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]

Verified step by step guidance

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).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

Recommended video:

6:04

Introduction to Polynomial Functions

Related Practice

Textbook Question

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

Textbook Question

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.

Textbook Question

Calculating Limits

Find the limits in Exercises 11–22.

limt→6 8(t−5)(t−7)

views

Textbook Question

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.

views