Problem 9 - The Calculus with Analytic Geometry

Question

\(f(x)= \begin{cases}x^{2} & \text { if } x<0 \\ -x^{2} & \text { if } x \geq 0\end{cases}\)

Step-by-Step Solution

Answer

For x < 0, \(f(x) = x^2\). For x \(\geq\) 0, \(f(x) = -x^2\). The function is continuous at x = 0.

Step 1 - Understand the piecewise function

The function is defined in two parts, based on the value of x. For values of x that are less than 0, the function is defined as \(x^2\). For values of x that are greater than or equal to 0, the function is defined as \(-x^2\).

Step 2 - Evaluate for x < 0

For any value of x that is less than 0, plug the value into the function \(f(x) = x^2\). For example, if x = -1, then \(f(-1) = (-1)^2 = 1\).

Step 3 - Evaluate for x \(\geq\) 0

For any value of x that is greater than or equal to 0, use the function \(f(x) = -x^2\). For instance, if x = 1, then \(f(1) = -(1)^2 = -1\).

Step 4 - Verify continuity at x = 0

To check if the function is continuous at x = 0, evaluate the left-hand limit and the right-hand limit. The left-hand limit as x approaches 0 from the negative side is \(0^2 = 0\). The right-hand limit as x approaches 0 from the positive side is \(-0^2 = 0\). Both limits equal 0, so the function is continuous at x = 0.

Step 5 - Summary

The function \(f(x)\) is a piecewise function with different expressions for x < 0 and x \(\geq\) 0. For x < 0, the function is \(x^2\). For x \(\geq\) 0, the function is \(-x^2\). The function is continuous at x = 0.

Key Concepts

Understanding Continuity

Continuity is a key concept in calculus that ensures a function behaves 'smoothly'. When we say a function is continuous at a point, it means there's no sudden jumps or breaks at that point. For our piecewise function, to determine if it's continuous at a specific point (like x = 0), we need to check:

• The function is defined at that point.
• The left-hand limit equals the right-hand limit as x approaches that point.
• Both these limits equal the function's value at that point.

In our example, we evaluated the limits as x approaches 0 from both sides and found they were indeed equal. Hence, our function is continuous at x = 0.

Limits in Calculus

Limits help us understand the behavior of functions as they approach a certain point. In calculus, we use limits to determine a function's value near a specific input, rather than exactly at that input. For a piecewise function, limits allow us to make sense of the transition between different parts of the function.
For our function:
When x approaches 0 from the negative side (\textcolor{red}{x < 0}), the limit is found using the part of the function where \( f(x) = x^2 \).
When x approaches 0 from the positive side (\textcolor{green}{x \bigcup_{0}}), we use the part where \( f(x) = -x^2 \).
As we saw in the step-by-step solution, both approaches yield a limit of 0. This matched the function's value at x = 0, confirming its continuity there.

Evaluating Piecewise Functions

Piecewise functions are defined by different rules for different parts of their domain. When evaluating such functions:

• Identify the correct 'piece' of the function based on the input value.
• Substitute the input into the corresponding function expression.

In our given piecewise function, for x less than 0, we use the expression \( x^2 \). For x greater than or equal to 0, we use \( -x^2 \).
Let's see an example:
For x = -2, since -2 is less than 0, we evaluate \( f(-2) = (-2)^2 = 4 \).
For x = 2, since 2 is greater than or equal to 0, we evaluate \( f(2) = -(2)^2 = -4 \).
This clear categorization ensures we correctly handle any value within the domain of the piecewise function.