Practice: Why area equals slope

Self-check

Six short questions. Answer each one in your head (or on paper) before opening the collapsible. Trying to retrieve the answer is where the learning sticks; rereading feels productive but does much less.

1. What is the “area function” A(x), and is it really a function?

Show answer

A(x) = ∫_a^x f(t) dt: the area under the curve f, accumulated from a fixed start a up to a movable right end x. Yes, it is genuinely a function: feed it a stopping point x, and it returns the total area built up by then. As x slides right, A(x) grows.

2. What is the one geometric move that gives A'(x) = f(x)?

Show answer

Slide the right end out by a tiny dx. The new area is a thin sliver against the right edge: width dx, height f(x) (the curve’s value there). So A(x + dx) - A(x) ≈ f(x)·dx. Divide by dx to get the difference-quotient ≈ f(x), and take the limit as dx -> 0: A'(x) = f(x). The derivative of the area function is the original curve.

3. Why is A'(x) = f(x) the same thing as the fundamental theorem?

Show answer

Because A'(x) = f(x) says A is an antiderivative of f. Any other antiderivative F differs from A by only a constant (which cancels in a difference), so F(b) - F(a) = A(b) - A(a) = ∫_a^b f(x) dx. The antiderivative trick works precisely because the area function is an antiderivative.

4. Why must any two antiderivatives of f differ by only a constant?

Show answer

If F and A both have derivative f, then (F - A)' = f - f = 0 everywhere. A function whose rate of change is zero at every point never changes, so it is a constant. Hence F - A = C, and the C cancels in F(b) - F(a). (This is also why the indefinite integral carries a + C.)

5. State the “everyday version” of the fundamental theorem.

Show answer

The rate at which a total accumulates is the thing being accumulated. A bucket under a tap: the rate the water level rises equals the tap’s flow at that moment (water = the integral, flow = the function, fill-rate = flow is A'(x) = f(x)). A car’s odometer and speedometer are the same pair: speed is how fast accumulated distance climbs.

6. What is the PDF-and-CDF instance of this in machine learning?

Show answer

The cumulative distribution function F(x) = ∫ f(t) dt is the accumulated probability up to x, an integral of the probability density f(x). By this lesson, F'(x) = f(x): the density is the derivative of the cumulative function, and the cumulative function is the integral of the density. Every PDF/CDF pair is an instance of A'(x) = f(x).

Try it yourself, part 1: verify A’(x) = f(x)

Pen and paper (a calculator helps for the numeric check), about 6 minutes.

(a) For f(x) = x³, the area function from 0 is A(x) = ∫_0^x t³ dt = x⁴/4. (i) Differentiate A(x) and confirm you get f(x). (ii) Check numerically at x = 2: compute the rate [A(2.01) - A(2)]/0.01 and compare to f(2).

(b) For f(x) = cos x, the area function from 0 is A(x) = ∫_0^x cos t dt = sin x. Differentiate A(x) and confirm you get f(x).

Show answer

(a)(i) A(x) = x⁴/4, so A'(x) = 4x³/4 = x³ = f(x). The area function’s slope is the curve.

(a)(ii) A(2) = 2⁴/4 = 4. A(2.01) = (2.01)⁴/4 ≈ 16.3224/4 ≈ 4.0806.

rate ≈ (4.0806 - 4) / 0.01 = 0.0806 / 0.01 = 8.06

That matches f(2) = 2³ = 8 (the small excess shrinks as the step does). The area is growing at the curve’s height, exactly A'(2) = f(2).

(b) A(x) = sin x, so A'(x) = cos x = f(x). The pattern holds for the trig functions too: the slope of the accumulated area is the curve.

Try it yourself, part 2: a density and its cumulative function

About 4 minutes. A probability density on the interval [0, 1] is f(x) = 2x. Its cumulative distribution function is F(x) = ∫_0^x 2t dt = x². Three checks:

1. Confirm f is a valid density (its total area over [0, 1] is 1).
2. Confirm F'(x) = f(x).
3. Find the probability that X lands in [0, 0.5].

Show answer

1. Total area: ∫_0^1 2x dx = [x²]_0^1 = 1 - 0 = 1. Yes, valid density (the area under it is 1).
2. F(x) = x², so F'(x) = 2x = f(x). The density is the derivative of the cumulative function, exactly A'(x) = f(x) from this lesson.
3. P(0 ≤ X ≤ 0.5) = F(0.5) - F(0) = 0.25 - 0 = 0.25. (Equivalently, the area under f from 0 to 0.5.)

This PDF/CDF pair is the fundamental theorem at work: the CDF accumulates the density, and the density is the CDF’s rate of change. Every continuous distribution you meet in machine learning is such a pair.

Flashcards

Nine cards. Click any card to reveal the answer. Use the Print flashcards button to lay out the full set as one card per page, ready to print or save as a PDF for offline review.

Print flashcards

Q. What is the area function A(x)?

A.

A(x) = ∫_a^x f(t) dt: the area under f accumulated from a fixed start a to a movable right end x. Feed it a stopping point, it returns the area built up by then. As x slides right, A(x) grows.

Q. What is the one geometric move that proves A'(x) = f(x)?

A.

Slide the right end by dx: the new area is a thin sliver of width dx and height f(x), so A(x+dx) - A(x) ≈ f(x)·dx. Divide by dx and take the limit: A'(x) = f(x). The derivative of the area function is the curve.

Q. Why is A'(x) = f(x) the fundamental theorem?

A.

It says A is an antiderivative of f. Any antiderivative F differs from A by a constant (which cancels), so F(b) - F(a) = A(b) - A(a) = ∫_a^b f(x) dx. The antiderivative trick works because the area function is an antiderivative.

Q. Why do two antiderivatives of f differ by only a constant?

A.

If F' = A' = f, then (F - A)' = 0 everywhere, and a function with zero rate of change never changes, so F - A is constant. That constant cancels in F(b) - F(a), and it is the + C on an indefinite integral.

Q. State the everyday version of the fundamental theorem.

A.

The rate a total accumulates is the thing being accumulated. A bucket under a tap fills at the tap’s flow rate; a car’s speed is how fast its odometer climbs. Water/distance is the integral, flow/speed is the function, and fill-rate = flow is A'(x) = f(x).

Q. Verify A'(x) = f(x) for f(x) = x².

A.

The area function is A(x) = ∫_0^x t² dt = x³/3. Differentiate: A'(x) = x² = f(x). The slope of the accumulated area is the original curve. (Likewise e^x gives A = e^x - 1, A' = e^x.)

Q. What is the PDF-and-CDF instance of A'(x) = f(x)?

A.

The cumulative distribution F(x) = ∫ f(t) dt (accumulated probability) has derivative F'(x) = f(x), the density. The CDF integrates the density; the density is the CDF’s slope. Every PDF/CDF pair is A'(x) = f(x).

Q. What is the difference between the curve f and its area function A?

A.

f(x) is the height of the graph (the rate); A(x) is the area accumulated under it (the total). They are different functions related by A' = f. Confusing them is a common error.

Q. Why is the FTC not a coincidence to memorize?

A.

Because it follows from one fact: extending an area by dx adds a rectangle of height f(x), so the area grows at rate f(x). Remember the sliver and you can reconstruct why integration and differentiation are inverses.