Textbook Question
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
f(x)=x³+1
a. [2, 3]
1. Limits and Continuity
Introduction to Limits
2:09 minutesProblem 2.1.6
Textbook Question
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 guidance1
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).
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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.
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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.
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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.
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