Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.2.71b

Capitolo 7, Problema 7.2.71b

71. Locate and identify the absolute extreme values of cos(ln x) on [1/2, 2]

Guida verificata passo dopo passo

First, understand that the function to analyze is \(f(x) = \cos(\ln x)\) on the interval \(\left[ \frac{1}{2}, 2 \right]\). We want to find the absolute maximum and minimum values on this closed interval.

Step 1: Find the derivative of the function to locate critical points. Using the chain rule, the derivative is \(f'(x) = -\sin(\ln x) \cdot \frac{1}{x}\).

Step 2: Set the derivative equal to zero to find critical points inside the interval: \(-\sin(\ln x) \cdot \frac{1}{x} = 0\). Since \(\frac{1}{x} \neq 0\) for \(x > 0\), this reduces to \(\sin(\ln x) = 0\).

Step 3: Solve \(\sin(\ln x) = 0\) for \(x\) in \(\left[ \frac{1}{2}, 2 \right]\). Recall that \(\sin t = 0\) when \(t = k\pi\) for any integer \(k\). So, set \(\ln x = k\pi\) and solve for \(x = e^{k\pi}\), then determine which values lie in the interval.

Step 4: Evaluate the function \(f(x) = \cos(\ln x)\) at all critical points found in the interval and also at the endpoints \(x = \frac{1}{2}\) and \(x = 2\). Compare these values to identify the absolute maximum and minimum.

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:

12m

Concetti chiave

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

Absolute Extreme Values

Absolute extreme values are the highest or lowest values a function attains on a given closed interval. These include absolute maxima and minima, which can occur at critical points or endpoints of the interval.

Video consigliato:

05:12

Finding Global Extrema (Extreme Value Theorem)

Critical Points and Derivatives

Critical points occur where the derivative of a function is zero or undefined. Finding these points helps identify potential locations of extreme values within the interval.

Video consigliato:

04:50

Critical Points

Chain Rule and Differentiation of Composite Functions

The chain rule is used to differentiate composite functions like cos(ln x). It involves differentiating the outer function and multiplying by the derivative of the inner function, essential for finding the derivative of cos(ln x).

Video consigliato:

03:58

The Chain Rule for 3+ Functions

Pratica correlata

Domanda del libro di testo

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

69. ∫(from 5/4 to 2)dx/(1-x²)

views

Domanda del libro di testo

31. The incidence of a disease (Continuation of Example 4.) Suppose that in any given year the number of cases can be reduced by 25% instead of 20%.

b. How long will it take to eradicate the disease—that is, reduce the number of cases to less than 1?

views

Domanda del libro di testo

b. Find the center of mass if, instead of being constant, the density function is δ(x)=4/√x.

views

Domanda del libro di testo

2. Express the following logarithms in terms of ln 5 and ln 7.

b. ln 9.8

views

Domanda del libro di testo

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.

67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3

views

Domanda del libro di testo

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

67. ∫(from 0 to 2√3)dx/√(4+x²)

views