Unit 2.4

Multiple Choice

Why is it important to understand the connection between differentiability and continuity in calculus?

• It helps solve real-world problems involving motion and change.

• It is only useful for theoretical mathematics.

• It is not relevant to practical applications.

• It is mainly used in statistics.

Open Ended

In your own words, explain the difference between continuity and differentiability of a function.

Multiple Choice

What is the 'secret enemy' of a differentiable function?

• A hole in the graph

• A sharp corner

• A continuous graph

• A tangent line

Open Ended

If a function is continuous at x = 5, does it have to be differentiable at x = 5? Explain your reasoning.

Multiple Select

Select all statements that are true about a continuous function.

• The graph can be drawn without lifting your pen.

• The function must have sharp corners.

• The point must exist (no holes in the graph).

• The limit must equal the point's value.

Multiple Choice

Which of the following is NOT a requirement for a function to be considered continuous?

• The point must exist (no holes in the graph).

• The limit must equal the point's value.

• You can draw the graph without lifting your pen.

• The function must have sharp corners.

Multiple Choice

Match the following terms to their correct definitions: 1. Continuous 2. Secant Line 3. Tangent Line 4. Differentiable

• 1-a, 2-d, 3-c, 4-b

• 1-b, 2-c, 3-d, 4-a

• 1-c, 2-b, 3-a, 4-d

• 1-d, 2-a, 3-b, 4-c

Multiple Select

Which of the following situations will cause a function to fail to be differentiable?

• Sharp corners

• Discontinuity

• Vertical tangents

• All of the above

Multiple Choice

At which x-value is f continuous but not differentiable?

• a
• b
• c
• d

Open Ended

Identify any x-values of the function that are not continuous and/or not differentiable in the given graphs.

Open Ended

Describe how you can visually determine if a function is not differentiable at a certain point on its graph.

Open Ended

Explain the relationship between continuity and differentiability using examples from the images provided.

Multiple Choice

Continuity and differentiability are both properties of functions. Which of the following statements best describes their relationship?

• Every continuous function is differentiable.

• Every differentiable function is continuous.

• A function can be differentiable but not continuous.

• Continuity and differentiability are unrelated.

Open Ended

Reflecting on today's lesson, how would you explain the connection between differentiability and continuity to someone new to calculus?