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一変数の微積分

微分

01微分係数とは何か：接線の傾きとして定義する 02割線から接線へ：極限による微分係数の定義 031/x の導関数：先に約分、極限はそのあと 041/x の接線と、面積がいつも2になる三角形 05微分の記法とべき関数の微分公式問題セット0/10 問題セット20/10 MIT 問題セット0/3

01The Derivative as an Instantaneous Rate of Change 02Rates of Change Without Time: Gradient and Sensitivity 03Easy Limits, 0/0, and One-Sided Limits 04Continuity, Jumps, and Removable Discontinuities 05Infinite Discontinuities: 1/x and Its Derivative 06Differentiable Implies Continuous 問題セット0/10 問題セット20/10

01Two Kinds of Formula, and the Derivative of Sine 02The Derivative of Cosine, and Two Limits at Zero 03Bow and Bowstring: Proving the Two Trig Limits 04Why the Trig Derivatives Hold Only in Radians 05A Geometric Proof: Sine as a Height on the Circle 06Finishing the Proof, and the Product and Quotient Rules 問題セット0/10 問題セット20/10 MIT 問題セット0/1

01Proving the Product Rule: Change One Factor at a Time 02The Quotient Rule, and the Power Rule for Negative Exponents 03The Chain Rule: Differentiating a Function of a Function 04Higher Derivatives: The Sine Cycle and Three Notations 05The nth Derivative of xⁿ Is n Factorial, by Induction 問題セット0/10 問題セット20/10 MIT 問題セット0/2

01The Power Rule for Rational Exponents 02The Slope of a Circle, Two Ways 03Implicit Differentiation of a Quartic Curve 04Inverse Functions and the Reflection Across y = x 05Derivatives of the Arctangent and Arcsine 問題セット0/10 問題セット20/10 MIT 問題セット0/1

01The Derivative of aˣ and the Mystery Number M(a)02Defining e by Slope One, and the Derivative of ln x 03M(a) = ln a: Differentiating aˣ and xˣ 04Why (1 + 1/n)ⁿ Approaches e 問題セット0/10 問題セット20/10 MIT 問題セット0/1

微分の応用

01Linear Approximation: A Curve Is Its Tangent Line 02Five Linear Approximations at Zero 03Combining Approximations, from Algebra to GPS Clocks 04Quadratic Approximation: The Best-Fit Parabola 問題セット0/10 問題セット20/10 MIT 問題セット0/1

01Linear Approximation as a Proportionality Rule 02Quadratic Approximation: Why the ½, and Products 03Curve Sketching: The Sign of f′ and Critical Points 04Curve Sketching: Ends, Concavity and Inflection 問題セット0/10 問題セット20/10 MIT 問題セット0/1

01Sketching Without Critical Points: (x+1)/(x+2)02A Five-Step Curve-Sketching Scheme, Applied to x / ln x 03x / ln x: Inflection Point and Where Max and Min Can Occur 問題セット0/10 問題セット20/10

01Two Squares From One Wire: The Critical Point Gives the Least Area 02The Open Box of Fixed Volume: Least Material When x = 2y 03Related Rates: The Police Radar and the Conical Tank 問題セット0/10 問題セット20/10 MIT 問題セット0/4

01The Hanging Ring: Why the String Makes Equal Angles 02Newton's Method: Tangent Lines That Find the Square Root of Five 03Why Newton's Method Doubles Its Digits, and How It Fails 04The Mean Value Theorem: Somewhere You Hit Your Average Speed 05Positive Derivative Means Increasing: Proving eˣ Exceeds 1 + x 問題セット0/10 問題セット20/10 MIT 問題セット0/2

積分

01Differentials: dy = f′(x) dx and the Cube Root of 64.1 02Antiderivatives Differ Only by a Constant 03Integration by Substitution and Advanced Guessing 問題セット0/10 問題セット20/10 MIT 問題セット0/1

01Separation of Variables: dy/dx = −xy and the Bell Curve 02General Solutions: Signs, Constants and Initial Conditions 03Orthogonal Trajectories: Parabolas and Crossing Ellipses 問題セット0/10 問題セット20/10 MIT 問題セット0/1

01The Definite Integral: Area as a Limit of Rectangle Sums 02Area Under y = x²: Squeezing the Sum of Squares 03Sigma Notation and the Area Pattern: b, b²/2, b³/3 04Riemann Sums: the Definite Integral and a Year of Loans 問題セット0/10 問題セット20/10 MIT 問題セット0/1

01The Fundamental Theorem of Calculus: Area in One Line 02Net Change and Signed Area: Integrating Velocity 03Six Rules for Definite Integrals and Bounds on eᵇ 04Substitution in Definite Integrals: Change the Limits 問題セット0/10 問題セット20/10 MIT 問題セット0/1

このページは日本語版を準備中です

The Hanging Ring: Why the String Makes Equal Angles

Why does a ring hanging on a string settle where its two pieces make equal angles, and what does that say about light in an ellipse?

日本語版の動画は準備中です（英語版を表示しています）

読み込み中…

←前へ Related Rates: The Police Radar and the Conical Tank 次へ Newton's Method: Tangent Lines That Find the Square Root of Five →

まとめノート

1. Setting up the minimization with a fixed-length constraint
  Place the ends at $(0,0)$ and $(a,b)$ and the ring at $(x,y)$ , write the length constraint as two Pythagorean distances summing to $L$ , and state that the lowest point satisfies $y'(x)=0$ .
2. Implicit differentiation at the critical point
  Differentiate the constraint implicitly in $x$ , set $y'=0$ , and reduce it to $\frac{x}{\sqrt{x^2+y^2}}=\frac{a-x}{\sqrt{(x-a)^2+(y-b)^2}}$ , noting that the ellipse equation is never needed.
3. Equal angles $\alpha=\beta$ and equal tension
  Read each side of the equation as $\sin\alpha$ and $\sin\beta$ from the two right triangles, with angles measured from the vertical, conclude $\alpha=\beta$ , and record the equal-tension interpretation.
4. The reflection property of the ellipse
  State that at every point of an ellipse the segments to the two foci make equal angles with the tangent line, and record the light-ray and elliptical golf-hole consequences.
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