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]

Verified step by step guidance

1. Identify the function g(t) = 2 + cos(t) and the interval [0, π].
2. Recall the formula for the average rate of change of a function g(t) over an interval [a, b], which is given by: \( \frac{g(b) - g(a)}{b - a} \).
3. Substitute the endpoints of the interval into the function: calculate g(0) and g(π).
4. Evaluate g(0) = 2 + cos(0) and g(π) = 2 + cos(π).
5. Substitute g(0) and g(π) into the average rate of change formula: \( \frac{g(\pi) - g(0)}{\pi - 0} \) and simplify the expression.

Key Concepts

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.

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.

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.