Ch. 3 - Derivatives

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

Hass 15th Edition

Ch. 3 - Derivatives

Problema 3.2.47a

Capitolo 3, Problema 3.2.47a

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

a. differentiable?

Give reasons for your answers.

Guida verificata passo dopo passo

Examine the graph to identify points where the function is continuous. A function is continuous at a point if there is no break, jump, or hole at that point.

Check for differentiability at each point where the function is continuous. A function is differentiable at a point if it has a defined tangent line, meaning the graph is smooth and not sharp or vertical at that point.

Identify any points where the graph has sharp corners or cusps, as these are typically points where the function is not differentiable.

Look for vertical tangents or discontinuities, as these indicate points where the function is not differentiable.

Based on the graph, determine the points within the interval [-3, 3] where the function is both continuous and smooth, indicating differentiability.

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:

Concetti chiave

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

Differentiability

A function is differentiable at a point if it has a defined derivative at that point, meaning the function's graph has a tangent line that is well-defined. This requires the function to be continuous at that point and for the left-hand and right-hand limits of the derivative to exist and be equal. Points where the graph has sharp corners, vertical tangents, or discontinuities indicate non-differentiability.

Video consigliato:

Percorso guidato

05:53

Finding Differentials

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph at that point. For a function to be differentiable, it must first be continuous; however, continuity alone does not guarantee differentiability.

Video consigliato:

05:34

Intro to Continuity

Closed Interval

A closed interval, denoted as [a, b], includes all numbers between a and b, including the endpoints a and b themselves. In the context of differentiability and continuity, analyzing a function over a closed interval allows for the examination of its behavior at the endpoints and within the interval, ensuring that all points are considered when determining differentiability and continuity.

Video consigliato:

Percorso guidato

11:11

Improper Integrals: Infinite Intervals

Pratica correlata

Domanda del libro di testo

Find y'' if:

a. y = csc x

299

views

Domanda del libro di testo

Interpreting Derivative Values

Effectiveness of a drug On a scale from 0 to 1, the effectiveness E of a pain-killing drug t hours after entering the bloodstream is displayed in the accompanying figure.

" style="" width="240">

a. At what times does the effectiveness appear to be increasing? What is true about the derivative at those times?

189

views

Domanda del libro di testo

Interpreting Derivative Values

Growth of yeast cells In a controlled laboratory experiment, yeast cells are grown in an automated cell culture system that counts the number P of cells present at hourly intervals. The number after t hours is shown in the accompanying figure.

" style="" width="250">

a. Explain what is meant by the derivative P'(5). What are its units?

181

views

Domanda del libro di testo

If the original 24 m edge length x of a cube decreases at the rate of 5 m/min, when x = 3 m at what rate does the cube’s

a. surface area change?

191

views

Domanda del libro di testo

Find y⁽⁴⁾ = d⁴y/dx⁴ if:

a. y = −2 sin x