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

Area Under y = x²: Squeezing the Sum of Squares

How can two pyramids of unit blocks trap 1² + 2² + ⋯ + n² tightly enough to prove the area under x² is b³/3?

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Your summary note

1. The staircase pyramid of unit blocks
  Draw the pyramid with levels $n$ by $n$ , $(n-1)$ by $(n-1)$ , down to $1$ by $1$ , and record that its volume equals $1^2 + 2^2 + \cdots + n^2$ .
2. Inner and outer pyramid bounds
  Write the inner-pyramid bound $\sum > \frac{1}{3}n^3$ and the outer-pyramid bound $\sum < \frac{1}{3}(n+1)^3$ , giving the base and height of each solid pyramid.
3. The squeeze and the result $\int_0^b x^2\,dx = \frac{b^3}{3}$
  Divide both bounds by $n^3$ , show both ends tend to $\frac{1}{3}$ , and carry the limit through the factored rectangle sum to the area $\frac{b^3}{3}$ .
4. Factoring out the easy part as the general method
  Record the strategy of pulling out $\frac{b^3}{n^3}$ and analysing the remaining sum on its own, and note that it applies systematically to sums of this kind.
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