Difficulty: Introductory
Prerequisites: Section 1.1 (Velocity and Distance).
This section steps back from graphs and introduces the algebra behind the distance-velocity relationship. The core insight: taking differences (to find velocities from distances) and taking sums (to recover distances from velocities) are inverse operations. This is a preview of the Fundamental Theorem of Calculus, shown here with plain numbers before any limits appear. The section also develops the equation of a straight line more carefully, using the income tax bracket as a real-world example of piecewise linear functions.
Differences and sums undo each other. If you have a list of distances and take the differences, you get the velocities. If you then sum those velocities, you recover the last distance minus the first. This is the Fundamental Theorem of Calculus in miniature, without limits.
Differences (of a sequence)
Given a sequence of values f_0, f_1, f_2, ..., the differences are v_j = f_j - f_(j-1). In simple terms, each difference tells you how much the function changed in one step. These are the discrete version of derivatives.
Sums (of differences)
Adding up the differences v_1 + v_2 + ... + v_j recovers f_j - f_0. The middle values cancel (telescoping). Think of it as: if you know all the small changes, you can reconstruct the total change. These are the discrete version of integrals.
Fundamental Theorem of Calculus (preview)
The relationship that sums undo differences, and differences undo sums. In full calculus this becomes: the integral of the derivative gives back the function, and the derivative of the integral gives back the function.
Derivative (informal)
The slope v(t) of the distance graph f(t). It measures how fast f is changing. In discrete terms, it is the difference v_j = f_j - f_(j-1).
Integral (informal)
The area under the velocity graph. It accumulates the total change in f. In discrete terms, it is the sum f_j = f_0 + v_1 + v_2 + ... + v_j.
Tangent line
The straight line that just touches a curve at one point and has the same slope as the curve there. Its equation is y = f(a) + f'(a)(x - a). This idea returns in Sections 2.3, 3.1, and 3.7.
Start with numbers: f = 3, 5, 9. The differences are v = 2, 4. Their sum is 2 + 4 = 6, which equals 9 - 3 (last minus first).
The middle values cancel out. In general: (f_1 - f_0) + (f_2 - f_1) = f_2 - f_0. This telescoping is the heart of the idea.
Taking differences gives you the "velocities." Summing those velocities gives you back the total change in "distance."
This is the Fundamental Theorem in its simplest form: sums undo differences.
The notation builds up in stages. First: specific numbers (3, 5, 9). Then: general numbers f_0, f_1, f_2. Then: a piecewise linear function f(t). Then: a general function.
Similarly for differences: first 5 - 3 and 9 - 5. Then v_1 = f_1 - f_0. Then a piecewise constant function v(t). Then the derivative df/dt.
Recognising this progression helps you see that the formal notation is simply a generalisation of arithmetic you already know.
Point-slope form: f(x) = f(a) + (slope)(x - a). This says "start at the known point (a, f(a)) and move along the line."
The income tax example makes this concrete: tax in the second bracket = tax on $20,350 + (0.28)(income over $20,350). Here a = 20,350, the slope is 0.28, and f(a) is the tax already owed on the first bracket.
Mathematics expresses patterns with symbols (a, x, slope). Specific numbers (20,350 and 0.28) are the special case. The pattern is the point.
If you know all the velocities v_1, v_2, ..., v_j and the starting value f_0, then f_j = f_0 + v_1 + v_2 + ... + v_j.
This is the discrete integral. It is the running total of all the changes, added to the starting position.
If the distances are f_j = j^2 (so 0, 1, 4, 9, 16, ...), then the velocities are v_j = 2j - 1 (so 1, 3, 5, 7, ...). The sum of the first j odd numbers is j^2.
v_j = f_j - f_{j-1}The difference (discrete derivative): how much f changed in one step.
f_j = f_0 + v_1 + v_2 + \cdots + v_jThe running sum (discrete integral): the starting value plus all the accumulated changes.
(f_1 - f_0) + (f_2 - f_1) + \cdots + (f_j - f_{j-1}) = f_j - f_0The telescoping sum. Middle terms cancel. This is the Fundamental Theorem preview.
f(x) = f(a) + f'(a)(x - a)Point-slope form / tangent line approximation. This formula recurs throughout the course.
The income tax system is a piecewise linear function. Each bracket has its own slope (the tax rate), and the formula f(x) = f(a) + (rate)(x - a) calculates the tax in each bracket. The same structure appears whenever pricing has tiers, such as electricity rates or shipping costs that change at certain thresholds.
Students sometimes think that knowing the differences is not enough to recover the original values. It is, provided you also know the starting value f_0. Without f_0, you can only recover the total change, not the actual values.
The telescoping cancellation is not a trick; it is the entire mechanism. Every middle term appears once with a plus sign and once with a minus sign. If you write it out fully, this becomes obvious.
When the text says "the derivative is like differences" and "the integral is like sums," these are not loose analogies. They are exact statements for piecewise linear functions, and the limit process of calculus extends them to curves.
⚠️ The formula f_j = f_0 + v_1 + ... + v_j is used repeatedly. Know it cold.
⚠️ Expect questions that give you a sequence of distances and ask for the velocities (differences), or give velocities and a starting distance and ask you to reconstruct the distances (sums).
⚠️ The telescoping property is tested both here and later in integration. Understand why the middle terms cancel.
True or False: The sum of the differences v_1 + v_2 + ... + v_j always equals f_j - f_0. True.
Fill in the blank: If f = 1, 6, 2, 5, the differences are v = ______, ______, ______. 5, -4, 3.
True or False: You can recover the original values f_0, f_1, ..., f_j from the differences alone, without knowing f_0. False. You need the starting value.
Fill in the blank: The sum of the first 10 odd numbers (1 + 3 + 5 + ... + 19) = ______. 100 (because the sum of the first j odd numbers is j^2).
Q: Given distances f_0 = 0, f_1 = 8, f_2 = 12, f_3 = 14, find the velocities v_1, v_2, v_3.
A: v_1 = 8 - 0 = 8, v_2 = 12 - 8 = 4, v_3 = 14 - 12 = 2.
Q: Using the velocities from the previous question, verify that v_1 + v_2 + v_3 = f_3 - f_0.
A: 8 + 4 + 2 = 14, and f_3 - f_0 = 14 - 0 = 14. They match.
Q: If f_j = j^2, what is v_j? Verify for j = 1, 2, 3.
A: v_j = j^2 - (j-1)^2 = 2j - 1. So v_1 = 1, v_2 = 3, v_3 = 5. Check: 0 + 1 = 1 = 1^2, 1 + 3 = 4 = 2^2, 4 + 5 = 9 = 3^2.
Q: Tax rates increase by 2 percentage points to 0.17, 0.30, and 0.33, but a $500 tax credit is applied. At what income x does the 2% rate increase exactly cancel the $500 credit?
A: At x = $25,000. The 2% increase costs 0.02 x $25,000 = $500, which exactly balances the credit.
The telescoping sum here is the discrete Fundamental Theorem of Calculus, which appears in its full form in Chapter 5. The point-slope equation f(x) = f(a) + f'(a)(x - a) returns as the tangent line (Section 2.3), the linear approximation (Section 3.1), and the starting point for Newton's method (Section 3.7). Recognising that all four appearances are the same formula is, as Strang puts it, "the key idea of differential calculus."
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