Free public beta until December 4, 2026:
every course is open, no card needed.
After that, the first chapter of each course stays free and full access is a one-time $39 ($19 for up to 100 founding members chosen for helpful feedback). Your progress is kept.
How the beta works
Chapter 0 wrap-up
Readiness review
Closing thought
Every tool this course leans on is now sharp and in reach. Most of them are old ones. Periodic functions, series, improper integrals, inner products, and projection all come from courses you have already taken. The one genuinely new tool is the complex exponential. Euler's formula, e^{i\theta}=\cos\theta+i\sin\theta, lets a single exponential carry both a cosine and a sine at once. The thread running through all of it is projection. A coordinate is an inner product divided by a length squared, and that idea will keep coming back.
You can now check that \int_{-\pi}^{\pi}\sin(2x)\cos(3x)\,dx=0 without looking anything up. You can rewrite \cos\theta as \tfrac12(e^{i\theta}+e^{-i\theta}), and project a vector onto a direction in \mathbb R^n. Next, Chapter 1 puts the toolbox down for a moment. It shows the picture this whole course is about: one signal, read once moment by moment and once frequency by frequency.
Recipes in this chapter
Step-by-step procedures for the problem types this chapter teaches.
One practice set per topic this chapter taught — same fixed, ungraded
ladders as the practice panels inside each section.
Practice: Chapter review: is the toolbox sharp?
(8 problems · doesn’t affect your progress)
Determine whether f(x) = x^5 - 2x is even, odd, or neither.
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Why
Both powers present, 5 and 1, are odd, so the power-pattern shortcut predicts odd — and indeed f(-x)=-x^5+2x=-f(x).
Odd
Sum the infinite geometric series \displaystyle\sum_{n=1}^{\infty}\left(\frac15\right)^n.
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Why
Geometric with a=\frac15, r=\frac15: \frac{a}{1-r}=\frac{1/5}{4/5}=\frac14.
Correct answer
Also accepted:
Why
1/4
Does \displaystyle\int_1^{\infty}\frac{1}{x^{3/2}}\,dx converge?
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Why
This is a p-integral with p=\frac32>1, which converges — the tail shrinks fast enough for a finite area.
Converges
Suppose the left- and right-hand limits at x=a agree at the value 3, yet f(a) itself is undefined. What kind of discontinuity is this?
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Why
Both one-sided limits exist, are finite, and agree with each other, but f(a) doesn't exist — exactly the removable case.
Removable
Find the imaginary part of e^{i\pi/3}.
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Why
e^{i\pi/3}=\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}, and \sin\frac{\pi}{3}=\frac{\sqrt3}{2}.
Correct answer
Also accepted:
Why
sqrt(3)/2
Is \displaystyle\int_{-\pi}^{\pi}\sin(x)\cos(x)\,dx equal to 0?
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Why
\sin x\cos x is an odd function (odd times even), and an odd integrand over the symmetric interval [-\pi,\pi] integrates to 0.
Yes
Take u=(0,4) and v=(3,-2). What is the y-component of \operatorname{proj}_u v?
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Why
v\cdot u=-8, u\cdot u=16, so \operatorname{proj}_uv=\frac{-8}{16}(0,4)=(0,-2) — the y-component is -2.
Correct answer
Also accepted:
Why
-2
For a real symmetric matrix, are eigenvectors from two different eigenvalues always orthogonal?
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Why
This is one of the guarantees symmetry buys: eigenvectors belonging to distinct eigenvalues of a real symmetric matrix are automatically orthogonal, with no extra work needed to arrange it.