Problem 2 - Thomas Calculus

Question

Find the average rate of change of the function over the given interval or intervals. \(g(x)=x^{2}-2 x\) a. \([1,3] \quad\) b. \([-2,4]\)

Step-by-Step Solution

Answer

a. 2; b. 0

Step 1: Understand the Average Rate of Change Formula

The average rate of change of a function over an interval \( [a, b] \) is calculated using the formula: \[ \text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a} \]where \( g(x) \) is the function and \( a \) and \( b \) are the endpoints of the interval.

Step 2: Apply the Formula to Interval [1, 3]

For interval [1, 3]: First, find \( g(1) \) and \( g(3)\).\( g(1) = 1^2 - 2(1) = 1 - 2 = -1 \)\( g(3) = 3^2 - 2(3) = 9 - 6 = 3 \)Now, substitute these into the formula:\[ \text{Average Rate of Change} = \frac{g(3) - g(1)}{3 - 1} = \frac{3 - (-1)}{2} = \frac{4}{2} = 2 \]

Step 3: Apply the Formula to Interval [-2, 4]

For interval [-2, 4]: First, find \( g(-2) \) and \( g(4) \).\( g(-2) = (-2)^2 - 2(-2) = 4 + 4 = 8 \)\( g(4) = 4^2 - 2(4) = 16 - 8 = 8 \)Now, substitute these into the formula:\[ \text{Average Rate of Change} = \frac{g(4) - g(-2)}{4 - (-2)} = \frac{8 - 8}{6} = \frac{0}{6} = 0 \]

Key Concepts

Quadratic Functions

A quadratic function is one of the simplest forms of polynomial functions you will come across in algebra. It follows the general form:
\[ y = ax^2 + bx + c \] where:

• \(a\), \(b\), and \(c\) are constants,
• \(x\) is the variable.

The function features a parabola when plotted on a graph. The parabola opens upwards if \(a\) is positive and downwards if \(a\) is negative.
This shape is symmetric around a vertical line called the axis of symmetry. The formula for the axis of symmetry is:
\[x = -\frac{b}{2a}\]Quadratic functions can model a wide range of real-world scenarios, such as projectile motion. Understanding these functions involves recognizing their form and analyzing the movement around their vertex—the highest or lowest point on the graph.

Interval Notation

Interval notation is a concise way of describing a set of numbers along a number line.
Intervals are sets that include all numbers between two endpoints.

• A closed interval [a, b] includes the endpoints \(a\) and \(b\).
• An open interval (a, b) excludes both endpoints.
• Semi-open intervals, like \([a, b)\) or \((a, b]\), mix these notations, including one endpoint and excluding the other.

This notation quickly communicates which parts of the graph or function are being examined. For instance, evaluating a function on the interval
\([1, 3]\) means you are looking at the behavior of the function from \(x = 1\) to \(x = 3\), including both values. This can be especially important for functions like quadratic equations that vary quite a bit across different intervals.

Rate of Change Formula

The average rate of change of a function is a key concept, helping understand how the function behaves over an interval.
It is calculated as:
\[ \text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a} \]
where:

• \(g(x)\) is the function given,
• \(a\) and \(b\) are the beginning and ending values of the interval.

This formula offers insight into how much a function increases or decreases on average between two points.
For example, using the function \(g(x) = x^2 - 2x\) over the interval \([1, 3]\), you plug in the values into the formula, finding that the average rate of change is \(2\).
This means that, on average, the function increases by \(2\) for every unit increase in \(x\) between \(1\) and \(3\). Understanding this formula helps predict trends and is widely used in calculus and data analysis.