My revision cards · sedifier

Variables

Cases: who or what. Variables: what is recorded.
Categorical: groups. Quantitative: meaningful numbers.

Watch out: numeric codes (ZIP, ID) are categorical.

Describing a distribution

Shape, center, spread, outliers, in context.

Watch out: skewed right means mean > median (usually).

Center

Mean: sum ÷ n; pulled by extremes.
Median: middle value; resistant.

Watch out: even count: average the two middle values.

Spread

Range, IQR (resistant), standard deviation.
$s$ divides by $n - 1$ ; $s^2$ is the variance.

Watch out: the SD is the square root of the variance.

Outliers and boxplots

Fences: $Q_1 - 1.5\,\text{IQR}$ and $Q_3 + 1.5\,\text{IQR}$ .
Boxplot: five-number summary plus outliers.

Watch out: 1.5 × IQR, not 1 × IQR.

Defining quantities

Name each quantity with its unit and period; compare with rates.

Watch out: a vague variable ("x = money") leads to a wrong model.

Converting units

To a smaller unit: multiply; to a larger unit: divide.

Watch out: check that the answer makes sense in size.

Percent error

Error ÷ actual × 100.

Watch out: divide by the actual value, not the measured one.

Linear equations

Do the same to both sides until $x$ is alone.

Watch out: distribute to every term.

Solving for a variable

Same moves as with numbers; the other letters are just numbers you do not know.

Watch out: divide every term.

Lines from equations

Intercepts: set the other variable to 0. Slope: solve for $y$ .

Watch out: slope of $Ax + By = C$ is $-A/B$ .

Substitution

Replace one variable by its expression, solve, then substitute back.

Watch out: answer the variable the question asks for.

Elimination

Equal coefficients: subtract. Opposite coefficients: add.

Watch out: multiply first when no coefficients match.

How many solutions?

Different slopes: one. Parallel: none. Same line: infinitely many.

Watch out: same slope alone is not enough; check the intercepts.

Inequalities

Solve as an equation; flip when multiplying or dividing by a negative.

Watch out: check with a number from your answer.

Absolute value equations and inequalities

|x − a| = b: a ± b. Less than: between. Greater than: outside.

Watch out: do not forget the negative case.

Inequalities in two variables

Dashed for $<, >$ ; solid for $\le, \ge$ . Test a point to choose the side.

Watch out: “at most” is $\le$ , “at least” is $\ge$ .

Two-way tables

Joint: cell / total. Marginal: margin / total.
Conditional: cell / its group total.

Watch out: name the group you condition on.

Association

Compare conditional proportions across groups.
Different: associated. Equal: not associated.

Watch out: association is not causation; counts are not rates.

Correlation

$-1 \le r \le 1$ ; sign = direction, size = linear strength.
Unitless; symmetric in $x$ and $y$ .

Watch out: $r$ only sees straight lines.

Least-squares line

$b = r\,s_y/s_x$ , $a = \bar{y} - b\bar{x}$ , through $(\bar{x}, \bar{y})$ .

Watch out: $s_y$ on top.

Residuals

$e = y - \hat{y}$ ; positive is above the line.
Residual plot: no pattern means a linear model is appropriate.

Watch out: observed minus predicted, in that order.

Interpreting regression

Slope: predicted change per unit of $x$ .
$r^2$ : share of variation explained.
No extrapolation.

Watch out: $r^2$ , not $r$ , is the explained fraction.

Function notation

$f(a)$ is the output for the input $a$ .
Replace every $x$ by $(a)$ , with parentheses: $f(-2)$ for $x^2$ is $(-2)^2 = 4$ .
On a graph: the height above $x = a$ .

Watch out: $f(a + b)$ is not $f(a) + f(b)$ .

Average rate of change

From $a$ to $b$ : $\dfrac{f(b) - f(a)}{b - a}$ , output change over input change.
Increasing: the graph rises from left to right.
Give intervals of increase with $x$ -values.

Watch out: change in $f(x)$ on top, change in $x$ below.

Domain and range

Domain: the inputs. Leave out zero denominators and negatives under a square root.
Range: the outputs.
$[a, \infty)$ includes $a$ ; $(a, \infty)$ does not.

Watch out: the range uses $y$ -values, not the $x$ of a vertex.

Piecewise functions

Find the piece whose condition the input satisfies.
Use that piece's formula.
$|x| = x$ for $x \ge 0$ , and $-x$ for $x < 0$ .

