# Calculus — The Mathematics of Change Canonical URL: https://shipslides.com/d/catalog-math-calculus Raw viewer URL: https://content.shipslides.com/d/catalog-math-calculus/raw Category: Mathematics Slides: 13 Updated: 2026-05-17T20:55:53.795Z Tags: catalog, math, calculus ## Summary 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. ## Slide Outline 1. CALCULUS / The mathematics of change 2. The Classical Problem 3. Limits — the Formal Foundation 4. The Derivative 5. Geometric Interpretation 6. The Derivative Rules 7. The Integral 8. The Fundamental Theorem 9. Newton vs. Leibniz 10. Into Many Dimensions 11. Applications 12. Beyond Elementary Calculus 13. Further Study ## Slide Transcript ### Slide 1: 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 2: 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 3: 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 4: 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 5: 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 6: 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 7: 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 8: 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 9: 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 — ## Related Decks - [Mathematical Cryptography](https://shipslides.com/d/catalog-math-cryptography) - [Game Theory — Strategy when others strategize too](https://shipslides.com/d/catalog-math-game-theory) - [Geometry — A Drafting Table Deck](https://shipslides.com/d/catalog-math-geometry) - [Linear Algebra — Vectors, Matrices, Transformations](https://shipslides.com/d/catalog-math-linear-algebra)