How can two pieces join both continuously and smoothly?
You will be able to: Solve separate value-matching and slope-matching conditions.
BC foundation: Unit 2 shares its derivative foundations with AB. Connect each rule to rates, graphs and its domain conditions. Use the written challenges to explain your reasoning. Chain-rule compositions, implicit and inverse differentiation, and advanced applications come later.
How can two pieces join both continuously and smoothly?
Two sections of a track must meet at the same height and point in the same direction. Matching only the height can leave a sharp turn.
A useful starting point: Why can a connected curve have no derivative? →
Words and symbols before equations
- Piecewise rule
- Different formulas assigned to different parts of the input domain.
- Join
- Input where the formula changes.
- Value matching
- Equality of both limiting heights and the assigned point.
- Slope matching
- Equality of finite left and right difference-quotient limits.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f=x² for x≤1 and mx+b for x>1. The left piece assigns f(1)=1. Only after continuity holds is m the right derivative at the join.
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.
- Left limit=f(1)=1; right limit=1. Continuous. Left slope=2; right slope=1. Unequal slopes give a corner.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Consider f(x)=x² for x≤1 and mx+b for x>1. The left limit and f(1) equal 1. Continuity first requires m+b=1.
The left derivative at 1 is 2. Once continuity is established, the right difference quotient has slope m, so differentiability requires m=2.
Combining these gives b=−1. Then both side quotients approach 2, and the pieces meet with a common tangent.
If m+b≠1, using m as the right derivative at the assigned join is invalid: the jump contributes a nonvanishing height difference divided by h.
A worked example, step by step
Choose m and b so x² (x≤1) and mx+b (x>1) are differentiable at 1.
- Set the right-hand height m+b equal to f(1)=1.
- The left limiting slope is 2.
- Match the right slope by choosing m=2.
- Substitute into m+b=1 to obtain b=−1; both continuity and slope conditions hold.
Matching slopes of formulas does not repair a jump. Check continuity before comparing their derivative formulas.
If m=2 and b=0, is the join differentiable?
Compare with an explanation
No. The right height is 2 but f(1)=1; equal formula slopes do not remove the jump.
Predict. Change one thing. Explain.
First make the endpoint heights match. Then adjust parameters to match slopes too. Find a continuous corner and a discontinuous pair with equal formula slopes.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Left limit=f(1)=1; right limit=1. Continuous. Left slope=2; right slope=1. Unequal slopes give a corner.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f=x² for x≤1 and mx+b for x>1. The left piece assigns f(1)=1. Only after continuity holds is m the right derivative at the join.
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 questionFor g(x)=x² when x≤2 and mx+b when x>2, find m and b for differentiability at 2.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Continuity requires 2m+b=4.
- 1 point: The left derivative at 2 is 4.
- 1 point: Match slopes: m=4.
- 1 point: Then b=4−8=−4; both one-sided derivatives equal 4.
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 1Which condition comes first?
Continuity, including the assigned value.
RECALL 2Can matching heights still leave a corner?
Yes, if the finite side slopes differ.
RECALL 3What makes the join differentiable?
Continuity and equal finite side difference-quotient limits.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How can two pieces join both continuously and smoothly?
- For this join: continuity m+b=1; differentiability also m=2.
- Solution m=2, b=−1.
Remember: Matching slopes of formulas does not repair a jump. Check continuity before comparing their derivative formulas.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f=x² for x≤1 and mx+b for x>1. The left piece assigns f(1)=1. Only after continuity holds is m the right derivative at the join.
Refresh Kid · AP Calculus BC 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 BC 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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