Watch out: at a boundary, read $<$ and $\ge$ carefully.

Inverse functions

Write $y = f(x)$ .
Solve for $x$ , undoing the last operation first.
Rename: $f^{-1}(x)$ . Check that $f(f^{-1}(x)) = x$ .

Watch out: $f^{-1}(x)$ is not $\dfrac{1}{f(x)}$ .

Shifts and reflections

$f(x - h)$ : $h$ units right. $f(x) + k$ : $k$ units up.
$-f(x)$ : flip across the $x$ -axis. $f(-x)$ : flip across the $y$ -axis.
Inside $f(\dots)$ acts on $x$ ; outside acts on $y$ .

Watch out: $f(x + 3)$ moves the graph left.

Sequences

Explicit: $a_n$ straight from $n$ .
Recursive: a first term and a rule from $a_n$ to $a_{n+1}$ .

Watch out: check your formula with $n = 1$ .

Linear or exponential?

Equal differences: linear. Equal factors: exponential.

Watch out: exponential growth wins in the end.

Arithmetic sequences

Add $d$ each time: $a_n = a_1 + (n - 1)d$ .
$d = \dfrac{a_q - a_p}{q - p}$ .

Watch out: $n - 1$ steps from $a_1$ to $a_n$ .

Geometric sequences

Multiply by $r$ each time: $a_n = a_1 r^{\,n-1}$ .
$r$ = a term divided by the one before.

Watch out: the exponent is $n - 1$ .

Rational exponents

$b^{\frac{1}{n}} = \sqrt[n]{b}$ ; $b^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m$ .
$b^{-r} = \dfrac{1}{b^{r}}$ : the value stays positive.
A factor $b$ per unit means a factor $b^{\frac12}$ per half-unit.

Exponential change: the same factor at every step

Start $a$ , factor $b$ per step: after $n$ steps, $a \cdot b^n$ .
Gain $p\%$ : factor $1 + \frac{p}{100}$ . Lose $p\%$ : factor $1 - \frac{p}{100}$ .

Watch out: adding the same amount each step is linear, not exponential.

Reading exponential functions

$a$ : value at $x = 0$ ; $b$ : factor per step; $b^{-1} = 1/b$ .

Watch out: negative exponent, positive value.

Writing an exponential function

$f(x) = a \cdot b^x$ : $a = f(0)$ , $b = \dfrac{f(1)}{f(0)}$ .
Multiplied by $b$ every $n$ units: $f(x) = a \cdot b^{\frac{x}{n}}$ .

Watch out: the starting value is $f(0)$ , and it goes in front, not in the base.

Percent change and interest

Increase $r$ : × $(1 + r)$ ; decrease: × $(1 - r)$ ; compounding: rate and exponent per period.

Watch out: 1.1, not 0.1, for a 10% increase.

Quadratic patterns

Constant second differences mean quadratic.

Watch out: growing differences are not linear.

Polynomial arithmetic

Add: combine like terms.
Multiply: every term by every term; degrees add.

Watch out: $x$ -terms add; constants multiply.

Standard and factored form

Factored form gives the $x$ -intercepts; standard form gives the $y$ -intercept.

Watch out: the factor $x + 5$ gives the intercept $-5$ .

Graphs of quadratics

Sign of $a$ : up or down; $c$ : $y$ -intercept; vertex midway between the zeros.

Watch out: the $y$ -intercept is $c$ , not $a$ .

Quadratic functions

Vertex at $x = -\dfrac{b}{2a}$ .
Its height $f\left(-\dfrac{b}{2a}\right)$ is the minimum ( $a > 0$ ) or maximum ( $a < 0$ ).
Vertex form: $a(x - h)^2 + k$ , vertex $(h, k)$ .

Watch out: the constant term $c$ is $f(0)$ , not the extreme value.

Vertex form

$y = a(x - h)^2 + k$ : vertex $(h, k)$ .

Watch out: $(x + h)$ means the vertex is at $-h$ .

Solving by square roots

Isolate the square; then $\pm\sqrt{\;}$ .

Watch out: two solutions, not one.

Zero product property

Product = 0: set each factor to 0.

Watch out: never divide by $x$ ; you lose $x = 0$ .

Solving by factoring

Multiply to $c$ , add to $b$ ; then each factor $= 0$ .

Watch out: the factor $x - 4$ gives $x = 4$ .

