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∞

The Derivative as an Instantaneous Rate of Change

Why does a falling pumpkin hit the ground at exactly twice the average speed of its whole four-second drop?

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Your summary note

1. Average versus instantaneous rate of change
  Write $\frac{\Delta y}{\Delta x}$ as the average rate of change over an interval and record that its limit as $\Delta x \to 0$ is the instantaneous rate $\frac{dy}{dx}$ .
2. Rates of change in physics
  Record that $\frac{dq}{dt}$ is electrical current and $\frac{ds}{dt}$ is speed as examples of derivatives interpreted as rates.
3. The pumpkin drop $h = 80 - 5t^2$
  Compute the average speed $\frac{\Delta h}{\Delta t} = \frac{0-80}{4-0} = -20$ m/s, differentiate to get $\frac{dh}{dt} = -10t$ , and evaluate $h'(4) = -40$ m/s at impact.
4. Average tells little about the moment that matters
  Record that the impact speed $-40$ m/s is twice the average $-20$ m/s, and note the power rule pattern extends to $n=0$ and $n=-1$ .
Attempt 1 of 2