Reference sheets: formulas with conditions

Use these pages for retrieval and method selection, not as a replacement for the worked teaching. Cover the right side of a formula, reconstruct it, and explain its restrictions before checking. All variables are real unless stated otherwise; trigonometric derivative formulas use radians.

1. Notation, algebra, and the first calculus chain#

A function $f$ is a rule with a specified domain. $f(a)$ is its output at $a$; $f'(a)$ is its derivative there when a finite derivative exists. $f^{-1}$ means an inverse function, not $1/f$. $\sin^{-1}x$ conventionally means $\arcsin x$, not $\csc x$.

The quotient is an average rate over the interval with endpoints $a$ and $a+h$; the limit is instantaneous. To form $f(a+h)$, replace every input with the whole expression $a+h$. In subtracting $f(a)$, distribute the negative sign over every term. Cancel common nonzero factors, not terms of sums.

Record denominator exclusions before simplifying. Even roots require nonnegative radicands; real logarithms require positive arguments. For $a,b>0$,

There is no comparable rule splitting $\ln(a+b)$ into $\ln a+\ln b$.

The tangent-line equation is exact as a line. The claim $f(x)\approx L(x)$ is a local approximation; differentiability says its error divided by $|x-a|$ tends to zero. A point’s height equals the vertical-axis intercept only when that point is on the vertical axis. Module 01–Module 08, Module 13.

2. Derivatives and their structure#

For differentiable $u,v$, where expressions are defined,

Inverse sine/cosine derivatives require $|x|<1$. The last inverse-cotangent formula uses the decreasing branch with values in $(0,\pi)$. For real powers with noninteger exponent, the safe general derivative statement uses $x>0$; larger real domains depend on the exponent and branch convention. Trigonometric quotient derivatives hold only away from their poles.

Implicit and inverse differentiation. Differentiate every term of an equation, attaching $y'$ when differentiating a $y$ expression with respect to $x$, then solve for $y'$. For a differentiable local inverse,

under the local invertibility and regularity assumptions taught in Module 11. The input $b$ of the inverse is the output of the original function.

Method selection. Identify outermost structure first: sum, product, quotient, or composition. In $x^2e^{\sin x}$, the top level is a product and one factor contains a composition. Use a product rule and then a chain rule; do not search for one memorized formula matching the entire expression. Module 09–Module 11.

3. Derivative applications and theorem gates#

For position $s(t)$, velocity $v=s'$ and acceleration $a=v'$. Speed is $|v|$, so when $v\ne0$, speed increases when $va>0$ and decreases when $va<0$. Total distance is $\int|v|\dd t$, not necessarily the magnitude of displacement $\left|\int v\dd t\right|$.

Related rates. Draw a time-specific diagram; write a relation valid during the motion; differentiate with respect to time; then substitute the instant’s data. Do not set a changing quantity equal to its instantaneous number before differentiating.

Closed-interval extrema. For continuous $f$ on $[a,b]$, evaluate endpoints and every interior critical number ($f'=0$ or $f'$ undefined where $f$ exists). Compare function values. For an unbounded or open domain, examine boundary behavior separately; existence of a global extremum is not automatic.

Sign information. On an interval, $f'>0$ implies strict increase and $f'<0$ implies strict decrease. A strictly increasing function may have derivative zero at isolated points. A sign change of $f'$ from $+$ to $-$ gives a local maximum; $-$ to $+$ gives a local minimum. Inflection requires a change of concavity, not merely $f''=0$.

Newton’s method.

It requires a usable nonzero denominator at each step and is not guaranteed to converge from every starting value. Check the final residual and, when possible, a bracket or derivative-based error bound.

Limits of derivative ratios are not identities between the original functions. A single application may still leave an indeterminate form. Module 12–Module 15. [O1MVT][O1LH]

4. Integrals: accumulation and geometry#

For continuous $f$ and an antiderivative $F$ on $[a,b]$,

For variable limits, continuity of $f$ and differentiability of $u,v$ give

A definite integral can be negative. Geometric area uses $\int|f|$ or upper-minus-lower pieces; an average value is $(b-a)^{-1}\int_a^bf$ when $a<b$. An indefinite integral represents a family of antiderivatives, including an arbitrary constant on each connected interval of its domain.