Completing the square

Add $(b/2)^2$ to both sides; then square roots.

Watch out: halve, then square; and do it on both sides.

The quadratic formula

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ ; the discriminant counts the solutions.

Watch out: $2a$ divides the whole numerator.

Modeling with quadratic equations

Translate, solve, then keep the solutions that fit the situation.

Watch out: a negative length or the start time is not the answer.

Rational and irrational

Ends or repeats: rational. Neither: irrational.

Watch out: a repeating decimal is a fraction over 9, 99, …

A line and a conic

Substitute the line into the conic.
Solve the quadratic: 0, 1 or 2 real solutions.
Find the other coordinate from the line.

Watch out: a negative square means no intersection.

Arithmetic series

$\displaystyle\sum_{k=p}^{q}$ has $q - p + 1$ terms.
$S_n = \dfrac{n(a_1 + a_n)}{2}$ .
$1 + 2 + \dots + n = \dfrac{n(n + 1)}{2}$ .

Watch out: count the terms with $+1$ .

Finite geometric series

$S_n = a_1\,\dfrac{r^n - 1}{r - 1}$ .
$n$ is the number of terms; the power is $r^n$ .

Watch out: the last term is $a_1 r^{\,n-1}$ , but the sum uses $r^n$ .

End behavior

Only the leading term (highest power) matters for large $|x|$ .
Even degree: both ends the same way. Odd degree: opposite ways.
A negative leading coefficient flips the picture.

Watch out: the leading term is not always written first.

Zeros and multiplicity

$(x - r)$ gives the zero $r$ ; $(x + q)$ gives the zero $-q$ .
Odd multiplicity: the graph crosses. Even: it touches and turns back.

Watch out: check the sign: $x + 3 = 0$ means $x = -3$ .

Polynomial division

$p = d \cdot q + r$ ; keep 0-coefficients for missing powers.

Watch out: check by multiplying back.

Remainder and factor theorems

Remainder of $p(x) \div (x - r)$ : $p(r)$ .
$x - r$ is a factor $\iff p(r) = 0$ .
For $x + r$ , evaluate at $-r$ .

Watch out: a factor needs a remainder of $0$ .

Polynomial identities

$a^2 - b^2$ , $a^3 \pm b^3$ , and the geometric-sum identity.

Watch out: the sign pattern in the cube identities.

Asymptotes of rational functions

Vertical: zeros of the denominator.
Horizontal: compare degrees. Top smaller: $y = 0$ . Equal: ratio of leading coefficients. Top larger: none.

Watch out: use the leading coefficients, not the constants.

Intercepts and holes

Cancel common factors: each one leaves a hole.
Hole height: the simplified rule at that $x$ .
$x$ -intercepts: remaining zeros of the numerator; $y$ -intercept: $f(0)$ .

Watch out: zeros of the denominator give asymptotes, not intercepts.

The binomial theorem

$(a + b)^n = \sum \dbinom{n}{k} a^{\,n-k} b^{\,k}$ .
Coefficients: a row of Pascal's triangle.
Include the powers of the constants.

Watch out: $(a + b)^2 \ne a^2 + b^2$ .

Multiplying and dividing rational expressions

Factor, cancel common factors, multiply; divide by multiplying by the reciprocal.

Watch out: cancel factors, never terms.

Adding rational expressions

Common denominator, then add the numerators.

Watch out: never add the denominators.

Rational equations

Clear denominators → solve → check in the original.

Watch out: reject values that make a denominator 0.

Radical equations

Isolate → square (or cube) → solve → check.

Watch out: squaring can add extraneous solutions.

The number i

$i^2 = -1$ ; $\sqrt{-a} = i\sqrt{a}$ ; powers of $i$ repeat every 4.

Watch out: $\sqrt{-16}$ is $4i$ , not $-4$ .

Complex arithmetic

Add parts separately; multiply out and use $i^2 = -1$ ; conjugates give a real product.

Watch out: $i^2$ is $-1$ , not $1$ .

Complex solutions

Discriminant < 0 → $x = p \pm qi$ (conjugates).

Watch out: divide the whole numerator by $2a$ .

Complex zeros

$i^2 = -1$ , so $\sqrt{-k^2} = ki$ .
$b^2 - 4ac < 0$ : no real zeros, two complex ones.
Real coefficients: $p + qi$ and $p - qi$ come together.

