Averaging over shorter and shorter intervals to get a speed at an instant

What to assume they know

• Reading a function given by a formula, and evaluating it at decimals
• Average speed as distance travelled divided by time taken
• That a graph of distance against time has a slope with units of distance per time
• The slope of a straight line joining two named points on a curve
• Arithmetic with four and five decimal places, done exactly rather than rounded
• Why writing a pair in order carries information a set cannot — a function as a rule with a domain

What they should be able to do

• State why an average velocity over an interval is computable and a velocity at an instant is not, in terms of what the quotient needs
• Compute the distance fallen from s = 4.9t² at any given time, and reproduce every entry of the chapter's distance table
• Compute the average velocity over an interval with one endpoint fixed at t = 2, from both the left and the right
• Read a shrinking family of average velocities as a sequence and say what it appears to approach
• State the bracket the chapter concludes with, and identify precisely which assumption lets it be stated
• Show algebraically that the average velocity over the interval from t₁ to 2 simplifies to 4.9(2 + t₁), and use that to explain why the bracket is symmetric about a single value
• Reinterpret each average velocity as the slope of a chord on the distance–time graph, and the limiting value as the slope of a tangent
• Identify what the chapter has and has not established at the end of §12.2

Where it usually goes wrong

• **"Average velocity over the last stretch just is the velocity at the end."** Table 12.2 shows every one of those averages sitting below 19.6 and none of them equal to it. The averages are correct answers to a different question.
• "The tables prove the velocity is 19.6." They do not, and the chapter is careful about this: it claims only that the value lies inside a bracket. The step from bracket to a single number needs the assumption stated on p. 218 and p. 219 — that nothing abrupt happens in the last hundredth of a second on either side. That assumption is doing real work.
• "You could get the exact answer by taking the interval to be zero." Then both the numerator and the denominator are 0, and 0/0 is not a number. The entire chapter exists because that shortcut is closed.
• "The left-hand list and the right-hand list are two different phenomena." Section 7 shows both are 4.9(2 + t); one list feeds t from below, the other from above. They were never two computations.
• "Slope of a chord and average velocity are separate ideas that happen to agree." They are the same quotient with different labels. The rise is distance covered, the run is time elapsed.
• "The chapter defines the derivative here." It uses the word on p. 219 and defines it twenty-one pages later, in §12.5, p. 240. Anything a student writes in an exam must come from the definition, not from this section's tables.

Questions to check understanding

• Given a distance law and a target instant, tabulate average velocities over intervals shrinking from both sides and state the bracket obtained
• Compute a single average velocity from two table entries, with correct units
• Show that an average-rate quotient simplifies after cancelling a common factor, and state the value of the variable at which the cancellation is not allowed
• Explain why an average rate over an interval cannot be quoted as a rate at an instant
• Identify the slope of which line, on a given distance–time graph, represents a stated average velocity
• Two-mark reasoning: name the assumption under which a bracket of computed averages is taken to pin down a single value

Examples worth working on the board

Values marked verified are worked out here on the chapter's printed data; the chapter prints its tables but does not show the working, and this brief never used any answer key.

