Problem 9 - Advanced Engineering Mathematics

Question

Give an interval over which \(f_{1}(x)=x^{2}\) and \(f_{2}(x)=x|x|\) are linearly independent. Then give an interval on which \(f_{1}\) and \(f_{2}\) are linearly dependent.

Step-by-Step Solution

Answer

Dependent on \([0, \infty)\), independent on \((-\infty, 0)\).

Step 1: Understand Linear Independence

Two functions are linearly independent over an interval if there are no constants \( c_1 \) and \( c_2 \), not both zero, such that \( c_1 f_1(x) + c_2 f_2(x) = 0 \) for all \( x \) in that interval. Otherwise, they are linearly dependent.

Step 2: Analyze Function Definitions

The functions given are \( f_1(x) = x^2 \) and \( f_2(x) = x|x| \). Note that \( f_2(x) = x|x| = x^2 \) when \( x \geq 0 \) but \( f_2(x) = -x^2 \) when \( x < 0 \).

Step 3: Find Interval of Linear Dependence

For \( x \geq 0 \), both \( f_1(x) = x^2 \) and \( f_2(x) = x^2 \), hence they are clearly linearly dependent. Therefore, an interval of linear dependence is \( [0, \, \infty) \).

Step 4: Find Interval of Linear Independence

For \( x < 0 \), \( f_1(x) = x^2 \) while \( f_2(x) = -x^2 \). Here, it is impossible to write one function as a constant multiple of the other since any linear combination \( c_1 x^2 + c_2 (-x^2) \) can only be zero if \( c_1 = c_2 = 0 \). Therefore, they are linearly independent over \( (-\infty, 0) \).

Key Concepts

Linear Dependence

Linear dependence is a critical concept in understanding how functions relate to one another. Two functions are considered to be linearly dependent if, over a given interval, there exist constants \( c_1 \) and \( c_2 \), not both zero, such that their weighted sum is equal to zero for all values of \( x \) within that interval. In simpler terms:

• If you can express one function as a constant multiple of the other, then they are dependent.
• If not, they are independent.

For example, let's consider the functions \( f_1(x) = x^2 \) and \( f_2(x) = x|x| \). On the interval \([0, \infty)\), both functions can be expressed as \( x^2 \) for all non-negative \( x \). Thus, they are linearly dependent in this region. Linear dependence provides insight into whether functions offer unique information compared to one another.

Functions

A function in mathematics is essentially a relationship that assigns each input to exactly one output. In our case, the functions \( f_1(x) \) and \( f_2(x) \) serve as perfect examples to understand this concept.Let's look closely:

• \( f_1(x) = x^2 \) is a quadratic function. For every value of \( x \), it squares the input.
• \( f_2(x) = x|x| \) is based on multiplying \( x \) by its absolute value. This means it behaves like \( x^2 \) for \( x \geq 0 \) and \(-x^2 \) for \( x < 0 \).

Functions can model diverse real-world scenarios by defining clear and precise input-output relationships. They are foundational in calculus and analysis, where these inputs (commonly represented by \( x \)) lead to various outputs, forming unique curves and shapes when graphed.

Interval Analysis

Interval analysis is the technique of exploring the behavior of functions over specific segments or intervals. By studying intervals, we can observe how functions like \( f_1(x) = x^2 \) and \( f_2(x) = x|x| \) behave differently or similarly in various sections of the number line.Interval analysis helps identify:

• Where functions might be dependent or independent, such as
  • dependent on \([0, \infty)\)
  • independent on \((-\infty, 0)\)
• The nature of any patterns or relationships within these intervals.

This analysis is not only vital when determining conditions under which functions are linearly independent or dependent, but it also plays a crucial role in solving and visualizing differential equations, optimization problems, and more. Understanding how different functions perform over varied intervals is pivotal for deeper mathematical insights.