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

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

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

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

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

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

Applications of Differentiation

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

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

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 Problem set0/10 Problem set 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 Problem set0/10 Problem set 20/10 MIT problem set0/4

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

Integration

01Differentials: dy = f′(x) dx and the Cube Root of 64.1 02Antiderivatives Differ Only by a Constant 03Integration by Substitution and Advanced Guessing Problem set0/10 Problem set 20/10 MIT problem set0/1

01Separation of Variables: dy/dx = −xy and the Bell Curve 02General Solutions: Signs, Constants and Initial Conditions 03Orthogonal Trajectories: Parabolas and Crossing Ellipses Problem set0/10 Problem set 20/10 MIT problem set0/1

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

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

Linear Approximation as a Proportionality Rule

Why is a linear approximation really a statement of proportionality, and why do engineers build GPS and forecasts out of such rules of thumb?

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Your summary note

1. Time dilation as $\frac{\Delta T}{T} \approx \frac{1}{2}\frac{v^2}{c^2}$
  Starting from $T' \approx T\left(1 + \frac{1}{2}\frac{v^2}{c^2}\right)$ , subtract $T$ and divide by $T$ , then record that both sides are dimensionless and that $\frac{1}{2}$ is the proportionality factor.
2. The pitching rule $\Delta v \approx a\,\Delta h$
  Write the rule with $a$ as the rate of change of pitch speed with mound height, and state the condition under which the approximation is treated as an equality for the problem.
3. Linear rules of thumb in multi-effect engineering
  List the effects in satellite timing design (relative speed by angle of sight, Earth's rotation, differing gravity) and the other fields named, and state the role simple linear rules play there.
4. Dependence of the constant $a$ on the base point
  Record that $a$ is a rate of change at a fixed base point, illustrate with pitch speeds of 80, 90 and 100 mph, and state that a small base-point change is second-order.
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