Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the average value of the function on the given interval. $$ f(x)=x+|x| ; \quad[-3,2] $$

Step-by-Step Solution

Answer

The average value is \( \frac{4}{5} \).

Step 1: Understanding the Function

The function given is \( f(x) = x + |x| \). This function behaves differently for positive and negative values of \( x \). For \( x \geq 0 \), \( |x| = x \), hence \( f(x) = 2x \). For \( x < 0 \), \( |x| = -x \), hence \( f(x) = 0 \).

Step 2: Dividing the Interval

The given interval is \([-3, 2]\). We'll divide this into two sub-intervals: \([-3, 0]\), where \( x < 0 \) and \([0, 2]\), where \( x \geq 0 \).

Step 3: Integrating Over [-3, 0]

On \([-3, 0]\), \( f(x) = 0 \). The integral of 0 over any interval is 0. Thus, \[ \int_{-3}^{0} 0\, dx = 0. \]

Step 4: Integrating Over [0, 2]

On \([0, 2]\), \( f(x) = 2x \). We integrate this function over \([0, 2]\):\[ \int_{0}^{2} 2x \, dx = \left[ x^2 \right]_{0}^{2} = 2^2 - 0^2 = 4. \]

Step 5: Calculating the Total Integral

Adding the integrals from the sub-intervals, the total integral over \([-3, 2]\) is:\[ 0 + 4 = 4. \]

Step 6: Determining the Average Value

The average value of \( f(x) \) over \([-3, 2]\) is given by the formula: \[ \frac{1}{b-a} \int_{a}^{b} f(x)\, dx. \] For our function, this is:\[ \frac{1}{2 - (-3)} \, (4) = \frac{1}{5} \, imes 4 = \frac{4}{5}. \]

Key Concepts

Piecewise Functions

Piecewise functions are an exciting area in mathematics, as they allow us to describe a function that behaves differently over distinct intervals. Think of them as a chameleon, changing according to the situation. For our function,
\( f(x) = x + |x| \), the behavior changes based on the value of \( x \).

• For \( x \geq 0 \): The absolute value \( |x| = x \), simplifying our function to \( f(x) = 2x \).
• For \( x < 0 \): The absolute value \( |x| = -x \), giving us \( f(x) = 0 \).

Understanding piecewise functions can be tricky at first. But remember, each "piece" responds to a condition. In our example, we essentially have two "pieces" based on whether \( x \) is non-negative or negative.
This is key when analyzing the overall function's behavior across an interval.
In our case, this helped us clearly determine which segment of the function to integrate over different sub-intervals.

Integration

Integration is a powerful tool in calculus that helps us determine areas under curves or, in our case, the overall accumulation of a function's values over a specific interval. Our goal here is to integrate the function \( f(x) \) over a piecewise-defined region.
Steps for Integration:

• Identify the segments of the function to integrate, such as \( f(x) = 0 \) or \( f(x) = 2x \) in their respective intervals.
• Apply the definite integral formula over each piece's intervals to find their areas. For \([-3, 0]\), since \( f(x)=0 \), its integral is 0. For \([0, 2]\), we compute \( \int_{0}^{2} 2x \, dx = [x^2]_{0}^{2} = 4 \).
• Add up the results from each integral to form a comprehensive picture of the function’s behavior across the whole interval.

Understanding integration involves thinking of merging pieces of a function's behavior into a singular expression of total change or growth over an interval.

Intervals in Calculus

Intervals in calculus are crucial, as they define where specific operations occur, like integration or differentiation. Recognizing which sub-intervals to use in piecewise functions can guide us in computations.
Main Tips for Handling Intervals:

• Carefully account for where changes in the function occur, such as \( x = 0 \) in our problem.
• Break the entire interval, like \([-3, 2]\), into specific parts that match the conditions of a piecewise function.
• Determine the start and endpoint for each piece to accurately apply rules of calculus such as integration or differentiation.

Understanding and using intervals correctly allow us to work efficiently with piecewise functions, ensuring calculations reflect the function's true behavior over its entire domain.

Absolute Value Functions

The concept of absolute value functions is foundational in mathematics, closely tied to real-world applications like determining distance.
Absolute value \( |x| \) represents the magnitude of \( x \) without considering its sign, effectively making all results non-negative.
Characteristics of Absolute Value Functions:

• For positive \( x \), \( |x| = x \).
• For negative \( x \), \( |x| = -x \), flipping the value to positive.
• This property makes the function inherently non-linear, critical in forming piecewise functions like \( f(x) = x + |x| \).

In our example, the absolute value transforms \( x \) based on its sign, drastically affecting the overall expression \( f(x) = x + |x| \). Understanding absolute value gets challenging when integrated with linear functions, yet it is an invaluable tool for mathematical modeling.