Calculus 1: Derivatives Definition Differentiability vs Continuity Corners Cusps Vertical Tangents

What Is This?

Differentiability vs Continuity concerns whether a function is smooth enough to have a derivative at a point, versus merely being continuous (no jumps or breaks). This topic appears in exams to test your understanding of function behavior, especially at corners, cusps, and vertical tangents. Questions typically involve identifying these points and determining differentiability.

Why It Matters

This topic is tested in calculus exams, particularly in AP Calculus, IB Mathematics, and university-level calculus courses. It appears frequently, often carrying 10-15% of the total marks. It tests your ability to analyze function graphs and apply theoretical concepts to practical scenarios.

Core Concepts

1. Continuity: A function is continuous at a point if you can draw the graph without lifting your pencil.
2. Differentiability: A function is differentiable at a point if it has a well-defined tangent line (not vertical).
3. Corners: Points where the function changes direction abruptly but is still continuous.
4. Cusps: Points where the function comes to a sharp point, like the tip of a needle.
5. Vertical Tangents: Points where the tangent line is vertical (slope is undefined).

Prerequisites

1. Basic Calculus: Understanding of limits and derivatives.
2. Graph Analysis: Ability to interpret and analyze function graphs.
3. Algebra: Basic algebraic manipulation skills.

The Rule-Book (How It Works)

• Primary Rule: A function is differentiable at a point if it is continuous and the tangent line is not vertical.
• Sub-rules and Exceptions:
• Corners: Continuous but not differentiable due to abrupt direction change.
• Cusps: Continuous but not differentiable due to sharp points.
• Vertical Tangents: Continuous but not differentiable due to undefined slope.
• Visual Pattern: Think of a smooth curve (differentiable) vs. a jagged or sharp curve (not differentiable).

Exam / Job / Audit Weighting

• Frequency: Common
• Difficulty Rating: Intermediate
• Question Type: Graph analysis, true/false, multiple-choice

Difficulty Level

Intermediate

Must-Know Rules, Formulas, Standards, or Principles

1. Continuity Condition: A function ( f(x) ) is continuous at ( x = a ) if ( \lim_{x \to a} f(x) = f(a) ).
2. Differentiability Condition: A function ( f(x) ) is differentiable at ( x = a ) if ( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} ) exists.
3. Vertical Tangent: Occurs when the derivative ( f'(x) ) is undefined due to a vertical slope.

Worked Examples (Step-by-Step)

Easy

Question: Is the function ( f(x) = |x| ) differentiable at ( x = 0 )? Step-by-Step: 1. Check continuity: ( \lim_{x \to 0} |x| = 0 ), so ( f(x) ) is continuous at ( x = 0 ).
2. Check differentiability: ( \lim_{h \to 0} \frac{|0+h| - |0|}{h} ) does not exist (different left and right limits).
Answer: No, ( f(x) ) is not differentiable at ( x = 0 ) due to a corner.

Medium

Question: Is the function ( f(x) = x^{2/3} ) differentiable at ( x = 0 )? Step-by-Step: 1. Check continuity: ( \lim_{x \to 0} x^{2/3} = 0 ), so ( f(x) ) is continuous at ( x = 0 ).
2. Check differentiability: ( \lim_{h \to 0} \frac{(0+h)^{2/3} - 0^{2/3}}{h} = \lim_{h \to 0} \frac{h^{2/3}}{h} = \lim_{h \to 0} h^{-1/3} ) does not exist.
Answer: No, ( f(x) ) is not differentiable at ( x = 0 ) due to a cusp.

Hard

Question: Is the function ( f(x) = \sqrt[3]{x} ) differentiable at ( x = 0 )? Step-by-Step: 1. Check continuity: ( \lim_{x \to 0} \sqrt[3]{x} = 0 ), so ( f(x) ) is continuous at ( x = 0 ).
2. Check differentiability: ( \lim_{h \to 0} \frac{\sqrt[3]{0+h} - \sqrt[3]{0}}{h} = \lim_{h \to 0} \frac{h^{1/3}}{h} = \lim_{h \to 0} h^{-2/3} ) does not exist.
Answer: No, ( f(x) ) is not differentiable at ( x = 0 ) due to a vertical tangent.

Common Exam Traps & Mistakes

1. Mistake: Assuming continuity implies differentiability.
2. Wrong Answer: A function is differentiable if it is continuous.
3. Correct Approach: Check both continuity and the existence of the derivative.
4. Mistake: Overlooking corners and cusps.
5. Wrong Answer: ( f(x) = |x| ) is differentiable at ( x = 0 ).
6. Correct Approach: Identify corners and cusps as points of non-differentiability.
7. Mistake: Ignoring vertical tangents.
8. Wrong Answer: ( f(x) = \sqrt[3]{x} ) is differentiable at ( x = 0 ).
9. Correct Approach: Recognize vertical tangents as points where the derivative is undefined.

