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When working with a function like \( g(t) = \frac{t}{t^2 + 2} + 3t \), understanding function values is key to solving calculus problems involving average rates of change.
The notion of function values refers to the output of the function for specific input values, which are determined by substituting numbers into the function.
For example:

• Evaluating at \( t=-1 \), you calculate \( g(-1) \).
• For \( t=1 \), it’s \( g(1) \).
• Similarly, compute \( g(0) \), \( g(2) \), and \( g(1+p) \).

Find these function values by substituting these into \( g(t) \) and simplifying each result. This provides the necessary data to analyze how the function behaves at different points.

An interval defines the range of inputs for which we'll calculate an attribute of the function, like the average rate of change.
Intervals are expressed as pairs of numbers, such as \([-1,1]\), \([0,2]\), or \([1,1+p]\). These numbers indicate the start and end of the interval.
Understanding intervals helps you see over what range you're evaluating changes in the function.

• For \([-1,1]\), we're looking at how the function behaves and changes from \( t=-1 \) to \( t=1 \).
• The interval \([0,2]\) considers the change from \( t=0 \) through \( t=2 \).
• \([1,1+p]\) includes a variable endpoint, allowing exploration of function behavior when the interval end is more flexible.

Grasping these intervals is crucial for solving the calculus problem effectively.

The problem of finding the average rate of change fits within the domain of calculus.
Calculus often deals with how functions change, and here, we're exploring the average rate which tells us how the function changes, on average, over specified intervals.
The average rate of change equation \[\frac{f(b) - f(a)}{b - a}\]deals with two aspects:

• \( f(b) \) and \( f(a) \) are the function values at the endpoints \( b \) and \( a \).
• \( (b - a) \) represents the length of the interval over which these changes are considered.

In the context of the function \( g(t) \), these rates tell us how steep or gentle the function's slope is across these inputs, providing deeper insights into its behavior.

Breaking down the problem into a step by step solution makes it more manageable and easier to understand.
Here’s the approach you take:- **Step 1**: Start by calculating the specific function values \( g(-1) \), \( g(1) \), \( g(0) \), etc., using substitutions. - **Step 2**: Apply these calculated values to the average rate of change formula for each interval. For instance, to find the change over \([-1, 1]\), compute: \[ \frac{g(1) - g(-1)}{1 - (-1)} \] - **Step 3**: Interpret these results to understand how the function changes over each interval. These easy steps allow you to systematically solve calculus problems involving average rates of change, ensuring a clear path from problem to solution.