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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

01瞬間の変化率としての導関数 ― 平均の速さの二倍 02時間のない変化率：気温の勾配と GPS の感度 03代入で求まる極限、0/0、片側極限 04連続性とは二つの数の比較：跳躍不連続点と除去可能な不連続点 05無限大に発散する不連続点：1/x とその導関数 06微分可能なら連続：一行で終わる証明と、0 でわらない理由問題セット0/10 問題セット20/10

01二種類の微分公式：和の微分とサインの導関数 02コサインの導関数と、ゼロでの二つの極限 03弓と弦：一枚の図で証明する二つの三角関数の極限 04どちらが速く縮むか：三角関数の微分がラジアンでしか成り立たない理由 05サインは高さ：小さな直角三角形で読むサインの導関数 06九十度の回転で証明を完成：積の微分法と商の微分法問題セット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

01Average Rate Times Time: FTC Beats the Mean Value Theorem 02Second Fundamental Theorem: Differentiate an Integral 03Proving FTC1 and FTC2 with the Mean Value Theorem 問題セット0/10 問題セット20/10 MIT 問題セット0/2

積分の応用

01The Natural Log Defined as an Area Under 1/t 02Functions With No Formula: erf, Fresnel and Li(x)03Setting Up the Area Between Two Curves 04Vertical vs Horizontal Slices: One Region, Two Integrals 問題セット0/10 問題セット20/10

01The Disk Method: Volume of a Ball and a Spherical Cap 02The Shell Method: Volume of a Parabolic Cauldron 03Units Matter: Why y = x² Is Not Scale-Invariant 問題セット0/10 問題セット20/10 MIT 問題セット0/1

01Average Value: Two Heights of a Semicircle 02Weighted Averages: The Cauldron's Average Temperature 03Probability as a Weighted Average: Part over Whole 問題セット0/10 問題セット20/10 MIT 問題セット0/2

このページは日本語版を準備中です

Average Value: Two Heights of a Semicircle

Why does the unit semicircle have average height π/4 measured along its diameter but only 2/π measured along its arc?

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 Units Matter: Why y = x² Is Not Scale-Invariant 次へ Weighted Averages: The Cauldron's Average Temperature →

まとめノート

1. From sample average to $\frac{1}{b-a}\int_a^b f(x)\,dx$
  Write the Riemann-sum derivation with $\Delta x = \frac{b-a}{n}$ , dividing both sides by $b-a$ so the left side becomes $\frac{y_1 + \cdots + y_n}{n}$ in the limit.
2. The average of a constant
  Compute $\frac{1}{b-a}\int_a^b c\,dx = c$ as a rectangle's area divided by its width, and record the matching discrete check $\frac{nc}{n} = c$ .
3. Semicircle average height over $x$ : $\frac{\pi}{4}$
  Set up $\frac12\int_{-1}^{1}\sqrt{1-x^2}\,dx$ and evaluate it using the semicircle's area $\frac{\pi}{2}$ .
4. Semicircle average height over arc length: $\frac{2}{\pi}$
  Work $\frac{1}{\pi}\int_0^{\pi}\sin\theta\,d\theta = \frac{2}{\pi}$ , show $\frac{2}{\pi} < \frac{\pi}{4}$ via $\pi^2 > 8$ , and state that the two answers describe two different random experiments.
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