Formula sheet
Every rule the problems here are built from. Knowing the rule is not the same as knowing the answer - you still have to do all of the work - so this is open in a run, in practice, and in the Daily. Nothing is hidden behind anything.
Getting started
The two that come before any calculus at all.
Evaluating a function
Not calculus. Just substitution, carefully.
Limits by substitution
If putting the number in gives a sensible answer, that is the answer.
Differentiation
Everything that turns a function into its rate of change.
Power rule
Any power of x, including roots and negative powers.
The standard derivatives
Worth knowing by heart. Everything else is built on these.
Product rule
Two things multiplied, where neither is just a number.
Quotient rule
One thing divided by another. Mind the order on top.
Chain rule
A function inside another function. Differentiate the outside, then multiply by the derivative of the inside.
Higher derivatives
Just do it again. And again, if asked.
Implicit differentiation
y is tangled up with x and cannot be separated out. Differentiate both sides, then rearrange.
Slope of a tangent
The derivative evaluated at a point IS the slope there.
Partial derivatives
More than one variable. One at a time, everything else frozen.
Gradient, divergence and curl
The gradient's length is the fastest rate of increase. Divergence adds three partials; each curl component subtracts two, such as Q_x - P_y.
Second derivative test
At a critical point: D > 0 with f_xx > 0 is a minimum, D > 0 with f_xx < 0 a maximum, D < 0 a saddle.
Integration
Going the other way, which is harder and always will be.
Power rule, backwards
Raise the power, divide by the new one. The exception is n = -1, which gives ln|x|.
Substitution
The chain rule in reverse. Look for a function and its own derivative in the same integral.
Integration by parts
A product where substitution does not help. Pick u so that differentiating it makes things simpler.
Partial fractions
A fraction with a factorisable bottom. Split it, then integrate each piece.
Improper integrals
An infinite limit, or a function that blows up inside the range. Take a limit.
Multiple integrals
One variable at a time, working outwards. The inner limits may depend on the outer variable.
Polar coordinates
Discs, rings and sectors of them. Do not forget the extra r.
Average value
Integrate over the region, then divide by its area.
Green's theorem and the divergence theorem
A closed curve or a closed surface: integrate the curl or the divergence over the inside instead.
Surface area
For a surface z = f(x, y). Planes, cones, paraboloids and spheres are the ones whose root simplifies.
Gaussian integral
Complete the square for a linear term in the exponent. For powers of x, differentiate with respect to a.
Beta integral
Powers of x and 1 - x on [0, 1]. Powers of sin and cos on [0, π/2] are the same thing after a substitution.
Frullani and Dirichlet integrals
An x underneath that will not go away. Put a parameter in and differentiate with respect to it (Feynman's trick).
Applications
What the machinery is actually for.
Area between curves
Find where they cross first. Those are your limits.
Volume by disks
Rotating around an axis the region touches.
Volume by shells
Rotating around an axis the region does not touch.
Arc length
The length of a curve. Only a few curves make that root simplify.
Related rates
Two quantities changing together. Differentiate the relationship with respect to time.
Optimisation
Write the thing being optimised as one variable first, then differentiate.
Differential equations
Separable ones split and integrate both sides. Linear ones need an integrating factor.
Exact and Bernoulli equations
Exact: find F with F_x = M and F_y = N, and the solution is F(x, y) = C. Bernoulli: the substitution makes it linear.
Undetermined coefficients
Guess a particular solution shaped like the right-hand side. If that shape already solves the left side, multiply the guess by x.
Laplace transform
Going back the other way: split into partial fractions, complete the square, and read each piece off a table.
Linear systems
Eigenvalues and eigenvectors of A, then fit the constants to the starting values.
Limits and series
The parts that decide whether anything above is allowed.
L'Hopital's rule
Only for those two forms. Check first, then differentiate top and bottom SEPARATELY.
Limits at infinity
Only the highest power on each side matters.
The standard limits
These come up constantly and are quicker than L'Hopital.
The definition of e
Anything of the form one-to-the-infinity. Not 1.
Multiplying by the conjugate
A difference of roots giving 0/0, or infinity minus infinity. The roots cancel on top.
One-sided limits
Approaching from below and from above. They can disagree.
Convergence
The usual first thing to try on a series.
Taylor coefficients
The nth coefficient needs the nth derivative at the centre.
Limits by Taylor series
When the first terms cancel, the answer is in the next ones. Expand every function that far.
Stirling's formula
nth roots of factorials. The square-root factor vanishes under the nth root.
Sums that collapse
Partial fractions until neighbouring terms cancel. A geometric series needs |r| < 1.
Known power series
Spot which series it is at which x. Differentiating or multiplying by x reaches the variants.
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