Average Rates of Change

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

g(t)=2+cos t

b. [0,π]

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Identify the function g(t) = 2 + cos(t) and the interval [0, π].

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

Substitute the endpoints of the interval into the function: calculate g(0) and g(π).

Evaluate g(0) = 2 + cos(0) and g(π) = 2 + cos(π).

Substitute g(0) and g(π) into the average rate of change formula: \( \frac{g(\pi) - g(0)}{\pi - 0} \) and simplify the expression.

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Average Rate of Change

The average rate of change of a function over an interval 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), where [a, b] is the interval. This concept helps in understanding how a function behaves on average over a specified range.

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Average Value of a Function

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. For the function g(t) = 2 + cos(t), evaluating it at the endpoints of the interval [0, π] means calculating g(0) and g(π). This step is crucial for finding the average rate of change, as it provides the necessary values to apply the average rate of change formula.

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Evaluating Composed Functions

Trigonometric Functions

Trigonometric functions, such as cosine, are periodic functions that relate angles to ratios of sides in right triangles. The function g(t) = 2 + cos(t) combines a constant and a cosine function, which oscillates between -1 and 1. Understanding the behavior of cosine over the interval [0, π] is essential for accurately calculating the average rate of change, as it influences the function's values at the endpoints.

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