Average Rates of Change

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

g(x)=x²−2x

a. [1, 3]

검증된 단계별 안내

Identify the function given: \( g(x) = x^2 - 2x \).

Recall the formula for the average rate of change of a function \( g(x) \) over an interval \([a, b]\): \( \frac{g(b) - g(a)}{b - a} \).

Substitute the interval \([1, 3]\) into the formula, where \( a = 1 \) and \( b = 3 \).

Calculate \( g(1) \) by substituting \( x = 1 \) into the function: \( g(1) = 1^2 - 2 \times 1 \).

Calculate \( g(3) \) by substituting \( x = 3 \) into the function: \( g(3) = 3^2 - 2 \times 3 \).

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 a specific value into a function to determine its output. For the function g(x) = x² - 2x, evaluating it at points 1 and 3 means calculating g(1) and g(3). This step is essential for finding the values needed to compute the average rate of change over the specified interval.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c. The function g(x) = x² - 2x is a quadratic function where a = 1, b = -2, and c = 0. Understanding the properties of quadratic functions, such as their parabolas' shape and vertex, is important for analyzing their behavior over intervals.

Average Rates of ChangeIn Exercises 1–6, find the average rate of change of the function over the given interval or intervals.g(x)=x²−2xa. [1, 3]