Writing
Notes on mathematics and statistics
I write these primarily for my students, though they do not always stay within the boundaries of an AP course. Some address a recurring difficulty in notation, interpretation, or method. Others follow an idea further than the syllabus requires.
AP Calculus
31 articles
AB & BCFoundations
What a limit claims, and what it does not
A limit describes what happens near a point, not necessarily what happens at the point itself.
AB & BCMechanical
Notation in AP Calculus
The notation and justification habits that make mathematical reasoning visible on free-response work
AB & BCMechanical
Indeterminate forms and the algebra that resolves them
When substitution produces 0/0, the next step is to rewrite the expression rather than treat the form as an answer
AB & BCFoundations
Breaking continuity one condition at a time
Continuity has three conditions. Each type of discontinuity records which part of the definition failed
AB & BCMechanical
Limits at infinity and end behavior
Limits at infinity describe end behavior. For rational functions, the leading terms determine what survives
AB & BCMechanical
Invoking the Intermediate Value Theorem
The theorem guarantees that a value is reached somewhere on an interval. It does not tell you where or how many times
AB & BCFoundations
The derivative as a limit
A derivative is the limit of secant slopes as the second point approaches the first
AB & BCFoundations
Where a function fails to be differentiable
Corners, cusps, vertical tangents, and discontinuities can all prevent a two-sided derivative from existing
AB & BCMechanical
Choosing a differentiation rule
Differentiation is often easier once the expression has been classified and simplified before any rule is applied
AB & BCMechanical
The chain rule, layer by layer
Composite functions are differentiated one layer at a time, with each layer contributing a factor.
AB & BCMechanical
Implicit differentiation
Implicit differentiation uses the chain rule to find slopes on relations that are not solved for y
AB & BCMechanical
Derivatives of inverse functions
Corresponding tangent slopes of a function and its inverse are reciprocals, with the derivative evaluated at the corresponding input
AB & BCFoundations
Why negative acceleration is not slowing down
A particle speeds up when velocity and acceleration have the same sign, and slows down when their signs differ
ABMechanical
Related-rates problems are mostly translation. Name the changing quantities, relate them, differentiate with respect to time, and only then use the snapshot values
AB & BCFoundations
Over or under: reading a linearization
A linearization uses the tangent line to approximate a nearby function value. Concavity determines whether the estimate is high or low
AB & BCMechanical
Checking the form before L'Hospital's rule
L'Hospital's rule applies only after the quotient has been shown to have an appropriate indeterminate form
ABMechanical
Reading the graph of f′
A graph of f' tells you where f increases, decreases, turns, and changes concavity. The main task is keeping the two functions separate.
AB & BCMechanical
Two existence theorems, and what they refuse to tell you
The Mean Value Theorem and Extreme Value Theorem guarantee that something exists under stated hypotheses. Neither theorem tells you where it occurs
AB & BCMechanical
Optimization, and the step that is not calculus
Optimization problems depend on a correct objective, constraint, and domain before the derivative is ever used
ABFoundations
Riemann sums and the definition of the integral
A definite integral is the limit of approximating sums. Left, right, midpoint, and trapezoidal estimates differ only in how each slice is represented
AB & BCFoundations
Reading an accumulation function off the graph of its integrand
If g(x) = ∫ₐˣ f(t) dt, then the graph of f tells you the slope and concavity of g, while signed area determines its values
ABFoundations
The Fundamental Theorem of Calculus from first principles
Build an accumulation function from area, then differentiate it. The result is the original integrand.
BCMechanical
Integration by parts and partial fractions
Two integration techniques for expressions that do not yield directly to substitution. One reverses the product rule. The other rewrites a rational function into simpler pieces
BCMechanical
When an unbounded region has finite area
An improper integral is defined through a limit. An infinite interval or an unbounded integrand does not by itself determine whether the integral converges
BCFoundations
Euler's method and the effect of step size
Euler's method follows a differential equation one tangent-line step at a time. Smaller steps usually reduce the accumulated error
BCFoundations
Logistic growth, read without solving it
The logistic differential equation reveals its equilibria, carrying capacity, fastest growth, and concavity before the equation is solved
AB & BCFoundations
Area between curves and average value
Average value and area between curves both depend on setting up the correct integrand before evaluating the integral
BCFoundations
Parametric, vector, and polar: three systems, one calculus
Parametric, vector-valued, and polar curves use familiar derivative and integral ideas with a different way of describing position
BCFoundations
The harmonic series and conditional convergence
Terms can approach zero while their series diverges. Alternating signs can restore convergence, and conditional convergence makes the order of terms matter
BCMechanical
Choosing a convergence test
The form of a series usually suggests which convergence test to try first. A reliable sequence keeps the tests from becoming a disconnected list
BCFoundations
Approximation by Taylor polynomials
A Taylor polynomial matches a function and its derivatives at one point. Increasing the degree improves the local approximation, while convergence determines how far that approximation extends.
