1}\).
Identify the domain and range of \(f^{-1}(x)\). Since the original function \(f(x) = x^3 + 1\) has domain \((-\infty, \infty)\) and range \((-\infty, \infty)\), the inverse function will have domain \((-\infty, \infty)\) and range \((-\infty, \infty)\) as well.
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:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Inverse Functions
An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. Finding the inverse involves solving y = f(x) for x in terms of y.
Video consigliato:
4:49
Inverse Cosine
Domain and Range of Functions and Their Inverses
The domain of a function is the set of all possible inputs, while the range is the set of all possible outputs. For an inverse function, the domain and range swap roles compared to the original function. Identifying these sets ensures the inverse is properly defined.
Video consigliato:
Percorso guidato
5:10Finding the Domain and Range of a Graph
Verification of Inverse Functions
To confirm two functions are inverses, compose them in both orders: f(f⁻¹(x)) and f⁻¹(f(x)). Both compositions should simplify to x, the identity function. This check validates the correctness of the inverse function found.
Video consigliato:
4:49Inverse Cosine
Pratica correlata
Domanda del libro di testo
Evaluate the integrals in Exercises 33–54.
49. ∫ e^(sec πt) sec πt tan πt dt
23
views
Domanda del libro di testo
15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.
20
views
Domanda del libro di testo
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
22. lim (x → 1) (x - 1) / (ln x - sin πx)
30
views
Domanda del libro di testo
7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.
a. lim(x→∞)A(t)
47
views
Domanda del libro di testo
20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.
b. Find the centroid of the region.
31
views