Why can a connected curve have no derivative?
You will be able to: Identify different failures of a finite two-sided derivative.
Why can a connected curve have no derivative?
A pencil can draw a connected V or a curve that becomes upright without lifting. Connectedness alone does not give a single finite local slope.
A useful starting point: Why does differentiability imply continuity? →
Words and symbols before equations
- Corner
- Two different finite limiting slopes meet.
- Cusp
- In this lesson, one-sided slopes grow without bound in opposite directions.
- Vertical tangent
- A tangent parallel to the output axis, with no finite slope.
- Unbounded quotient
- A ratio whose magnitude grows beyond every finite bound.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Nonzero h=10⁻ᵖ. Quotient limits, not a finite graph, establish derivative failure. Cube root is real on both sides.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- h magnitude=0.1. Left quotient=-1; right quotient=1. Limits −1 and 1 disagree. All examples are continuous at zero but have no finite derivative there.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For abs(x) at 0, the difference quotient is −1 on the left and 1 on the right: a corner.
For x^(2/3)=abs(x)^(2/3), the quotient at 0 is h^(2/3)/h. It tends to −∞ from the left and +∞ from the right: a cusp.
For the real cube root of x, the quotient at 0 is 1/abs(h)^(2/3), which tends to +∞ from both sides: a vertical tangent.
All three functions are continuous at 0. None has an ordinary finite derivative there. A large finite displayed quotient is only a sample; the formulas establish the unbounded limits.
A worked example, step by step
Compare one-sided difference quotients for f(x)=cube root of x at h=±0.001.
- f(0)=0.
- At h=0.001 the quotient is 0.1/0.001=100.
- At h=−0.001 the quotient is −0.1/−0.001=100.
- As abs(h)→0, 1/abs(h)^(2/3)→+∞, so there is a vertical tangent but no finite derivative.
An infinite limiting slope is not a real derivative value. Distinguish it from unequal finite slopes.
Does a readout of 100 mean the cube-root derivative at zero is 100?
Compare with an explanation
No. Smaller nonzero intervals produce larger quotients; the limit is not finite.
Predict. Change one thing. Explain.
Choose each continuous example and shrink the nonzero h. Compare left and right quotients; explain which finite-derivative condition fails.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
h magnitude=0.1. Left quotient=-1; right quotient=1. Limits −1 and 1 disagree. All examples are continuous at zero but have no finite derivative there.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Nonzero h=10⁻ᵖ. Quotient limits, not a finite graph, establish derivative failure. Cube root is real on both sides.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using a difference quotient, tangent slope, or derivative rule with its domain conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionClassify abs(x), abs(x)^(2/3) and cube root of x at zero. Explain what they share and how their side quotients differ.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: All three are continuous at zero.
- 1 point: abs(x) has side quotients −1 and 1: corner.
- 1 point: abs(x)^(2/3) has limits −∞ and +∞: cusp.
- 1 point: Cube root has +∞ on both sides: vertical tangent; none has a finite derivative.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Does a continuous corner have a derivative?
No, if its one-sided slopes disagree.
RECALL 2Is infinity a finite slope?
No.
RECALL 3Why label the cube root as real?
It is defined for negative as well as positive inputs.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Why can a connected curve have no derivative?
- Corner: unequal finite side slopes.
- Cusp example: opposite unbounded side slopes.
- Vertical tangent example: same-direction unbounded slopes.
Remember: An infinite limiting slope is not a real derivative value. Distinguish it from unequal finite slopes.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Nonzero h=10⁻ᵖ. Quotient limits, not a finite graph, establish derivative failure. Cube root is real on both sides.
Refresh Kid · AP Calculus AB Unit 2 · Objectives FUN-2.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.4, FUN-2.A. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 2 has 10 official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative definitions require finite two-sided limits at interior inputs. Graphs and finite tables supply evidence and estimates, not universal proofs. Continuity is necessary but insufficient for differentiability. Rules retain their original domain restrictions, and trigonometric inputs use radians. Chain-rule compositions, implicit differentiation, inverse-function differentiation and L’Hôpital’s rule are deferred to later units. Piecewise joins are checked for continuity before their side slopes are compared.
The Organic Chemistry Tutor video titles, creators and descriptions were checked; full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.1, 3.3 and 3.5 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. The optional 3D model uses separate planes for f and f′. Depth distinguishes panels, not an additional variable; camera rotation does not change their values. Use the complete labeled 2D graphs and text to read precise coordinates and slopes.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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