Watch out: the conjugate changes only the sign of the $i$ part.

Logarithms are exponents

$\log_b x = y$ means $b^y = x$ .
$\log x$ means $\log_{10} x$ .
The solution of $b^x = c$ is $x = \log_b c$ .

Watch out: $\log_b x$ is never $x \div b$ .

The number $e$

$e \approx 2.718$ , the limit of $\left(1 + \frac{1}{x}\right)^x$ .
Continuous growth at rate $r$ : $A(t) = A_0 \, e^{rt}$ .
In $a \, e^{kt}$ , the factor per unit of $t$ is $e^{k}$ .

Watch out: "continuously at $r$ " is not "by $1 + r$ once a year".

Solving $e^{kx} = a$ (with $a > 0$ )

Take $\ln$ of both sides: $kx = \ln a$ .
Divide by $k$ : $x = \dfrac{\ln a}{k}$ .

Watch out: $e$ is not a factor. Never divide by $e$ .

Logarithmic functions

$y = \log_b x$ is the inverse of $y = b^x$ .
Through $(1, 0)$ and $(b, 1)$ ; domain $x > 0$ ; asymptote $x = 0$ .
$y = \log_b(x - h)$ : asymptote $x = h$ ; crosses the $x$ -axis at $x = h + 1$ .

Properties of logarithms ( $M, N > 0$ )

Product: $\log_b(MN) = \log_b M + \log_b N$ .
Quotient: $\log_b \frac{M}{N} = \log_b M - \log_b N$ .
Power: $\log_b(M^p) = p \log_b M$ .

Watch out: there is no rule for $\log_b(M + N)$ .

Change of base

$\log_a c = \dfrac{\ln c}{\ln a}$ : the base goes below.

Check: $\log_2 5$ must be between 2 and 3, because $2^2 = 4$ and $2^3 = 8$ .

Solving $a \cdot b^x = c$

Isolate the power: $b^x = \dfrac{c}{a}$ .
Take $\ln$ : $x \ln b = \ln \dfrac{c}{a}$ .
Divide: $x = \dfrac{\ln(c/a)}{\ln b}$ .

Watch out: never multiply $a$ into the base.

Solving logarithmic equations

Combine into one logarithm (product, quotient, power rules).
Rewrite as a power, or compare arguments.
Solve, then check that every argument is positive.

Half-life and doubling time

Half-life $h$ : $A_0 \left(\tfrac{1}{2}\right)^{\frac{t}{h}}$ . Doubling time $d$ : $P_0 \cdot 2^{\frac{t}{d}}$ .
$\frac{t}{h}$ = number of half-lives.

Watch out: $n$ half-lives divide by $2^n$ , not by $2n$ .

Asymptotes after shifts

$y = a \cdot b^{x} + k$ : horizontal asymptote $y = k$ .
$y = \log_b(x - h) + k$ : vertical asymptote $x = h$ , domain $x > h$ .

Stretches, even and odd

$a f(x)$ : vertical stretch by $a$ . $f(bx)$ : horizontal compression by $b$ .
Even: $f(-x) = f(x)$ , mirror image in the $y$ -axis.
Odd: $f(-x) = -f(x)$ , symmetric about the origin.

Watch out: compute $f(-x)$ ; the signs of the coefficients do not decide.

Composition

$f(g(x))$ : apply $g$ first, then $f$ .
Work from the inside out.
The order matters: $f(g(x))$ and $g(f(x))$ usually differ.

Watch out: $f(g(x))$ is not $f(x) \cdot g(x)$ .

Function families

Base or exponent, denominator, log, periodic: each family has its sign.

Watch out: x in the exponent means exponential, not polynomial.

Radians

$180^\circ = \pi$ rad.
Degrees → radians: $\times \dfrac{\pi}{180}$ . Radians → degrees: $\times \dfrac{180}{\pi}$ .
Arc length: $s = r\theta$ , $\theta$ in radians.

Watch out: never put degrees into $s = r\theta$ .

The unit circle

The point at angle $\theta$ is $(\cos\theta, \sin\theta)$ .
$\dfrac{\pi}{6}$ : $\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)$ ; $\dfrac{\pi}{4}$ : $\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)$ ; $\dfrac{\pi}{3}$ : $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$ .

Watch out: cosine is $x$ , sine is $y$ .

