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Problem 50 · One Knob, Infinitely Many HistoriesPrint one blank worksheet now, download its PDF, or prepare a family or classroom packet.

Problem 50

One Knob, Infinitely Many Histories

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One Knob, Infinitely Many Histories

Complete to collectAlexandrite CrystalLevel 2 · 4 Glints

Turn one knob; bend a whole history

A puck starts at position 0 at time 0 and ends at position 1 at time 1. Turning the parameter creates infinitely many histories with these same endpoints.

Changing changes the entire motion while keeping its endpoints fixed.

The symbols play two different roles

Time moves along one curve

Hold fixed. The derivative is that history’s velocity.

The knob selects a curve

Hold the endpoints fixed. Changing changes the whole history at once.

First study the family and its velocities. On page 2, you will assign one action number to each curve and then differentiate with respect to .

A position-versus-time graph shows five histories x sub a of t sharing the endpoints (0,0) and (1,1). Positive a curves rise early, negative a curves rise late, and a equals zero is the straight steady history.
The display shows five settings; the formula defines a history for every real value of .

Page 1 · Understand the family

Show derivatives, integrals, and algebra on separate paper.

  1. Verify that every history in the family has the required endpoints. Differentiate to find its velocity. Which sign of the parameter makes the puck rush early, and which makes it rush late?

Page 2 · Find the stationary history

Show derivatives, integrals, and algebra on separate paper.

Action of one history

After integration, is a function of the knob setting. For this family, stationary action means .

  1. Substitute the velocity into the action. Expand the square, use the three supplied integrals, and show that infinitely many histories reduce to one action function.

    Action function

  2. Differentiate the action function with respect to the knob setting. Find the stationary parameter, identify the corresponding history, and use the action formula to decide whether it is a minimum.

    stationary

  3. Explain why opposite knob settings have equal action even though one rushes early and the other rushes late.

Next idea: Calculus of variations tests arbitrary endpoint-fixing changes , not only this one knob.

At a glance

About this problem

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Before you choose this problem

A plain-language look at readiness, thinking, time, and the expected response.

Math readiness

Designed for Grade 12

Uses HS CALC math.

Thinking

Standard challenge

The solver chooses and coordinates familiar methods.

Time & response

About 13–27 minutes

Moderate amount of written work · Short Explanation response.

How to support: The useful support is to separate the two differentiations: first with respect to time inside each history, then with respect to the parameter after integration. Do not identify the stationary setting.

Worksheet arc · Synthesis · 3 of 3 on Complete arc

Discover Action

Compress infinitely many endpoint-sharing histories into one action function and find its stationary member.

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Technical reference

Rating, skills & curriculum

Designed for Grade 12 · Uses HS CALC math · About 13–27 minutes

Grade 12 · Calculus and continuous change · Derivatives · Integrals and the Fundamental Theorem

MJ HS:CALC.8 · C2 · W2

Precise MJD coordinates, factor scores, skill weights, calibration evidence, and curriculum crosswalks. This reference is available to everyone and is intentionally last.

Content level

MJ HS:CALC.8

High school

HS CALC · point 8

Likely range
MJ HS:CALC.6–HS:CALC.8
Confidence
72%

Challenge

C2 · Standard

The solver chooses and coordinates familiar methods.

  • Method selection1
  • Reasoning depth2
  • Novelty2
  • Constraint management2
  • Proof or justification2

Workload

W2 · Moderate

Several coordinated calculations, representations, or explanation steps.

Active time
13–27 minutes · typically 19
Response
Short Explanation
Known-method steps
18
  • Mechanical execution2
  • Written output2
  • Representation production1
  • Bookkeeping1

Learning focus

Skills this problem practices

What a successful response demonstrates, with each skill’s share of the content rating.

  • Use derivatives for motion, approximation, related rates, and optimization

    The learner can use derivatives for motion, approximation, related rates, and optimization. This capability is distinct from mere exposure to or isolated use of derivative rules; modeling.

    • Independent mastery
    • 60% focus
    • MJ HS:CALC.5 milestone

  • Interpret definite integrals as accumulation and signed area

    The learner can interpret definite integrals as accumulation and signed area. This capability is distinct from mere exposure to or isolated use of limits; area; functions.

    • Independent mastery
    • 40% focus
    • MJ HS:CALC.6 milestone

Readiness

Helpful prerequisites

Skills the rating assumes are already available to the learner.

  • Compute derivatives using standard rules

    The learner can compute derivatives using standard rules. This capability is distinct from mere exposure to or isolated use of algebra; derivative definition.

    • Fluent readiness
    • Required
    • MJ HS:CALC.6 milestone

  • Find basic antiderivatives and indefinite integrals

    The learner can find basic antiderivatives and indefinite integrals. This capability is distinct from mere exposure to or isolated use of derivative rules.

    • Fluent readiness
    • Required
    • MJ HS:CALC.8 milestone

  • Add, subtract, and multiply polynomials

    The learner can add, subtract, and multiply polynomials. This capability is distinct from mere exposure to or isolated use of distributive property; exponent laws.

    • Fluent readiness
    • Required
    • MJ HS:A1.8 milestone

External alignment

Curriculum and standards crosswalks

Approved evidence-backed locations that support discovery. They do not define the MathJewels rating.

Common Core State Standards for Mathematics2010 · 1 location
Practice locations1 location · Representative: Construct viable arguments and critique the reasoning of others.