AP Precalculus
6 articles
Foundations
Increasing at a decreasing rate
Whether a function is increasing and whether its rate of change is increasing are separate questions. A table of average rates makes the distinction clear.
Foundations
What the factored form tells you
Factored form reveals zeros, multiplicity, holes, vertical asymptotes, and much of a function's end behavior before the graph is drawn
Foundations
The four parameters of transformation
In g(x) = a f(b(x - h)) + k, the outside parameters act on outputs and the inside parameters act on inputs. That distinction explains the direction and scale of each transformation
Foundations
Functions inside functions
Composition sends the output of one function into another. Order matters, and the domain of the composite has to satisfy both functions
Mechanical
Logarithms undo exponentials
A logarithm is an exponent. Exponential and logarithmic statements describe the same relationship in opposite directions
Foundations
The unit circle and the sine curve
The sine graph records the vertical coordinate of a point moving around the unit circle. Its amplitude, period, zeros, and symmetry all follow from that motion.
AP Statistics
16 articles
Foundations
Describing a distribution in the exam's own words
Shape, center, variability, and unusual features should be described in context. The choice of summary statistics depends on the shape of the distribution.
Foundations
What moves the least-squares line
Residuals, leverage, influence, correlation, and least squares become easier to distinguish when one point can be moved by hand
Foundations
Simpson's paradox and the lurking variable
An overall association can reverse after data are separated into meaningful subgroups. The reversal comes from unequal weighting across those groups
Foundations
Sampling and bias
A larger sample reduces sampling variability. It does not repair a biased sampling method
Foundations
Independence and mutual exclusivity
Mutual exclusivity is about overlap. Independence is about whether learning one event changes the probability of the other
Foundations
Conditional probability and the base rate
Sensitivity and the probability of disease given a positive test are different conditional probabilities. Prevalence determines how far apart they can be.
Foundations
Expected value is not a value you expect
Expected value is a probability-weighted long-run average. It need not be a possible outcome of a single trial
Foundations
The Central Limit Theorem in simulation
Sample means from a skewed population become increasingly normal as sample size grows, while their spread decreases according to 1/√n
Mechanical
A distribution explorer
Enter a distribution and a region, then compare the probability, cutoff, graph, and TI-84 command in one place
Mechanical
Writing parameters in AP Statistics
Define the population quantity before beginning inference. The symbol, variable, and population should all be explicit
Foundations
The meaning of 95% confidence
A confidence level describes the long-run success rate of the interval-producing method, not a probability attached to one finished interval.
Foundations
One template for every interval
Most inference procedures in the course are built from two general forms. The procedure changes mainly through the standard error
Foundations
What a p-value cannot tell you
A p-value measures how unusual the observed result would be if the null hypothesis were true. It is not the probability that the null is true
Foundations
The two ways a test can be wrong
Type I error is a false rejection of the null. Type II error is a failure to detect a false null. Power is the probability of detecting the effect when it is real
Mechanical
Which chi-square test? Independence or homogeneity
The two chi-square tests use the same calculation. The study design determines whether the question is about association within one sample or distributions across several groups
Mechanical
Which inference procedure?
Choose an inference procedure from the response type, the number and relationship of groups, and whether the goal is estimation or hypothesis testing.
Looking ahead
9 articles
AP Calculus AB & BC
Newton's method and its basins of attraction
Newton's method turns tangent lines into an iterative root-finding algorithm. For functions with several roots, the starting value can determine which root the method finds.
AP Statistics
Probability against intuition
Monty Hall, the birthday problem, and Simpson's paradox all become less mysterious once the conditioning information is written explicitly
AP Statistics
Buffon's needle and the estimation of pi
Drop a needle across parallel lines, count the crossings, and a geometric probability produces an estimate of π
AP Statistics
Benford's law and the distribution of first digits
In many datasets spanning several orders of magnitude, smaller leading digits occur more often than larger ones. The pattern is logarithmic rather than uniform
All courses
An introduction to Fourier series
A periodic function can be represented by a sum of sine and cosine waves. Adding harmonics shows how increasingly complex shapes can be built from simple frequencies
AP Calculus BC
After BC: multivariable calculus
Functions become surfaces, derivatives become directional, and integrals extend over regions and volumes. Much of BC reappears in a higher-dimensional setting.
AP Calculus BC
After BC: differential equations
Differential equations model systems by specifying how their state changes. The college course moves from single equations to systems, phase analysis, oscillation, and numerical solutions
All courses
A preview of linear algebra
Linear algebra studies vectors, matrices, transformations, and systems. Its geometry underlies regression, machine learning, differential equations, and much of modern applied mathematics
AP Statistics
After AP Statistics: the upper division
Probability theory, mathematical statistics, regression, Bayesian inference, stochastic processes, and computation extend the ideas introduced in AP Statistics
Diagnostics
1 tool
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