Reference angles

Find the quadrant.
Reference angle: acute angle to the $x$ -axis ( $180^\circ - \theta$ , $\theta - 180^\circ$ or $360^\circ - \theta$ ).
Value of the reference angle, with the quadrant's sign.

Watch out: cosine is negative on the left (II, III), sine below (III, IV).

The Pythagorean identity

$\sin^2 x + \cos^2 x = 1$ for every $x$ .
From one value: square, subtract from 1, take the root, choose the sign by quadrant.

Watch out: the identity needs the squares.

Sinusoidal graphs $y = A\sin(Bx) + D$

Amplitude $|A|$ , midline $y = D$ .
Period $\dfrac{2\pi}{B}$ .
Maximum $D + |A|$ , minimum $D - |A|$ .

Watch out: a bigger $B$ means a shorter period.

Phase shift and models

$A\sin\big(B(x - h)\big) + D$ : shift $h$ right.
$\sin(Bx - C)$ : shift $\dfrac{C}{B}$ , after factoring out $B$ .
Model: $A$ = half the range, $D$ = middle value, $B = \dfrac{2\pi}{\text{period}}$ .

Watch out: factor out $B$ before reading the shift.

Vectors and components

$\overrightarrow{PQ} = \langle x_2 - x_1,\ y_2 - y_1 \rangle$ : end minus start.
Same components, same vector, wherever it starts.

Watch out: start minus end gives the opposite vector.

Vectors in context

Write each velocity or force in components.
Add them: the resultant.
Magnitude = speed or total force; $\tan\theta$ = direction.

Watch out: speeds in different directions do not simply add.

Matrices

Size is rows × columns; add entry by entry; a scalar multiplies every entry.

Watch out: rows first.

Multiplying matrices

(m × n)(n × p) = m × p; entry = row · column.

Watch out: AB is usually not BA.

Matrices as transformations

Av moves vectors; columns are the images of (1, 0) and (0, 1); areas scale by |det|.

Watch out: the area factor is the determinant, not a sum of entries.

Population and sample

Population → parameter (usually unknown).
Sample → statistic (an estimate).

Watch out: a number from the sample is a statistic.

Bias

Voluntary response, convenience, undercoverage, nonresponse, response bias.

Watch out: a bigger biased sample is still biased.

Comparing two groups

Two proportions: pooled $\hat{p}$ in the SE.
Paired: test the differences.

Watch out: pool counts, not proportions.

Studies

Observational: record. Experiment: assign.
Confounding hides cause in observational data.

Watch out: association is not causation.

Experimental design

Compare · randomize · replicate · block. Treatments = combinations of factor levels.

Watch out: blocks (experiments) vs strata (samples).

z-scores and the normal model

$z = (x - \mu)/\sigma$ ; $x = \mu + z\sigma$ .
68–95–99.7 within 1, 2, 3 SDs.

Watch out: the percentages are for both sides together.

Sampling variability

Statistics vary from sample to sample.
Sampling distribution: a statistic over all samples.
Larger $n$ : less variability.

Watch out: data distribution vs sampling distribution.

Interval estimates

Estimate ± ME, ME = critical value × SE; ME = half the width.

Watch out: width = 2 × ME.

Interval for p

$\hat{p} \pm z^*\sqrt{\hat{p}(1 - \hat{p})/n}$ ; check random and large counts.

Watch out: "confident", not "probability", for a computed interval.

Discrete distributions

Values with probabilities in $[0, 1]$ summing to 1.
Add probabilities for events.

Watch out: values are not automatically equally likely.

Expected value and SD

$E(X) = \sum x P(x)$ ; $\text{SD} = \sqrt{\sum (x - \mu)^2 P(x)}$ .

Watch out: weight by probabilities; include losses.

Long run

Probability = long-run relative frequency.
Simulate with random numbers.

Watch out: no memory, no "due" outcomes.

Trees

Multiply along paths; add paths to the same end.
$P(A \mid B) = P(A \text{ and } B)/P(B)$ .

Watch out: with rare events, most positives can be false.

The idea of a limit

$\displaystyle\lim_{x \to a} f(x)$ : where $f(x)$ heads as $x$ approaches $a$ .
$f(a)$ itself does not matter (it may not even exist).
$\dfrac{0}{0}$ means: simplify, then look again.

Watch out: the limit is not always $f(a)$ .

Limits with algebra

Substitute. A number? That is the limit.
$\dfrac{0}{0}$ ? Factor and cancel, expand, or use conjugates.
Substitute again.

