Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. \(f(x)=2 x^{2}+1\) on \([x, x+h]\)

Step-by-Step Solution

Answer

The average rate of change is \(4x + 2h\).

Step 1: Understand 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 \( \frac{f(b) - f(a)}{b - a} \). In this question, the interval is \([x, x+h]\), so \(a = x\) and \(b = x + h\).

Step 2: Calculate \(f(x)\) and \(f(x + h)\)

First, calculate \(f(x)\) using the function definition: \(f(x) = 2x^2 + 1\). Next, calculate \(f(x + h)\): substitute \(x + h\) into the function, yielding \(f(x + h) = 2(x + h)^2 + 1\).

Step 3: Expand \(f(x + h)\)

Expand \(f(x + h) = 2(x + h)^2 + 1\):\[2(x + h)^2 + 1 = 2(x^2 + 2xh + h^2) + 1 = 2x^2 + 4xh + 2h^2 + 1 \].

Step 4: Apply the Average Rate of Change Formula

Substitute \(f(x + h)\) and \(f(x)\) back into the average rate of change formula:\[\frac{f(x + h) - f(x)}{h} = \frac{(2x^2 + 4xh + 2h^2 + 1) - (2x^2 + 1)}{h}\].

Step 5: Simplify the Expression

Simplify the expression:\[\frac{2x^2 + 4xh + 2h^2 + 1 - 2x^2 - 1}{h} = \frac{4xh + 2h^2}{h} = \frac{h(4x + 2h)}{h} = 4x + 2h\].

Step 6: Conclusion

The average rate of change of the function \( f(x) = 2x^2 + 1 \) on the interval \([x, x+h]\) is \(4x + 2h\).

Key Concepts

Quadratic Functions

Quadratic functions are fundamental in mathematics and appear in various applications, particularly in physics and engineering. A quadratic function is typically in the form of \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In the given problem, the function \( f(x) = 2x^2 + 1 \) is a quadratic function with \( a = 2 \), \( b = 0 \), and \( c = 1 \). This specific function has a parabolic graph opening upwards due to the positive \( a \) value.

Key properties of quadratic functions include:

• The graph is a parabola, which may open upwards or downwards depending on the sign of \( a \).
• The vertex of the parabola provides the function's maximum or minimum value, critical in optimization problems.
• The axis of symmetry is a vertical line that passes through the vertex, helping in finding peaks and troughs in real-world applications.

Understanding these characteristics is crucial when working with quadratic functions, including analyzing their rates of change over specific intervals.

Difference Quotient

The difference quotient is a crucial concept used to find the average rate of change of a function over a given interval, analogous to finding the slope of a line connecting two points on a graph. For a function \( f(x) \) over the interval \([x, x+h]\), the difference quotient is defined as:
\[\frac{f(x+h) - f(x)}{h}\]
The difference quotient mimics the derivative's role in calculus but is not precisely the derivative unless the limit as \( h \) approaches zero is considered. However, calculating this quotient over a non-zero interval provides valuable insight into how the function behaves between two points.
When calculating the difference quotient in the given exercise, these steps are crucial:

• Substitute \( x+h \) into the function \( f(x) \) to find \( f(x+h) \).
• Find \( f(x+h) - f(x) \).
• Divide by \( h \) to complete the quotient.

The outcome, \( 4x + 2h \), represents the average rate of change indicating how the function changes over \([x, x+h]\).

Function Intervals

Function intervals are periods or ranges over which you examine how a function behaves. In this exercise, the function \( f(x) = 2x^2 + 1 \) is evaluated over the interval \([x, x+h]\). Such intervals help in understanding the function's growth or decline between two points.

Exploring intervals consists of several aspects:

• Determining whether the function is increasing or decreasing over the interval by examining the sign of the average rate of change.
• Analyzing endpoints to understand behavior at specific points.
• Predicting future behavior or trends by observing the rate of change over different intervals.

Specifically, for a quadratic function like \( f(x) = 2x^2 + 1 \), examining intervals can show how quickly the function grows since it is symmetric around its vertex. In practical applications, such insights can support decisions in fields like physics, where trajectories and optimizations are examined.