Problem 13 - The Calculus with Analytic Geometry

Question

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 \(x_{1}\). $$ f(x)=\left\\{\begin{aligned} 5-6 x & \text { if } x \leq 3 \\ -4-x^{2} & \text { if } x>3 \\ & x_{1}=3 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The function is continuous and differentiable at \(x_1 = 3\), and \(f'(3) = -6\).

Step 1: Sketch the Graph

Draw the graph of the function \(f(x)\). For \(x \leq 3\), use the linear function \(5-6x\), and for \(x > 3\), use the quadratic function \(-4-x^2\). Plot these on the same set of axes and observe the behavior at \(x = 3\).

Step 2: Determine Continuity at \(x_1 = 3\)

To check if \(f\) is continuous at \(x_1 = 3\), verify if \(\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3)\). Calculate these limits and compare the results to \( f(3) \).

Step 3: Calculate \(f(3)\)

Using the piecewise definition of \(f\), evaluate \(f(3)\). Since \(3 \leq 3\), use the first piece: \(f(3) = 5 - 6 \times 3 = -13\).

Step 4: Calculate \(\lim_{x \to 3^-} f(x)\)

For \(x \leq 3\), use \(5-6x\). Compute the limit as \(x\) approaches 3 from the left: \(\lim_{x \to 3^-} f(x) = 5 - 6 \times 3 = -13\).

Step 5: Calculate \(\lim_{x \to 3^+} f(x)\)

For \(x > 3\), use \(-4 - x^2\). Compute the limit as \(x\) approaches 3 from the right: \(\lim_{x \to 3^+} f(x) = -4 - (3^2) = -13\).

Step 6: Determine Continuity

Since \(\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3)\), the function \(f\) is continuous at \(x_1 = 3\).

Step 7: Calculate \(f'_{-}(3)\)

To find the left-hand derivative \(f'_{-}(3)\), use the function \(5-6x\) and compute the derivative: \(f'_{-}(x) = -6\). Evaluate it at \(x = 3\), giving \(f'_{-}(3) = -6\).

Step 8: Calculate \(f'_{+}(3)\)

To find the right-hand derivative \(f'_{+}(3)\), use the function \(-4 - x^2\) and compute the derivative: \(f'_{+}(x) = -2x\). Evaluate it at \(x = 3\), giving \(f'_{+}(3) = -6\).

Step 9: Determine Differentiability

Since \(f'_{-}(3) = f'_{+}(3)\), \(f\) is differentiable at \(x_1 = 3\).

Key Concepts

Continuity

A function is continuous at a point if there are no breaks, jumps, or holes in the graph at that point. To determine if a function is continuous at some point, say at \( x = 3 \), we need to check three things:

• The function’s value at that point, \( f(3) \), must be defined.
• The limit of the function as it approaches from the left, \( \lim_{x \to 3^-} f(x) \), must exist.
• The limit of the function as it approaches from the right, \( \lim_{x \to 3^+} f(x) \), must exist.

Additionally, the left and right-hand limits should equal the function’s value at that point:
\( \lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3) \)

In our example, since \( \lim_{x \to 3^-} f(x) = -13 \), \( \lim_{x \to 3^+} f(x) = -13 \), and \( f(3) = -13 \), the function is continuous at \( x = 3 \).

Left-Hand Derivative

The left-hand derivative at a point measures how the function is changing as it approaches from the left. It is defined as:
\(\frac{d}{dx} f(x) \bigg|_{x \to c^-} = \frac{f(c+h) - f(c)}{h} \bigg|_{h \to 0^-}\)

For \( f(x) \) in the given problem, the function for \( x \leq 3 \) is \( 5 - 6x \). Differentiating this piece gives us a constant slope:
\( f'_{-}(x) = -6 \)
Therefore, the left-hand derivative at \( x = 3 \) is:
\( f'_{-}(3) = -6 \)
This tells us that as we approach \( x = 3 \) from the left, the function descends at a rate of \( -6 \) per unit change in \( x \).

Right-Hand Derivative

The right-hand derivative at a point measures how the function changes as it approaches the point from the right. It is defined as:
\(\frac{d}{dx} f(x) \bigg|_{x \to c^+} = \frac{f(c+h) - f(c)}{h} \bigg|_{h \to 0^+}\)

For the piecewise function above, for \( x > 3 \), the function is \( -4 - x^2 \). Differentiating this piece gives:
\( f'_{+}(x) = -2x \)
Evaluating at \( x = 3 \) gives us:
\( f'_{+}(3) = -6 \)
Hence, as we approach \( x = 3 \) from the right, the function also descends at a rate of \( -6 \) per unit change in \( x \).

Differentiability

A function is differentiable at a point if it has a well-defined derivative at that point. This means both the left-hand and right-hand derivatives at that point must not only exist but also be equal:

• If \( f''_{-}(c) = f''_{+}(c) \), then \( f \) is differentiable at \( c \).

In our example, both the left-hand and right-hand derivatives at \( x = 3 \) are \( -6 \). Therefore, the function is differentiable at \( x = 3 \). Remember, if a function is differentiable at a point, it is also continuous at that point. However, the reverse is not always true.