Shortcut Strategies & Exam Hacks

• Memory Aid: "Continuous but not differentiable: corners, cusps, verticals."
• Elimination Strategy: If a function has a sharp point or abrupt change, it's not differentiable there.
• Pattern Recognition: Look for ( |x| ), ( x^{2/3} ), and ( \sqrt[3]{x} ) as common examples.

Question-Type Taxonomy

1. Graph Analysis: Identify points of non-differentiability on a graph.
2. Example: Is the function ( f(x) = |x| ) differentiable at ( x = 0 )?
3. Exams: AP Calculus, IB Mathematics
4. True/False: Determine if a statement about differentiability is true or false.
5. Example: ( f(x) = x^{2/3} ) is differentiable at ( x = 0 ).
6. Exams: University-level calculus
7. Multiple-Choice: Choose the correct option based on function behavior.
8. Example: Which of the following functions is not differentiable at ( x = 0 )?
9. Exams: All calculus exams

Practice Set (MCQs)

Question 1

Question: Is the function ( f(x) = |x| ) differentiable at ( x = 0 )? Options: A) Yes B) No C) Only if ( x \neq 0 ) D) Only if ( x = 0 ) Correct Answer: B) No Explanation: The function has a corner at ( x = 0 ), making it non-differentiable.
Why the Distractors Are Tempting: A) Assumes continuity implies differentiability. C) and D) confuse the condition.

Question 2

Question: Is the function ( f(x) = x^{2/3} ) differentiable at ( x = 0 )? Options: A) Yes B) No C) Only if ( x \neq 0 ) D) Only if ( x = 0 ) Correct Answer: B) No Explanation: The function has a cusp at ( x = 0 ), making it non-differentiable.
Why the Distractors Are Tempting: A) Assumes continuity implies differentiability. C) and D) confuse the condition.

Question 3

Question: Is the function ( f(x) = \sqrt[3]{x} ) differentiable at ( x = 0 )? Options: A) Yes B) No C) Only if ( x \neq 0 ) D) Only if ( x = 0 ) Correct Answer: B) No Explanation: The function has a vertical tangent at ( x = 0 ), making it non-differentiable.
Why the Distractors Are Tempting: A) Assumes continuity implies differentiability. C) and D) confuse the condition.

Question 4

Question: Which of the following functions is not differentiable at ( x = 0 )? Options: A) ( f(x) = x^2 ) B) ( f(x) = |x| ) C) ( f(x) = x^{2/3} ) D) ( f(x) = \sqrt[3]{x} ) Correct Answer: B) ( f(x) = |x| ) Explanation: The function has a corner at ( x = 0 ), making it non-differentiable.
Why the Distractors Are Tempting: A) and C) are differentiable at ( x = 0 ). D) has a vertical tangent.

Question 5

Question: Is the function ( f(x) = x^{1/3} ) differentiable at ( x = 0 )? Options: A) Yes B) No C) Only if ( x \neq 0 ) D) Only if ( x = 0 ) Correct Answer: B) No Explanation: The function has a vertical tangent at ( x = 0 ), making it non-differentiable.
Why the Distractors Are Tempting: A) Assumes continuity implies differentiability. C) and D) confuse the condition.

30-Second Cheat Sheet

• Continuity does not imply differentiability.
• Corners, cusps, and vertical tangents are not differentiable.
• Check for abrupt changes or sharp points in the graph.
• Remember: ( |x| ), ( x^{2/3} ), ( \sqrt[3]{x} ) are common examples.
• Differentiability requires a well-defined, non-vertical tangent line.

Learning Path

1. Beginner Foundation: Review basic calculus concepts, especially limits and derivatives.
2. Core Rules: Understand the definitions of continuity and differentiability.
3. Practice: Work through examples of corners, cusps, and vertical tangents.
4. Timed Drills: Practice identifying non-differentiable points under time pressure.
5. Mock Tests: Take full-length practice exams to solidify your understanding.

Related Topics

1. Limits: Understanding how functions approach a point.
2. Relation: Limits are used to determine continuity.
3. Derivatives: Calculating the rate of change of a function.
4. Relation: Derivatives are used to determine differentiability.
5. Graph Analysis: Interpreting and analyzing function graphs.
6. Relation: Graph analysis helps identify corners, cusps, and vertical tangents.