Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, find the average rate of change of each function on the interval specified. \(f(x)=x^{2}\) on [1,5]

Step-by-Step Solution

Answer

The average rate of change is 6.

Step 1: Identify the Function and Interval

The function given is \( f(x) = x^2 \). We need to find the average rate of change on the interval [1, 5].

Step 2: Apply the Average Rate of Change Formula

The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is given by the formula: \( \frac{f(b) - f(a)}{b - a} \). Here, \( a = 1 \) and \( b = 5 \).

Step 3: Evaluate the Function at the Endpoints

Compute \( f(1) \) and \( f(5) \): - \( f(1) = 1^2 = 1 \) - \( f(5) = 5^2 = 25 \)

Step 4: Substitute Values into the Formula

Use the values computed from the previous step in the average rate of change formula:\[\frac{f(5) - f(1)}{5 - 1} = \frac{25 - 1}{4}\]

Step 5: Perform the Calculation

Calculate the result from the substitution: \[\frac{25 - 1}{4} = \frac{24}{4} = 6\]

Step 6: Conclusion

The average rate of change of the function \( f(x) = x^2 \) on the interval [1, 5] is 6.

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. This typically involves substituting the input into the function's equation and performing the necessary calculations. In the original exercise, we were tasked with finding the average rate of change of the function \( f(x) = x^2 \) over the interval [1, 5]. To do this, we needed to evaluate the function at the endpoints of the interval:

• Calculate \( f(1) \): Substitute 1 into the function equation: \( f(1) = 1^2 = 1 \).
• Calculate \( f(5) \): Substitute 5 into the function equation: \( f(5) = 5^2 = 25 \).

By finding \( f(1) \) and \( f(5) \), we obtain the necessary values to compute the average rate of change. Understanding function evaluation helps us know exactly how a function behaves at specific points.

Interval Notation

Interval notation is a way of writing subsets of the real number line, often used to describe the domain or range in mathematics. It provides a concise way to specify a range of values between two endpoints. In our exercise, we were given the interval [1, 5], which includes all numbers between 1 and 5, inclusive. Here's how it works:

• "[" and "]": These brackets indicate that the endpoints are included in the interval. Thus, both 1 and 5 are part of the set.
• "(" and ")": If used, these would indicate that the endpoints are not included.

Understanding interval notation is essential when we're dealing with functions, as it informs us precisely which values are considered in a given calculation.

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of a variable, with each term having a coefficient. They form an essential class of functions in mathematics, due to their simple construction and wide applicability. A polynomial function like \( f(x) = x^2 \) consists of:

• Terms: In our case, \( f(x) = x^2 \) has only one term, \( x^2 \).
• Degree: The highest power of the variable in the polynomial, which is 2.
• Coefficients: The number preceding each term. Here the coefficient is 1 for \( x^2 \).

Polynomial functions like this one are generally easy to evaluate and work with, which makes them a common choice in problems involving intervals and rate of change. Recognizing and understanding polynomial functions allows us to predict their behavior across different ranges, which is crucial when calculating things like the average rate of change.