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∞

Continuity, Jumps, and Removable Discontinuities

How many separate conditions hide inside the one equation lim f(x) = f(x0), and which of them fails at a jump but not at a hole?

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Your summary note

1. Definition of continuity at $x_0$
  Write $\lim_{x \to x_0} f(x) = f(x_0)$ and list the three ingredients: the limit exists (left and right agree), $f(x_0)$ is defined, and the two are equal.
2. Avoiding $x_0$ keeps the definition non-tautological
  Record that the left side is computed by avoiding $x_0$ while the right side plugs in $x_0$ , and state that continuous functions are exactly the easy plug-in limits.
3. Jump discontinuity
  State that both one-sided limits exist but differ, and give the antenna example where one limit is $1$ and the other is $2$ .
4. Removable discontinuity
  State that the one-sided limits agree but $f(x_0)$ is undefined or set to another value, and record $g(x)=\frac{\sin x}{x}$ with limit $1$ and $h(x)=\frac{1-\cos x}{x}$ with limit $0$ at $x=0$ .
Attempt 1 of 2