Math · 04
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
Using only \(f'(a)=\lim_{h\to 0}(f(a+h)-f(a))/h\), prove that the derivative of \(x^3\) is \(3x^2\).
Timer not started. Twenty minutes of real work before a solution.
\(((x+h)^3-x^3)/h=(3x^2 h+3xh^2+h^3)/h=3x^2+3xh+h^2\), which tends to \(3x^2\) as \(h\to 0\).
Attack 2
A particle has position \(s(t)=t^3-6t^2+9t\) for \(t\ge 0\). On which interval is it moving in the negative direction?
Timer not started. Twenty minutes of real work before a solution.
\(s'(t)=3t^2-12t+9=3(t-1)(t-3)\). The velocity is negative when the quadratic is negative, which is between the roots: \(t\in(1,3)\).
Attack 3
From the definition, why is the derivative of a constant function equal to 0?
Timer not started. Twenty minutes of real work before a solution.
If \(f(x)=c\), the difference quotient is \((c-c)/h=0\) for every \(h\ne 0\), so the limit is 0.
Single-variable calculus is the study of limits, and of the two constructions that limits make rigorous: instantaneous rate of change, and accumulated change. The derivative is a limit of difference quotients. The integral is a limit of sums. The fundamental theorem says these are inverse operations when the hypotheses hold. Everything else in a first course is a consequence or a technique: mean value theorem, L’Hôpital, sequences, series, Taylor polynomials with a remainder you can estimate.
Stewart is the book most colleges assign. It is paid, and I am not pinning an edition because I could not verify one cleanly. You do not need it. APEX Calculus is free, complete, and enough for computational control; MIT 18.01 supplies the problem sets and the exams with solutions. Spivak is the ceiling, not the first book: its exercises are where a correct computation stops being the standard and a proof becomes the standard. Demidovich is the computational grind, thousands of problems, and it is on the Internet Archive legally as a library scan.
Learn Anything’s calculus page exists and is a junk drawer. The two links from it worth keeping are Thompson’s Calculus Made Easy, which is a preview and not a course, and Keisler’s infinitesimal calculus, which is a legitimate alternate foundation. They are not better than 18.01 plus a problem book if your goal is to be able to sit an exam.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Limit and continuity. \(\lim_{x\to a}f(x)=L\) means: for every \(\varepsilon>0\) there exists \(\delta>0\) such that \(0<|x-a|<\delta\) implies \(|f(x)-L|<\varepsilon\). \(f\) is continuous at \(a\) when the limit equals \(f(a)\).
This is a definition. The work is using it. Sum and product rules for limits follow by algebraic estimates and the triangle inequality; write the product rule once, in ε–δ, before you trust it.
Derivative. \(f'(a)\) is the limit of \((f(a+h)-f(a))/h\) when it exists. Product, quotient, and chain rules hold when the pieces are differentiable. If \(f\) has a local extremum at an interior point and \(f'\) exists there, then \(f'=0\).
The product rule is the usual add-and-subtract of \(f(a+h)g(a)\). Fermat’s interior extremum theorem: the difference quotient cannot approach a positive number and a negative number at once, because the function lies on one side of its value.
Mean value theorem. If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then some \(c\) in \((a,b)\) has \(f'(c)=(f(b)-f(a))/(b-a)\). Consequence: if \(f'=0\) everywhere on an interval then \(f\) is constant.
Reduce to Rolle by subtracting the chord. Rolle: a continuous function on a compact interval attains its max and min; if it is not constant an interior extremum exists, and there the derivative vanishes. The constant-function corollary is immediate.
Fundamental theorem. If \(f\) is continuous on \([a,b]\) and \(F(x)=\int_a^x f\), then \(F'=f\). If \(G'=f\), then \(\int_a^b f=G(b)-G(a)\).
The difference quotient of \(F\) is the average of \(f\) over a short interval, which approaches \(f(x)\) by continuity. The second form is the mean value theorem applied to \(F-G\), whose derivative is zero.
Sequences and series. A series \(\sum a_n\) converges when its partial sums converge. Absolute convergence implies convergence. The ratio test, the comparison test, and the alternating series test are the working tests. A power series has a radius of convergence. Taylor’s theorem supplies a remainder, Lagrange form \(f^{(n+1)}(c)(x-a)^{n+1}/(n+1)!\) under the usual hypotheses.
Absolute convergence uses comparison with the series of absolute values and completeness of \(\mathbb{R}\), which you may treat carefully now or again in analysis. The ratio test compares with a geometric series. Taylor with Lagrange remainder is repeated integration by parts or repeated application of the mean value theorem; write one of those once.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
3Blue1Brown, Essence of Calculus
Watch once, do not study from this. It shows what a derivative is buying you. It does not make you able to compute one under pressure.
The thing you work through.
APEX Calculus, together with MIT 18.01SC
APEX is free and complete; the download page is https://www.apexcalculus.com/downloads . MIT 18.01SC is the lecture, problem set, and exam spine: https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/ . Stewart is the usual paid textbook; no edition is pinned here. APEX plus 18.01 is the better working pair because the exams are public.
The problem book.
Demidovich, Problems in Mathematical Analysis (Mir, 1970)
About three thousand problems, ordered by technique. Apostol, Calculus, Volume 1, ISBN 978-0-471-00005-1, is the paid proof-flavored alternate. Do Demidovich for speed and Apostol when you want the theorem proved before the drill.
The hardest questions that exist at this level.
Spivak, Calculus, ISBN 978-0-914098-91-1, and Putnam analysis problems
Spivak is paid; the ISBN is the catalogued one for his Calculus. Its end-of-chapter problems are the English-language ceiling for a first course. The 2024 Putnam paper is one concrete ceiling exam; pick the problems that are limits, inequalities, or series. Keisler’s free infinitesimal book is a side door, not this ceiling: https://people.math.wisc.edu/~hkeisler/calc.html
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: one full MIT 18.01 exam, closed book, in the allotted time, score at the level of the published solutions on every problem you claim, plus five Spivak problems from three different chapters, written as proofs. Fail if the Taylor remainder is a formula you cannot derive.
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
Define the derivative, prove the product rule, state the fundamental theorem, and explain what a radius of convergence means. No notes.