Single Variable Calculus

Differentiation

01The Derivative as the Slope of the Tangent Line 02Secant Lines and the Limit Definition of the Derivative 03Differentiating 1/x Straight From the Definition 04Tangents to 1/x and the Triangle of Area 2 05Newton and Leibniz Notation, and the Power Rule Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

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 Problem set0/10 Problem set 20/10 Practice∞

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 Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

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 Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

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 Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Bow and Bowstring: Proving the Two Trig Limits

How does a bow and its bowstring on the unit circle prove that sin(θ)/θ tends to 1 while (1 − cos θ)/θ tends to 0?

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Your summary note

1. The bow-and-bowstring picture on the unit circle
  Draw the unit circle with angle $\theta$ above and below the axis, label the chord $2\sin\theta$ and the arc $2\theta$ , and cancel to $\sin\theta/\theta$ .
2. The straightness principle for short pieces of curves
  Write down that short pieces of curves are nearly straight, and carry the chord-against-arc comparison through to $\sin\theta/\theta\to1$ as $\theta\to0$ , which is limit $B$ .
3. The sagitta $1-\cos\theta$
  Flip the sign to study $(1-\cos\theta)/\theta$ , redraw the sector with a distant vertex, and mark the unit radius, the distance $\cos\theta$ to the bowstring, and the remaining gap.
4. The flattening bow, and limit $A$
  Send the vertex far away, record that the gap shrinks far faster than the arc, and conclude $(1-\cos\theta)/\theta\to0$ , which is limit $A$ .
Attempt 1 of 2