How do you choose a parameter to make pieces meet?
You will be able to: Solve matching conditions while checking the point assignment.
BC foundation: Unit 1 shares its limits-and-continuity objectives with AB. If you have studied these ideas before, use the explanations and written challenges to check your reasoning. Later BC units build on these foundations; this unit does not require series or advanced integration.
How do you choose a parameter to make pieces meet?
A pricing model uses one formula below a threshold and another at and above it. To remove a jump, the limiting charges from the two formulas must meet.
A useful starting point: How do domain restrictions determine continuous intervals? →
Words and symbols before equations
- Parameter
- A constant chosen to change a family of functions.
- Boundary
- The input separating pieces.
- Matching condition
- Equality of the two finite side limits and the point value.
- Redefinition
- Changing an assigned value or formula intentionally.
What this picture assumes
Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. f(x)=x+1 for x<2 and kx−1 for x≥2. The right piece assigns the boundary value. Changing k changes the whole right branch.
Read the picture in three steps
- Read the axes, coordinates and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Left limit=3; right limit=f(2)=2k−1=1. Solve 2k−1=3 to obtain k=2.
- 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 f(x)=x+1 when x<2 and f(x)=kx−1 when x≥2, the left limit is 3. The right limit and assigned value are both 2k−1.
Set 2k−1=3 and solve k=2. This makes the value and both side limits agree at the boundary.
If instead only a single point is missing from a curve whose limit is L, fill that point with L. These are related but distinct repairs: changing a branch parameter can change many inputs.
A single point assignment cannot repair an existing jump or an unbounded break because it does not alter their side behavior.
A worked example, step by step
Choose k for the stated piecewise function to be continuous at x=2.
- Use the left formula to get 2+1=3.
- Use the right formula to get 2k−1.
- Equate: 2k−1=3, so 2k=4 and k=2.
- Because the right formula includes x=2, f(2)=3 too; all checks hold.
Matching two limits is not enough if a separate third piece assigns a different value at the boundary.
If a separate definition assigns f(2)=9, does k=2 finish the repair?
Compare with an explanation
No. The value must also be changed to the common limit 3.
Predict. Change one thing. Explain.
Adjust k until the branch endpoints meet. Check the exact point value as well as both approached heights.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Left limit=3; right limit=f(2)=2k−1=1. Solve 2k−1=3 to obtain k=2.
Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. f(x)=x+1 for x<2 and kx−1 for x≥2. The right piece assigns the boundary value. Changing k changes the whole right branch.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using nearby function values, graph coordinates, or the hypotheses of a limit law or theorem. 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 questionLet g(x)=2x+1 for x<1 and g(x)=kx+2 for x≥1. Choose k and verify the value as well as both limits.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: The left limit is 3.
- 1 point: The right limit is k+2.
- 1 point: Set k+2=3 to get k=1.
- 1 point: Then g(1)=3 from the included right branch, so continuity holds.
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 1What equation determines the parameter?
Set the side limits equal.
RECALL 2What still needs checking?
The value assigned exactly at the boundary.
RECALL 3Can a lone dot repair a jump?
No.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you choose a parameter to make pieces meet?
- For continuity at a boundary: left limit = right limit = assigned value.
Remember: Matching two limits is not enough if a separate third piece assigns a different value at the boundary.
Conditions: Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. f(x)=x+1 for x<2 and kx−1 for x≥2. The right piece assigns the boundary value. Changing k changes the whole right branch.
Refresh Kid · AP Calculus BC Unit 1 · Objectives LIM-2.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.13, LIM-2.C. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 1 has 16 official topics. Focused lesson titles and questions are original Refresh Kid teaching material. Topics 1.7 and 1.9 integrate earlier objectives and skills rather than introducing new numbered learning objectives.
Formal epsilon-delta proofs are not assessed in this unit. L’Hôpital’s rule, derivative rules and differentiation tests belong later in the course and are not used to bypass limit reasoning here. Trigonometric limits use radians. Finite graphs and tables provide evidence, not a proof of all nearby behavior. Infinity describes unbounded behavior, not a number to substitute. The Intermediate Value Theorem requires continuity on a closed interval and an intermediate output; it guarantees existence, not uniqueness.
The Organic Chemistry Tutor video title, creator and description were checked; the full video was not reviewed. Khan Academy’s BC unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Section 2.2 was 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. Camera rotation does not add a variable or change slope; use labeled coordinates and axis scales.
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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