Real-power domains still matter. Reverse differentiation is the fastest independent check of a proposed antiderivative.

Applications arise from a small contribution.

Radii are distances to the actual rotation axis, not automatically $x$ or $y$. Arc/surface formulas require suitable smoothness and no unintended multiple tracing. Work units are force times distance; volume units are length cubed. Module 16–Module 19. [O1FTC][O2Apps]

5. Choosing an integration method#

First simplify. Expand a short polynomial product, split sums, complete a square, or divide an improper rational function. Then inspect structure.

Substitution: look for an inner expression whose derivative appears as a factor. Set $u=g(x)$ and replace the differential as well as the expression. Change definite bounds or return to $x$ before evaluating—not a mixture of both.

Parts: a product with a factor that simplifies on differentiation, or a lone logarithm/inverse trigonometric function. The product rule gives

Repeated or cyclic parts may be necessary. The goal is a simpler integral, not loyalty to a mnemonic.

Trigonometric powers: reserve a sine/cosine factor when its power is odd; use $\sin^2x=(1-\cos2x)/2$ and $\cos^2x=(1+\cos2x)/2$ for even powers. For tangent/secant, a $\sec^2x\dd x$ or $\sec x\tan x\dd x$ factor often reveals a substitution.

Radicals: $\sqrt{a^2-x^2}$ suggests $x=a\sin\theta$; $\sqrt{a^2+x^2}$ suggests $x=a\tan\theta$; $\sqrt{x^2-a^2}$ suggests a secant substitution on a carefully specified branch. Square roots produce absolute values; choose the angle interval so the sign simplification is valid.

Rational functions: divide first if needed, factor the denominator, and use partial fractions with every repeated linear power and a general linear numerator over each irreducible quadratic factor. Integrate on intervals avoiding poles.

No useful elementary simplification: a definite integral may be approximated numerically or expanded by a justified series method. Not finding an elementary antiderivative does not imply that the integral is undefined. Module 20–Module 24.

Numerical integration error bounds#

Let $h=(b-a)/n$. If $|f''|\le M_2$ throughout $[a,b]$ and the needed derivatives are continuous,

For composite Simpson, $n$ must be even; with continuous fourth derivative and $|f^{(4)}|\le M_4$,

The bound is not the actual error. A fine-looking table alone does not supply a derivative bound between nodes. Improper integrals must be split at each singularity and interpreted as separate one-sided limits; finite cancellation across a pole does not establish convergence. [O2Num][O2Improper]

6. Series and approximation: select the claim carefully#

A sequence is a list $a_n$. A series is a limit of partial sums $\sum_{n=1}^Na_n$. The necessary condition $a_n\to0$ does not suffice for convergence. A convergence test need not determine the value of a convergent sum.

Absolute convergence implies convergence. Conditional convergence means the signed series converges while its absolute-value series diverges. Endpoint tests for a power series must be done separately after finding its radius.

requires the Taylor remainder to tend to zero; existence of all derivatives alone is insufficient. If $f$ is $C^{n+1}$ on the interval between $a$ and $x$ and $|f^{(n+1)}|\le M$, then

Inside a power series’ radius, termwise differentiation and integration preserve that radius; endpoints may change. Module 26–Module 30. [O2Tests][O2Alt][O2Power][O2Taylor]

7. Curves, coordinates, and local multivariable change#

For a regular parametric plane curve,

In polar coordinates, $x=r\cos\theta$, $y=r\sin\theta$; signed $r$ allows multiple representations. For a correctly traced region/curve,

Between polar boundaries use outer squared radius minus inner squared radius on the angular ranges where both describe the intended region.

