Problem 1 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 14, do each of the following: (a) Draw a sketch of the graph of the function; (b) determine if \(f\) is continuous at \(x_{1} ;\) (c) find \(f^{\prime}-\left(x_{1}\right)\) and \(f_{+}^{\prime}\left(x_{1}\right)\) if they exist; (d) determine if \(f\) is differentiable at \(x_{1}\). $$ \begin{gathered} f(x)=\left\\{\begin{aligned} x+2 & \text { if } x \leq-4 \\ -x-6 & \text { if } x>-4 \end{aligned}\right. \\ x_{1}=-4 \end{gathered} $$

Step-by-Step Solution

Answer

The function is continuous but not differentiable at \( x = -4 \).

Step 1 - Sketch the Graph

First, identify the function pieces: For \( x \leq -4 \), the function is \( f(x) = x + 2 \). For \( x > -4 \), the function is \( f(x) = -x - 6 \). Sketch both pieces on the same graph, ensuring the correct piecewise boundary at \( x = -4 \).

Step 2 - Evaluate Continuity at \( x_1 = -4 \)

To check continuity at \( x = -4 \), determine the left-hand limit and right-hand limit at \( x = -4 \): \( \text{Left-hand limit} = \ \lim_{{x \to -4^-}} f(x) = (-4) + 2 = -2 \).\( \text{Right-hand limit} = \ \lim_{{x \to -4^+}} f(x) = -(-4) - 6 = -2 \). Since these limits are equal and \( f(-4) = -2 \), the function is continuous at \( x = -4 \).

Step 3 - Find the Derivatives

The derivative of the function for \( x \leq -4 \) is \( f'(x) = 1 \).The derivative for \( x > -4 \) is \( f'(x) = -1 \). Now, evaluate these derivatives at \( x = -4 \):\( f'_{-}(-4) = 1 \) and \( f'_{+}(-4) = -1 \).

Step 4 - Determine Differentiability at \( x_1 = -4 \)

For differentiability at a point, the left-hand and right-hand derivatives must match.Here, \( f'_{-}(-4) = 1 \) and \( f'_{+}(-4) = -1 \) do not match.Therefore, the function is not differentiable at \( x = -4 \).

Key Concepts

continuity

To understand if a function is continuous at a point, we need to check the left-hand limit, the right-hand limit, and the value of the function at that point itself. For a function to be continuous at a point, these three values must all be equal.

In our exercise, we see that for \(x = -4\), both the left-hand limit and the right-hand limit are equal to -2. Additionally, the function's value at \(x = -4\) (\(f(-4)\)) is also -2. Thus, we conclude that the function is continuous at \(x = -4\).

differentiability

Differentiability of a function at a point means the function has a defined tangent at that point, implying the slope of the function is consistent from both the left and right sides. For this, the left-hand and right-hand derivatives at that point need to be equal.

In this example, at \(x = -4\), the derivative of the function from the left side, \(f'_{-}(-4) = 1\), and the derivative from the right side, \(f'_{+}(-4) = -1\). Since these derivatives aren't equal, the function is not differentiable at \(x = -4\).

graphical representation

Graphical representation of piecewise functions can be very illuminating. You need to plot each piece of the function and ensure they are correctly joined at the specified boundary.

For our function:

• Piece 1: \(f(x) = x + 2\) for \(x \leq -4\)
• Piece 2: \(f(x) = -x - 6\) for \(x > -4\)

At \(x = -4\), we join these pieces ensuring that both start and end points connect correctly. This helps in understanding the behavior and the continuity of the function visually.

left-hand limit

The left-hand limit of a function as it approaches a point gives the anticipated value of the function from the left side. It is denoted as \( \lim_{x \to c^{-}} f(x) \).

For \(x = -4\) in our function, calculate the left-hand limit:

• Using \(f(x) = x + 2\) for \(x \leq -4\)
• Thus, \( \lim_{x \to -4^{-}} f(x) = -4 + 2 = -2\).

This value helps determine continuity and behavior as \(x\) approaches \(-4\) from the left.

right-hand limit

The right-hand limit of a function as it approaches a point gives the anticipated value of the function from the right side. It is denoted as \( \lim_{x \to c^{+}} f(x) \).

For \(x = -4\) in our function, calculate the right-hand limit:

• Using \(f(x) = -x - 6\) for \(x > -4\)
• Thus, \( \lim_{x \to -4^{+}} f(x) = -(-4) - 6 = -2\).

This value, when comparing with the left-hand limit, helps in analyzing the overall behavior at \(x = -4\).