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The rate of change measures how a function's output changes as its input changes. In simpler terms, it tells us how quickly a function is increasing or decreasing. When you determine the average rate of change over an interval for a function, you're essentially finding the slope of the line that connects the two points on the function at the ends of the interval.
For instance, if a function is represented by the equation \( f(x) = \frac{1}{4} x^3 + 1 \), and you want to find the average rate of change over the interval \([0,2]\), you would use the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Here, \(a = 0\) and \(b = 2\). You calculate the function values at these points, \( f(2) \) and \( f(0) \), and then plug them into the formula to find the average rate of change.
This step is crucial for understanding how the function behaves between the given points.

An interval refers to a continuous range of values, often used to specify the region over which certain calculations are performed. In this case, our interval is \[ [0, 2] \]. This means we're looking at all the points between 0 and 2, inclusive.
Intervals can be open, closed, or half-open, which affects whether the endpoints are included.

• A \[ [a, b] \] is a closed interval where both ends are included.
• A \[ (a, b) \] is an open interval where neither endpoint is included.
• A \[ [a, b) \] or \[ (a, b] \] is a half-open interval that includes only one endpoint.

Here, \[ [0, 2] \] is a closed interval, meaning we include both 0 and 2 in our calculations. This is important for finding the average rate of change, as we need to evaluate the function at both endpoints.

Function evaluation is the process of determining the output of a function for specific input values. In our scenario, we need to evaluate the functions \( f(x) = \frac{1}{4} x^3 + 1 \) and \( g(x) = \frac{3}{4} x^2 \) at particular points.
For example, to calculate the average rate of change of \( f(x) \) over the interval \[ [0, 2] \], we start by finding \( f(2) \) and \( f(0) \):
\[ f(2) = \frac{1}{4} (2)^3 + 1 = 3 \]
\[ f(0) = \frac{1}{4} (0)^3 + 1 = 1 \]
These evaluations allow us to compute the average rate of change, which is then set equal to \( g(c) \) to find the specific value of \( c \) that matches this rate of change within the given interval.
Without accurate function evaluation, we wouldn’t be able to reliably determine the behavior of the function over an interval or find specific values that meet certain criteria.