Skip to main content
Ch. 2 - Limits and Continuity
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. 2 - Limits and ContinuityProblema 2.1.6
Capitolo 2, Problema 2.1.6

Average Rates of Change


In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.


P(θ)=θ³ − 4θ² + 5θ; [1,2]

Guida verificata passo dopo passo
1
Identify the function P(θ) = θ³ − 4θ² + 5θ and the interval [1, 2].
Recall that the average rate of change of a function over an interval [a, b] is given by the formula: (P(b) - P(a)) / (b - a).
Substitute the endpoints of the interval into the function to find P(1) and P(2).
Calculate P(1) by substituting θ = 1 into the function: P(1) = 1³ − 4(1)² + 5(1).
Calculate P(2) by substituting θ = 2 into the function: P(2) = 2³ − 4(2)² + 5(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.

Average Rate of Change

The average rate of change of a function over an interval [a, b] is defined as the change in the function's value divided by the change in the input value. Mathematically, it is expressed as (f(b) - f(a)) / (b - a). This concept is crucial for understanding how a function behaves over a specific range and is often used to analyze the function's growth or decline.
Video consigliato:
Percorso guidato
06:37
Average Value of a Function

Function Evaluation

Function evaluation involves substituting specific values into a function to determine its output. For the function P(θ) = θ³ − 4θ² + 5θ, evaluating it at the endpoints of the interval [1, 2] means calculating P(1) and P(2). This step is essential for finding the average rate of change, as it provides the necessary values to compute the difference in the function's output.
Video consigliato:
Percorso guidato
4:26
Evaluating Composed Functions

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function P(θ) = θ³ − 4θ² + 5θ is a cubic polynomial, which can exhibit various behaviors such as increasing, decreasing, and having local maxima or minima. Understanding the properties of polynomial functions helps in analyzing their rates of change and overall behavior.
Video consigliato:
6:04
Introduction to Polynomial Functions
Pratica correlata
Domanda del libro di testo

Limits of Average Rates of Change


Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.


f(x) = 3x - 4, x = 2

312
views
Domanda del libro di testo

Theory and Examples


Once you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.

323
views
Domanda del libro di testo

Calculating Limits


Find the limits in Exercises 11–22.


limt→6 8(t−5)(t−7)

228
views
Domanda del libro di testo

A function value Show that the function F(x) = ( x − a)²(x − b)² + x takes on the value (a + b)² for some value of x.

257
views
Domanda del libro di testo

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?

<IMAGE>

240
views
Domanda del libro di testo

Slope of a Curve at a Point


In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.


y=√7−x, P(−2,3)

325
views