Skip to main content
Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.7.5a
Capitolo 4, Problema 4.7.5a

Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
1 / x²

Guida verificata passo dopo passo
1
Recognize that the function given is \(\frac{1}{x^2}\), which can be rewritten as \(x^{-2}\) to make it easier to apply the power rule for antiderivatives.
Recall the power rule for antiderivatives: for any function \(x^n\) where \(n \neq -1\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
Apply the power rule to \(x^{-2}\) by increasing the exponent by 1: \(-2 + 1 = -1\), so the antiderivative will be \(\frac{x^{-1}}{-1} + C\).
Simplify the expression to get the antiderivative in a more standard form: \(-x^{-1} + C\), which can also be written as \(-\frac{1}{x} + C\).
Verify your answer by differentiating \(-\frac{1}{x} + C\) and checking that the derivative is \(\frac{1}{x^2}\).

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:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Antiderivatives (Indefinite Integrals)

An antiderivative of a function is another function whose derivative equals the original function. It represents the reverse process of differentiation and is expressed as an indefinite integral with a constant of integration, C, since differentiation loses constant terms.
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals

Power Rule for Integration

The power rule for integration states that for any real number n ≠ -1, the integral of x^n dx is (x^(n+1)) / (n+1) + C. This rule is essential for finding antiderivatives of polynomial and power functions like 1/x², which can be rewritten as x^(-2).
Video consigliato:
Percorso guidato
04:04
Power Rule for Indefinite Integrals

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the antiderivative and helps identify any mistakes in the integration process.
Video consigliato:
Percorso guidato
05:53
Finding Differentials
Pratica correlata
Domanda del libro di testo

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = |x³ − 9x|.


a. Does f'(0) exist?

167
views
Domanda del libro di testo

52. Two masses hanging side by side from springs have positions s_1 = 2 sin t and s_2 = sin 2t,

respectively.

a. At what times in the interval 0 < t do the masses pass each other? (Hint: sin 2t = 2 sint cost.)

174
views
Domanda del libro di testo

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = x / 2 − 2sin (x/2), 0 ≤ x ≤ 2π

180
views
Domanda del libro di testo

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = sin 2x, 0 ≤ x ≤ π

165
views
Domanda del libro di testo

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


k(x) = x³ + 3x² + 3x + 1, −∞ < x ≤ 0

210
views
Domanda del libro di testo

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

sec²x

31
views