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∞

Rates of Change Without Time: Gradient and Sensitivity

When the independent variable is not time, what does a derivative measure — and how does it turn a ranging error into a GPS position error?

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Your summary note

1. Derivatives with a non-time variable
  State that the independent variable need not be time, and record the temperature gradient $\frac{dT}{dx}$ as the quantity that drives air flow and weather change.
2. GPS flat-earth setup
  Describe the model where measured distance $h$ to the satellite is used to deduce horizontal distance $L$ , and note that $h$ carries an uncertainty $\Delta h$ .
3. Sensitivity formula $\frac{\Delta L}{\Delta h} \approx \frac{dL}{dh}$
  Write the approximation $\Delta L \approx \frac{dL}{dh}\cdot\Delta h$ and record that it converts the measurement error in $h$ into a bound on the deduced error in $L$ .
Attempt 1 of 2