Math13 slides0 views

Calculus — The Mathematics of Change

Newton + Leibniz, ~1665–1684

Standalone Download

Shared with ShipslidesCreate your own deck →

About this HTML presentation

This Shipslides page presents Calculus — The Mathematics of Change as an interactive HTML presentation deck in the Mathematics catalog with 13 slides. The share page keeps the uploaded deck sandboxed while exposing readable context, topics, and a slide outline for viewers and search engines.

Newton + Leibniz, ~1665–1684 Key sections include: CALCULUS / The mathematics of change; The Classical Problem; Limits — the Formal Foundation; The Derivative; Geometric Interpretation; The Derivative Rules; The Integral; The Fundamental Theorem; Newton vs. Leibniz; Into Many Dimensions.

Key sections

01CALCULUS / The mathematics of change
02The Classical Problem
03Limits — the Formal Foundation
04The Derivative
05Geometric Interpretation
06The Derivative Rules
07The Integral
08The Fundamental Theorem
09Newton vs. Leibniz
10Into Many Dimensions
11Applications
12Beyond Elementary Calculus
13Further Study

Topics covered

catalog math calculus

Related decks

Mathematical Cryptography

Game Theory — Strategy when others strategize too

Geometry — A Drafting Table Deck

Linear Algebra — Vectors, Matrices, Transformations

Slide outline

01CALCULUS / The mathematics of change
02The Classical Problem
03Limits — the Formal Foundation
04The Derivative
05Geometric Interpretation
06The Derivative Rules
07The Integral
08The Fundamental Theorem
09Newton vs. Leibniz
10Into Many Dimensions
11Applications
12Beyond Elementary Calculus
13Further Study

Page data

Canonical: https://shipslides.com/d/catalog-math-calculus
Category: Mathematics
Size: 24.5 KB
Updated: 2026-05-17
LLM text: https://shipslides.com/d/catalog-math-calculus/llms.txt

Presentation Transcript

Detailed slide-by-slide text content extracted from this presentation.

Slide 01

CALCULUS / The mathematics of change

Anno Domini MDCLXV
Newton + Leibniz, ~1665–1684
A method of fluxions & fluents — the secret arithmetic of motion, area, and the infinitesimal.

Slide 02

The Classical Problem

Given a curve y = f(x), how steep is it at a single point P? The slope between two points is easy:
slope = ( y2 − y1 ) / ( x2 − x1 )
But at a single point there is no second point. The Greeks could draw tangents to circles and parabolas; no one had a general method.
how to find the slope when the second point is the first?

Slide 03

Limits — the Formal Foundation

The trick: do not arrive at the point. Approach it.
limx→a f(x) = L
For every ε > 0 there exists a δ > 0 such that whenever 0 < |x − a| < δ, we have |f(x) − L| < ε.
Cauchy & Weierstraß gave this rigour two centuries after Newton — the calculus worked long before it was understood.
ε–δ was scandalously late!

Slide 04

The Derivative

Let the second point slide toward the first. The slope of the chord becomes the slope of the tangent in the limit.
f′(x) = limh→0 f(x + h) − f(x)h
Newton's notation: ỹ (a dot above for time-derivatives, his fluxion).
Leibniz's notation: dydx — a ratio of infinitesimals.
Both notations survive. Leibniz's, more flexible, dominates analysis. Newton's lingers in physics.

Slide 05

Geometric Interpretation

The derivative f′(x) at a point is the slope of the line that just kisses the curve there — the tangent.
a line that touches but does not cross

Slide 06

The Derivative Rules

Power rule
ddx xn = n xn−1
Product rule
(fg)′ = f′g + f g′
Chain rule
ddx f(g(x)) = f′(g(x)) · g′(x)
From these three, with sin, cos, ex, ln, every elementary function may be differentiated. The whole machinery is mechanical.
three rules, one universe of functions

Slide 07

The Integral

The other ancient problem: find the area under a curve. Slice it into rectangles, sum, then take the limit.
∫ab f(x) dx = limn→∞ Σi=1n f(xi) Δx
∫ is a stretched S — for summa

Slide 08

The Fundamental Theorem

The deep miracle: the slope problem and the area problem are inverse operations.
If F′(x) = f(x), then ∫ab f(x) dx = F(b) − F(a)
ddx ∫ax f(t) dt = f(x)
Two seemingly unrelated geometric problems — the tangent and the quadrature — turn out to be the same problem read in opposite directions. This is the central jewel of the calculus.
the bridge between two ancient problems

Slide 09

Newton vs. Leibniz

Two men, two countries, the bitterest priority dispute in mathematics.
Isaac Newton (1665–66)
Developed the "method of fluxions" during the plague years at Woolsthorpe. Did not publish for two decades. Used the dot notation ỹ.
Motivated by physics: motion, gravity, the Principia (1687).
Gottfried Leibniz (1684)
Published first, in Acta Eruditorum. Invented dx, dy, ∫ — notation so superior it is still ours today.
A philosopher's calculus: symbolic, algorithmic, teachable.
"Second inventors have no rights." — Newton (anonymously)
The Royal Society's 1712 report ruled for Newton. Both, we now know, discovered it independently.

Slide 10

Into Many Dimensions

Functions of several variables need partial derivatives — slope along one axis at a time.
∂f∂x, ∂f∂y, ∂f∂z
The gradient ∇f points in the direction of steepest ascent. Divergence ∇·F measures spread; curl ∇×F measures rotation.
∇ — "nabla", a Hebrew harp

Slide 11

Applications

Physics. Newton's F = m a is a differential equation; Maxwell's electromagnetism, Einstein's general relativity, Schrödinger's wave equation — all calculus.
Engineering. Stress, flow, heat, signals — every continuous system is described by derivatives or integrals.
Economics. Marginal cost, marginal utility — derivatives in disguise. Optimisation under constraint via Lagrange multipliers.
Probability & statistics. Densities integrated, expected values, moment-generating functions.
Machine learning. Gradient descent: take the derivative of the loss, step opposite the gradient. Backpropagation is the chain rule, applied at scale.
Three centuries on, every quantitative science speaks the language Newton and Leibniz invented.
backprop = chain rule + bookkeeping

Slide 12

Beyond Elementary Calculus

Differential equations
Equations relating a function to its derivatives. Solve them and you predict orbits, populations, voltages, epidemics.
Calculus of variations
Find the function — not the number — that minimises a quantity. The brachistochrone, geodesics, the principle of least action.
Real analysis
Cauchy, Riemann, Weierstrass, Lebesgue — the rigorous re-foundation. Measure theory, uniform convergence, the Lebesgue integral that handles wilder functions than Riemann's.
Complex & functional analysis
Calculus over ℂ, then over spaces of functions. Modern physics and modern probability live here.
the calculus has many calculuses

Slide 13

Further Study

"If I have seen further it is by standing on the shoulders of Giants." — Newton, 1675
Reading
Spivak, Calculus — the rigorous undergraduate classic.
Courant & John, Introduction to Calculus and Analysis.
Stewart, Calculus: Early Transcendentals — the standard service text.
Bell, Men of Mathematics, ch. on Newton & Leibniz (with caution).
Edwards, The Historical Development of the Calculus.
Watch
Essence of Calculus — 3Blue1Brown (YouTube)
Fundamental Theorem of Calculus (YouTube)
— finis —

Remove this deck