Calculus

The Gradient of a Curve #

A different steepness at every point.

A curve has a different gradient at every point, so one number cannot describe it

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A straight line has one gradient, the same everywhere along it.

A curve is flat near the bottom and steep further out. So what is its gradient?

Ask instead for the gradient at one point, and draw the line that just touches the curve there: the tangent.

Limits #

What a value closes in on but never reaches.

A limit is the value a sequence or a function closes in on, even when it never quite arrives

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Halve the gap each time and you close in on 2 without ever reaching it.

The gradient of a chord through two nearby points is close to the gradient at one point.

Slide the two points together and the chord settles onto the tangent.

From First Principles #

Slide two points together and read the gradient.

The derivative is the limit of the chord gradient as the two points merge

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Take two points h apart. Rise over run gives the gradient of the chord.

For $f\left(x\right)=x^2$f(x) = x², expand the numerator, cancel, and divide by h, leaving 2x + h.

Now shrink h. Each chord lies closer to the tangent than the one before.

At the limit only the tangent is left, and its gradient is 2x.

Differentiating Powers #

Multiply by the power, then drop it by one.

Multiply by the power and drop the power by one

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First principles gave 2x for $x^2$x². Now look at the pattern.

Run the same first principles on $x^3$x³: expand, cancel, divide, shrink h.

And $x^3$x³ gives $3x^2$3x². The power comes down in front.

So the rule is: multiply by the power, then take one off it.

Tangents and Normals #

The derivative is the tangent gradient.

The derivative at a point gives the gradient of the tangent there

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At x = 2 on $y=x^2$y = x², the derivative 2x gives a gradient of 4.

The curve passes through (2, 4), so the tangent is y = 4x − 4.

Perpendicular gradients multiply to −1, so the normal has gradient $-1/m$−1 / m.

The normal crosses the tangent at a right angle, with gradient $-\frac{1}{4}$−1/4.

Stationary Points #

Where the curve levels off into a peak or trough.

Where the gradient is zero the curve has leveled off into a peak or a trough

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At the very bottom of this curve the tangent is horizontal, so its gradient is zero.

So set the derivative to zero and solve. Here that gives x = 0.

The tangent is also horizontal at a peak. Whether a stationary point is a peak or a trough depends on the curve.

Integration #

Differentiation run backwards.

Integration undoes differentiation, so the power goes up instead of down

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Differentiating $x^3$x³ gave $3x^2$3x². Now run it backwards.

Add one to the power, then divide by the new power — the rule run in reverse.

The c shifts the curve up. Both tangents at x = 1 still have gradient 2.

Area Under a Curve #

Thinner and thinner rectangles, taken to a limit.

Integrating between two values gives the area under the curve between them

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Slice the region under $y=x^2$y = x² into rectangles and add their areas. The total is close, but not exact.

Thinner rectangles fit the curve better, and the error keeps shrinking.

As the rectangles get thinner without limit, the sum becomes the integral, and that gives the area exactly.

Integrating $x^2$x² gives $\frac{x^3}{3}$x³/3, so take its value at 3 and subtract its value at 0.

Two Infinite Sums, Opposite Fates #

One settles on a number, one never stops.

One endless sum settles on a number while another, with terms just as small, grows without limit

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Halve the gap each time and the total closes in on exactly 1.

Now add the reciprocals of the whole numbers instead. These terms also shrink toward 0.

But group them and each block adds more than a half, so the total grows without limit.