Calculus

Author

Chao Ma

Published

February 8, 2026

Notes

Short, structured notes on calculus fundamentals, focused on change, accumulation, and geometric intuition.

Gilbert Strang’s Calculus: Power Series and Euler’s Great Formula Build Taylor series from derivatives at zero, derive the series for $e^x$, $\sin x$, and $\cos x$, then combine them into Euler’s formula and the geometric/logarithmic series.

Gilbert Strang’s Calculus: Linear Approximation and Newton’s Method Use tangent-line linearization to approximate nearby values, then turn the same idea into Newton’s root-finding iteration, with examples from $\sqrt{9.06}$ and $e^{0.01}$.

Gilbert Strang’s Calculus: Growth Rates and Log Graphs Compare linear, polynomial, exponential, factorial, and logarithmic growth, then use log scales and log-log plots to reveal power laws and numerical error rates.

Gilbert Strang’s Calculus: Derivatives of ln y and arcsin(y) Use inverse functions and the chain rule to derive $\frac{d}{dy}(\ln y)=\frac{1}{y}$ and $\frac{d}{dy}(\arcsin y)=\frac{1}{\sqrt{1-y^2}}$, with the logarithm filling the missing $y^{-1}$ power.

Gilbert Strang’s Calculus: Limits and Continuous Functions Formal $\epsilon$-$N$ and $\epsilon$-$\delta$ definitions, indeterminate forms, a $1^\infty$ limit example, L’Hopital’s rule, and the core link between limits and continuity.

Gilbert Strang’s Calculus: Chains and the Chain Rule Chain rule for composite functions with clean examples, intuition from delta-ratio decomposition, and Gaussian first/second derivatives via chain + product rules.

Gilbert Strang’s Calculus: Product Rule and Quotient Rule From rectangle-area increments to formula derivation: product rule, quotient rule, chain-like power patterns for f(x)^n, and worked checks with sqrt(x) and 1/x^n.

Gilbert Strang’s Calculus: Derivative of sin x and cos x A geometric-and-limit proof of $\frac{d}{dx}(\sin x)=\cos x$ and $\frac{d}{dx}(\cos x)=-\sin x$, including the key limit $\lim_{h\to0}\frac{\cos h-1}{h}=0$.

Gilbert Strang’s Calculus: Big Picture Integral Connect derivatives and integrals through sums, differences, and area accumulation; then verify with a worked example from slope to distance.

Gilbert Strang’s Calculus: The Exponential Function Build $e^x$ from the condition $y'=y$, derive the factorial series coefficients, and connect the limit $(1+1/N)^N$ to continuous compounding.

Gilbert Strang’s Calculus: Max, Min, and Second Derivative Use first and second derivatives to classify extrema and find inflection points, with worked examples from $\sin x$ and a cubic.

Gilbert Strang’s Calculus: Big Picture on Derivatives A compact map of derivatives through slopes, limits, and the core function families: power, trigonometric, and exponential.

Gilbert Strang’s Calculus: Highlights A concise tour of derivatives, slopes, second derivatives, exponential growth, extrema, and the integral as accumulation, guided by graphs and intuition.

--- title: "Calculus" author: "Chao Ma" date: "2026-02-08" --- ## Notes Short, structured notes on calculus fundamentals, focused on change, accumulation, and geometric intuition. --- ::: {.content-grid} ::: {.content-card} **[Gilbert Strang's Calculus: Power Series and Euler's Great Formula](power-series-eulers-great-formula.qmd)** Build Taylor series from derivatives at zero, derive the series for $e^x$, $\sin x$, and $\cos x$, then combine them into Euler's formula and the geometric/logarithmic series. ::: ::: {.content-card} **[Gilbert Strang's Calculus: Linear Approximation and Newton's Method](linear-approximation-newtons-method.qmd)** Use tangent-line linearization to approximate nearby values, then turn the same idea into Newton's root-finding iteration, with examples from $\sqrt{9.06}$ and $e^{0.01}$. ::: ::: {.content-card} **[Gilbert Strang's Calculus: Growth Rates and Log Graphs](growth-rates-log-graphs.qmd)** Compare linear, polynomial, exponential, factorial, and logarithmic growth, then use log scales and log-log plots to reveal power laws and numerical error rates. ::: ::: {.content-card} **[Gilbert Strang's Calculus: Derivatives of ln y and arcsin(y)](derivatives-ln-arcsin.qmd)** Use inverse functions and the chain rule to derive $\frac{d}{dy}(\ln y)=\frac{1}{y}$ and $\frac{d}{dy}(\arcsin y)=\frac{1}{\sqrt{1-y^2}}$, with the logarithm filling the missing $y^{-1}$ power. ::: ::: {.content-card} **[Gilbert Strang's Calculus: Limits and Continuous Functions](limits-continuous-functions.qmd)** Formal $\epsilon$-$N$ and $\epsilon$-$\delta$ definitions, indeterminate forms, a $1^\infty$ limit example, L'Hopital's rule, and the core link between limits and continuity. ::: ::: {.content-card} **[Gilbert Strang's Calculus: Chains and the Chain Rule](chain-rule.qmd)** Chain rule for composite functions with clean examples, intuition from delta-ratio decomposition, and Gaussian first/second derivatives via chain + product rules. ::: ::: {.content-card} **[Gilbert Strang's Calculus: Product Rule and Quotient Rule](product-quotient-rule.qmd)** From rectangle-area increments to formula derivation: product rule, quotient rule, chain-like power patterns for f(x)^n, and worked checks with sqrt(x) and 1/x^n. ::: ::: {.content-card} **[Gilbert Strang's Calculus: Derivative of sin x and cos x](derivative-sin-cos.qmd)** A geometric-and-limit proof of $\frac{d}{dx}(\sin x)=\cos x$ and $\frac{d}{dx}(\cos x)=-\sin x$, including the key limit $\lim_{h\to0}\frac{\cos h-1}{h}=0$. ::: ::: {.content-card} **[Gilbert Strang's Calculus: Big Picture Integral](big-picture-integral.qmd)** Connect derivatives and integrals through sums, differences, and area accumulation; then verify with a worked example from slope to distance. ::: ::: {.content-card} **[Gilbert Strang's Calculus: The Exponential Function](exponential-function.qmd)** Build $e^x$ from the condition $y'=y$, derive the factorial series coefficients, and connect the limit $(1+1/N)^N$ to continuous compounding. ::: ::: {.content-card} **[Gilbert Strang's Calculus: Max, Min, and Second Derivative](max-min-second-derivative.qmd)** Use first and second derivatives to classify extrema and find inflection points, with worked examples from $\sin x$ and a cubic. ::: ::: {.content-card} **[Gilbert Strang's Calculus: Big Picture on Derivatives](big-picture-derivatives.qmd)** A compact map of derivatives through slopes, limits, and the core function families: power, trigonometric, and exponential. ::: ::: {.content-card} **[Gilbert Strang's Calculus: Highlights](highlights-of-calculus.qmd)** A concise tour of derivatives, slopes, second derivatives, exponential growth, extrema, and the integral as accumulation, guided by graphs and intuition. ::: :::