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

01Average Rate Times Time: FTC Beats the Mean Value Theorem 02Second Fundamental Theorem: Differentiate an Integral 03Proving FTC1 and FTC2 with the Mean Value Theorem Problem set0/10 Problem set 20/10 MIT problem set0/2

Applications of Integration

01The Natural Log Defined as an Area Under 1/t 02Functions With No Formula: erf, Fresnel and Li(x)03Setting Up the Area Between Two Curves 04Vertical vs Horizontal Slices: One Region, Two Integrals Problem set0/10 Problem set 20/10

01The Disk Method: Volume of a Ball and a Spherical Cap 02The Shell Method: Volume of a Parabolic Cauldron 03Units Matter: Why y = x² Is Not Scale-Invariant Problem set0/10 Problem set 20/10 MIT problem set0/1

01Average Value: Two Heights of a Semicircle 02Weighted Averages: The Cauldron's Average Temperature 03Probability as a Weighted Average: Part over Whole Problem set0/10 Problem set 20/10 MIT problem set0/2

Weighted Averages: The Cauldron's Average Temperature

Why is the water in a cauldron that runs from 100 °C at the bottom to 70 °C at the top only 80 °C on average, not 85?

Loading…

←Previous Average Value: Two Heights of a Semicircle Next Probability as a Weighted Average: Part over Whole →

Your summary note

1. Weighted average $\frac{\int_a^b f w\,dx}{\int_a^b w\,dx}$
  State the formula, show that a constant $c$ averages to $c$ for any weight, and give the share-price example as its discrete version.
2. Heating energy by horizontal disks
  Set up the disk volume $\pi y\,dy$ for the cauldron $y = x^2$ , integrate $\int_0^1 (100 - 30y)\pi y\,dy = 40\pi$ , and convert to about $125{,}000$ kcal.
3. Volume-weighted average temperature of $80^\circ$ C
  Divide $40\pi$ by the volume $\int_0^1 \pi y\,dy = \frac{\pi}{2}$ , and compare with the height average $85^\circ$ C, noting where most of the water sits.
Attempt 1 of 2