For $\vv r(t)$ in space, $\vv v=\vv r'$, $\vv a=\vv r''$, speed $=\|\vv v\|$, and

These regular-curve formulas require nonzero speed at the point. A singular parametrization need not mean the geometric curve itself is singular.

For differentiable $f(x,y)$,

where the final error divided by $\sqrt{h^2+k^2}$ tends to zero. Existence of partial derivatives alone is not enough. Continuous first partials in a neighborhood are a useful sufficient condition.

At a nonzero gradient, the maximum unit-direction rate is $\|\grad f\|$, along the gradient. A zero gradient is not automatically an extremum. For two-variable stationary points, $D=f_{xx}f_{yy}-f_{xy}^2$: $D>0$ with $f_{xx}>0$ gives a local minimum, with $f_{xx}<0$ a local maximum; $D<0$ gives a saddle; $D=0$ is inconclusive. Use sufficient smoothness. Constrained candidates satisfy $\grad f=\lambda\grad g$ at regular points of $g=c$; also inspect singular constraint points and boundaries. Module 31–Module 38.

8. Multiple integrals and coordinate changes#

Draw the region, identify the outer variable’s range, then describe one slice for the inner bounds. Reversing order changes the description, not the region. A Jacobian is part of the area/volume element:

Here $\phi$ is measured down from the positive $z$ axis, $0\le\phi\le\pi$, and

Under a suitably regular, one-to-one change of variables (apart from harmless boundary exceptions),

The absolute value corrects area scaling, not multiple covering. Transform the function and the bounds and the differential. Mass is $\int\rho\dd V$, a first moment is $\int x\rho\dd V$, and an average over a region is the integral divided by its volume. Here density $\rho$ is a different use of the same letter as spherical radius; the context must identify which is meant. Module 39–Module 41. [O3Jac]

9. Vector integrals and orientation#

Reversing orientation leaves a scalar arc-length integral unchanged but reverses the sign of work/circulation. For a parameterized surface,

when the cross product matches the requested orientation. Reversing orientation reverses flux, not scalar surface area. For an upward graph $z=g(x,y)$, use vector area $(-g_x,-g_y,1)\dd A$, without another square-root factor.

Conservative field. A potential $\vv F=\grad\psi$ gives work $\psi(B)-\psi(A)$. For a $C^1$ field, zero curl is necessary for a potential. It is sufficient on an appropriate simply connected open domain, but not on arbitrary domains with holes. Check the domain before replacing a closed-loop integral by zero.

Green. For a positively oriented planar boundary and a $C^1$ field $(P,Q)$ on a neighborhood of the region,

Outer boundaries are counterclockwise; hole boundaries are clockwise so the region remains on the left. The second form is outward flux.

Stokes. For a suitably smooth oriented surface and a $C^1$ field on its neighborhood,

Match boundary orientation to the normal by the right-hand rule. A singular field on a proposed replacement surface blocks that replacement.

Divergence. For a piecewise smooth closed boundary oriented outward and a $C^1$ field on a neighborhood of the entire solid,

An open cap needs its missing faces added and their signed fluxes subtracted. On a cavity boundary, outward from the material points into the cavity. A field defined only on the outer boundary may fail the theorem’s interior hypothesis. Module 42–Module 46. [O3Conservative][O3Green][O3Stokes][O3Div]

10. A final method-and-error checklist#

Name the object. Number, function, sequence, series, scalar field, vector field, curve, region, or oriented surface? A derivative of a scalar function and a divergence of a vector field are different objects.

Name the requested result. Value, rate, approximation, equation, convergence decision, area, volume, work, circulation, or flux? A convergence proof is not an exact sum, and a line equation is not just its slope.

Check the gate. Domains; nonzero denominators; differentiability; positivity/monotonicity; interval endpoints; theorem neighborhoods; unit directions; regularity; multiplicity; orientation.

Check independently. Substitute algebraic roots; differentiate antiderivatives; insert ODE solutions and initial data; estimate signs and units; compare direct and theorem-based integrals; inspect endpoint behavior; bound an error rather than equating a numerical match with proof.