Question

this is section 2.4 problem 20: the function is represented by the graph in the textbook. use the g
answer: the function ---select--- differentiable at ( x_0 ) due to ---select--- .
the function ---select--- differentiable at ( x_1 ) due to ---select--- .
the function ---select--- differentiable at ( x_2 ) due to ---select--- .
the function ---select--- differentiable at ( x_3 ) due to ---select--- .
the function ---select--- differentiable at ( x_4 ) due to ---select--- .
the function ---select--- differentiable at ( x_5 ) due to ---select--- .
the function ---select--- differentiable at ( x_6 ) due to ---select--- .
the function ---select--- differentiable at ( x_7 ) due to ---select--- .
the function ---select--- differentiable at ( x_8 ) due to ---select--- .
the function ---select--- differentiable at ( x_9 ) due to ---select--- .
the function ---select--- differentiable at ( x_{10} ) due to ---select--- .
hint: refer to figure 2.4.4 in the textbook.
resources
ebook

Explanation:

To solve this problem, we need to analyze the differentiability of a function at various points \( x_0, x_1, \dots, x_{10} \) using the graph (Figure 2.4.4 in the textbook). The key concepts for differentiability at a point are:

Key Rules for Differentiability:

• A function is differentiable at a point if:

1. It is continuous at that point.
2. The graph has no “corner” (sharp turn), “cusp,” “vertical tangent,” or “discontinuity” at that point.
3. The left - hand derivative and right - hand derivative exist and are equal.

Common Reasons for Non - Differentiability:

• Discontinuity: If the function has a jump, hole, or infinite discontinuity at \( x = a \), it is not differentiable there.
• Corner/Sharp Turn: If the graph changes direction abruptly (e.g., a “V - shape” corner), the left and right derivatives differ, so it is not differentiable.
• Vertical Tangent: If the tangent line is vertical (slope is infinite), the derivative does not exist.
• Cusp: A sharp point where the function “pinches” (similar to a corner but with both sides approaching the point in a specific way), leading to non - differentiability.

How to Analyze Each \( x_i \):

For each \( x_i \) (from \( x_0 \) to \( x_{10} \)):

1. Examine the graph at \( x = x_i \).
2. Check for continuity: Does the function have a defined value, and do the left - hand and right - hand limits equal the function’s value?
3. Check for smoothness: Is there a corner, vertical tangent, cusp, or discontinuity?

Example Analysis (Hypothetical, Based on Typical Textbook Graphs):

Suppose Figure 2.4.4 is a standard graph with:

• \( x_0 \): Discontinuity (e.g., jump discontinuity) → Not differentiable (reason: Discontinuity).
• \( x_1 \): Corner (sharp turn) → Not differentiable (reason: Corner).
• \( x_2 \): Smooth, continuous, and no sharp features → Differentiable (reason: Smooth, continuous, equal left/right derivatives).

Since the actual graph (Figure 2.4.4) is not provided here, you must:

1. Open the textbook to Figure 2.4.4.
2. For each \( x_i \):

• Observe if the function is continuous.
• Observe if there is a corner, vertical tangent, cusp, or discontinuity.

1. Use the dropdown menus to select “is” or “is not” differentiable, and the corresponding reason (e.g., “discontinuity,” “corner,” “smooth and continuous,” etc.).

If you can describe the graph (e.g., “At \( x_0 \), the graph has a jump discontinuity; at \( x_1 \), there is a sharp corner; at \( x_2 \), the graph is smooth,” etc.), we can provide more specific guidance.

Answer:

To solve this problem, we need to analyze the differentiability of a function at various points \( x_0, x_1, \dots, x_{10} \) using the graph (Figure 2.4.4 in the textbook). The key concepts for differentiability at a point are:

Key Rules for Differentiability:

• A function is differentiable at a point if:

1. It is continuous at that point.
2. The graph has no “corner” (sharp turn), “cusp,” “vertical tangent,” or “discontinuity” at that point.
3. The left - hand derivative and right - hand derivative exist and are equal.

Common Reasons for Non - Differentiability:

• Discontinuity: If the function has a jump, hole, or infinite discontinuity at \( x = a \), it is not differentiable there.
• Corner/Sharp Turn: If the graph changes direction abruptly (e.g., a “V - shape” corner), the left and right derivatives differ, so it is not differentiable.
• Vertical Tangent: If the tangent line is vertical (slope is infinite), the derivative does not exist.
• Cusp: A sharp point where the function “pinches” (similar to a corner but with both sides approaching the point in a specific way), leading to non - differentiability.

How to Analyze Each \( x_i \):

For each \( x_i \) (from \( x_0 \) to \( x_{10} \)):

1. Examine the graph at \( x = x_i \).
2. Check for continuity: Does the function have a defined value, and do the left - hand and right - hand limits equal the function’s value?
3. Check for smoothness: Is there a corner, vertical tangent, cusp, or discontinuity?

Example Analysis (Hypothetical, Based on Typical Textbook Graphs):

Suppose Figure 2.4.4 is a standard graph with:

• \( x_0 \): Discontinuity (e.g., jump discontinuity) → Not differentiable (reason: Discontinuity).
• \( x_1 \): Corner (sharp turn) → Not differentiable (reason: Corner).
• \( x_2 \): Smooth, continuous, and no sharp features → Differentiable (reason: Smooth, continuous, equal left/right derivatives).

Since the actual graph (Figure 2.4.4) is not provided here, you must:

1. Open the textbook to Figure 2.4.4.
2. For each \( x_i \):

• Observe if the function is continuous.
• Observe if there is a corner, vertical tangent, cusp, or discontinuity.

1. Use the dropdown menus to select “is” or “is not” differentiable, and the corresponding reason (e.g., “discontinuity,” “corner,” “smooth and continuous,” etc.).

If you can describe the graph (e.g., “At \( x_0 \), the graph has a jump discontinuity; at \( x_1 \), there is a sharp corner; at \( x_2 \), the graph is smooth,” etc.), we can provide more specific guidance.