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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

Positive Derivative Means Increasing: Proving eˣ Exceeds 1 + x

How does the Mean Value Theorem prove that a positive derivative means increasing, and turn that into a proof that eˣ > 1 + x?

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 The Mean Value Theorem: Somewhere You Hit Your Average Speed 次へ Differentials: dy = f′(x) dx and the Cube Root of 64.1 →

まとめノート

1. Sign of $f'$ and increasing, decreasing or constant
  Rewrite the MVT as $f(b)=f(a)+f'(c)(b-a)$ with $a<b$ , and prove from it the three facts for $f'>0$ , $f'<0$ and $f'=0$ .
2. The MVT versus linear approximation
  Contrast $\frac{\Delta f}{\Delta x}\approx f'(a)$ with the exact MVT equation, and write the bound $\min f'\le\frac{f(b)-f(a)}{b-a}\le\max f'$ over $a\le x\le b$ as a statement about speeds.
3. Proving $e^x>1+x$ for $x>0$
  Write out the proof with $f(x)=e^x-(1+x)$ : its value at $0$ , its derivative, the sign of that derivative, and the conclusion.
4. Iterating to $e^x>1+x+\frac{x^2}{2}$ and beyond
  Prove the second inequality with $g(x)=e^x-(1+x+\frac{x^2}{2})$ , using the first inequality for $g'$ , then write the next polynomial bound and the limiting series for $e^x$ .
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