• The falling-body law (§12.2, p. 217). A body dropped from a tall cliff covers 4.9t² metres in t seconds, so distance is s = 4.9t². The chapter states this as an experimental fact, not a derivation.
• Table 12.1 (p. 218) — the distance table. Pairs (t, s): (0, 0), (1, 4.9), (1.5, 11.025), (1.8, 15.876), (1.9, 17.689), (1.95, 18.63225), (2, 19.6), (2.05, 20.59225), (2.1, 21.609), (2.2, 23.716), (2.5, 30.625), (3, 44.1), (4, 78.4). Verified: every one of these is 4.9t². Three worth doing — 4.9 × 2.25 = 11.025, 4.9 × 3.8025 = 18.63225, 4.9 × 6.25 = 30.625.
• Table 12.2 (p. 218) — average velocity over the interval from t₁ up to t = 2. Pairs (t₁, v): (0, 9.8), (1, 14.7), (1.5, 17.15), (1.8, 18.62), (1.9, 19.11), (1.95, 19.355), (1.99, 19.551). Verified: (19.6 − 0)/2 = 9.8; (19.6 − 4.9)/1 = 14.7; (19.6 − 11.025)/0.5 = 17.15; (19.6 − 15.876)/0.2 = 18.62; (19.6 − 17.689)/0.1 = 19.11; (19.6 − 18.63225)/0.05 = 19.355. The last column cannot be checked from Table 12.1, which stops at 1.95 on that side. You have to compute s(1.99) = 4.9 × 3.9601 = 19.40449 first, and then (19.6 − 19.40449)/0.01 = 19.551.
• Table 12.3 (p. 219) — average velocity over the interval from t = 2 up to t₂. Pairs (t₂, v): (4, 29.4), (3, 24.5), (2.5, 22.05), (2.2, 20.58), (2.1, 20.09), (2.05, 19.845), (2.01, 19.649). Verified: (78.4 − 19.6)/2 = 29.4; (44.1 − 19.6)/1 = 24.5; (30.625 − 19.6)/0.5 = 22.05; (23.716 − 19.6)/0.2 = 20.58; (21.609 − 19.6)/0.1 = 20.09; (20.59225 − 19.6)/0.05 = 19.845. Again the last entry needs s(2.01) = 4.9 × 4.0401 = 19.79649, giving (19.79649 − 19.6)/0.01 = 19.649.
• The bracket (p. 219). The chapter concludes that the instantaneous velocity at t = 2 lies between 19.551 m/s and 19.649 m/s, and calls that number the derivative of s = 4.9t² at t = 2.
• The algebra that explains both tables — this is section 7, and it is the brief's own derivation, not printed in §12.2. For an interval with endpoints t and 2, the average velocity is 4.9(2² − t²)/(2 − t) = 4.9(2 − t)(2 + t)/(2 − t) = 4.9(2 + t), valid for every t ≠ 2. Verified against the printed tables: 4.9(2 + 1.99) = 19.551 and 4.9(2 + 2.01) = 19.649 — the two ends of the chapter's own bracket. It follows that the bracket is centred on 4.9 × 4 = 19.6, and that the two tables are one formula sampled on either side of t = 2. Say plainly when explaining it that §12.2 stops short of this cancellation; it is held back until §12.3.2, where it becomes the standard method.
• Fig 12.1 (p. 219). Distance-axis vertical, time-axis horizontal, the curve s = 4.9t² rising through the point A at t = 2. Two later times are marked on the time-axis, 2 + t₂ and 2 + t₁, with B₂ and B₁ the corresponding points on the curve and C₂, C₁ the feet of the verticals dropped to the level of A. The chords AB₁ and AB₂ are drawn, and one further line through A that lies below both of them. Read off the page image.
• The ratios read off the figure (p. 220). C₁B₁/AC₁, C₂B₂/AC₂, and so on. In each of them the top line is a vertical rise on the curve, s₁ − s₀ for the first, and the bottom line is the horizontal run beneath it, which is how long the interval lasted. Each is an average velocity of exactly the kind Table 12.3 tabulates — but none of them is one of that table's entries. The figure hangs no numbers on its intervals at all, and it marks two later instants against the table's seven. Treat the figure as the same idea drawn rather than as the table redrawn.

Figures to have open

• The distance–time curve s = 4.9t² with the point at t = 2 and a family of chords from it to points on both sides. This is the chapter's Fig 12.1 (p. 219) extended to the left, which the printed figure does not show; redraw as a schematic rather than reproducing the printed art.
• A two-column table movement: Table 12.2 filling downward and Table 12.3 filling upward, converging on a shaded band.
• A number line marked 19.551, 19.6 and 19.649, used twice — once with the middle mark hidden (the chapter's position) and once with it revealed (section 7's).
• No photograph is needed. The portrait of Newton on p. 217 is decorative.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 12 "Limits and Derivatives", §12.1 Introduction and §12.2 Intuitive Idea of Derivatives, printed pp. 217–220. Tables 12.1, 12.2 (p. 218) and 12.3 (p. 219); Fig 12.1 (p. 219).
• Forward pointers inside the chapter, both deliberate: the cancellation used in section 7 is the method of §12.3.2, Example 2, pp. 230–232; the definition of the derivative that this section anticipates is Definition 1, §12.5, p. 240.
• Printed stale cross-references, both confirmed on the page images. The sentence on p. 217 introducing the distance table calls it Table 13.1 while the caption above the table on p. 218 reads Table 12.1; the sentence on p. 218 introducing the second table calls it Table 13.2 while its caption reads Table 12.2. These are leftovers from the pre-2022 edition, in which this was Chapter 13. Cite the captions. Anyone sent to "Table 13.1" will not find it in this book.
• One further wording slip on p. 219, worth knowing but not worth teaching: the paragraph summarising the second set of computations describes those intervals as ending at t = 2, where the tables and the sentence introducing them on p. 218 both have them starting at t = 2. Table 12.3 is the authority.
• Chapter Summary, p. 254, and Historical Note, pp. 255–256.