Checkpoint: change and accumulation

Where we are on the map

This closes the calculus block; every formula must carry meaning, conditions, units, and an independent check.

Begin with a familiar image

An engineer does not accept a graph because its line looks smooth. Initial data, rate, accumulated total, and boundary behavior must agree.

Precise meaning

Mastery means seeing one process as a verbal law, table or graph, formula, and checkable calculation.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

meaning → model → limit/derivative/integral → calculation → units → independent check

Worked example

For motion, differentiating position gives velocity, integrating velocity returns displacement, and a differential equation states a law of change.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Choosing an operation mechanically from a symbol is wrong. First decide whether the question asks for local rate, total accumulation, limiting behavior, or a function from its change law.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. Complete ten problems and show meaning, calculation, and a check.

1. Find lim(x→4)(x²−16)/(x−4) and explain the hole.
2. Let f(x)=−1 for x<0 and f(x)=2 for x≥0. Find both one-sided limits and decide whether the limit at zero exists.
3. Use the definition to differentiate f(x)=x²+3x and find f′(2).
4. For s(t)=t³−6t²+9t, find rest times and intervals of forward and backward motion.
5. Among rectangles of perimeter 32, find the maximum area and justify it with the derivative sign.
6. Evaluate ∫[0,4](3x²−2x)dx and check by differentiating the antiderivative.
7. Find the signed integral ∫[−2,2]x dx and explain by symmetry.
8. Velocity is v(t)=6t−2 m/s. Find displacement from t=1 to t=4 and state units.
9. Solve y′=0.2y, y(0)=30; check the initial condition and derivative.
10. Build a cooling model for a drink in a room; name its parameters, limiting temperature, and two model limitations.

Pass target: at least 8/10, including problem 2 or 10. After seven days, complete a changed version with a target of at least 7/10.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

The next block turns geometric vectors into systems, matrices, and linear transformations.

Course contents