Watch out: $\dfrac{0}{0}$ is never the answer.

One-sided limits

$x \to a^-$ : from the left; $x \to a^+$ : from the right.
The limit exists only if both sides agree.

Watch out: one side alone is not the two-sided limit.

Special limits

$\dfrac{\sin u}{u} \to 1$ as $u \to 0$ (radians); so $\dfrac{\sin(ax)}{x} \to a$ .
Squeeze: trapped between two functions with the same limit.

Watch out: $\dfrac{\sin(ax)}{x}$ tends to $a$ , not 1.

Limits at infinity

Divide by the highest power in the denominator.
Top degree smaller: 0. Equal: ratio of leading coefficients. Larger: $\pm\infty$ .
A finite limit $L$ gives the asymptote $y = L$ .

Watch out: constants do not decide the limit at infinity.

Infinite limits

Denominator → 0, numerator not: $\pm\infty$ , a vertical asymptote.
Test a value just beside $a$ for the sign.

Watch out: left and right of the asymptote, the signs can differ.

Continuity

$f(a)$ exists.
$\displaystyle\lim_{x \to a} f(x)$ exists.
They are equal.

Watch out: a limit that exists is not enough; $f(a)$ must match it.

The Intermediate Value Theorem

Check that $f$ is continuous on $[a, b]$ .
Compare $f(a)$ and $f(b)$ with $N$ (often 0).
If $N$ is between them, $f(c) = N$ for some $c$ in $(a, b)$ .

Watch out: no continuity, no conclusion.

Velocity

Average: $\dfrac{s(a + h) - s(a)}{h}$ , change in position ÷ change in time.
Instantaneous: the limit of that as $h \to 0$ .

Watch out: divide by the time; a change in position alone is not a velocity.

The derivative at a point

Write $f(a + h) - f(a)$ and simplify.
Divide by $h$ .
Let $h \to 0$ : the result is $f'(a)$ .

Watch out: $f'(a)$ is a slope, not the value $f(a)$ .

The tangent line

Point: $(a, f(a))$ . Slope: $f'(a)$ .
$y = f(a) + f'(a)(x - a)$ .

Watch out: $y = f'(a)\,x$ has the right slope but usually misses the point.

The derivative function

$f'(x)$ is the slope of $f$ at each $x$ .
$f$ rises: $f' > 0$ . $f$ falls: $f' < 0$ . Horizontal tangent: $f' = 0$ .

Watch out: $f' = 0$ (flat) is not the same as $f = 0$ (on the axis).

Reading a derivative

Units: output units per input unit.
$f'(a) = r$ : at $x = a$ , $f$ changes by about $r$ per extra unit of $x$ .

Watch out: $f'(a)$ is a rate at one moment, not a value and not a total change.

The second derivative

$f'' > 0$ : slopes grow, concave up. $f'' < 0$ : slopes shrink, concave down.
Inflection point: the concavity changes; there $f'$ has a maximum or minimum.
Acceleration is the derivative of velocity.

Watch out: $f'$ says rising or falling; $f''$ says how it bends.

Differentiability

Differentiable ⇒ continuous; not the other way round.
No derivative at a corner, a cusp, a vertical tangent, or a break.
To join two pieces smoothly, match the heights and the slopes.

Watch out: an unbroken graph can still have no tangent line.

Local linearization

$L(x) = f(a) + f'(a)(x - a)$ , the tangent line used as an estimate.
Good near $a$ ; concave up underestimates, concave down overestimates.

Watch out: multiply the step $x - a$ by the slope $f'(a)$ .

Basic rules

$(x^n)' = n x^{n-1}$ ; rewrite roots and fractions as powers first.
Constants factor out; sums split; a constant alone gives 0.

Watch out: with negative exponents, $-2 - 1 = -3$ .

Exponentials

$(e^x)' = e^x$ .
$(b^x)' = b^x \ln b$ .

Watch out: $x^n$ (variable base) and $b^x$ (variable exponent) have different rules.

Sine and cosine

$(\sin x)' = \cos x$ .
$(\cos x)' = -\sin x$ .

Watch out: the minus sign goes with the derivative of cosine; radians only.

Product rule

$(fg)' = f'g + fg'$ : differentiate one factor at a time, keep the other.

Watch out: $(fg)' \ne f'g'$ .

Quotient rule

$\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$ : "low d-high minus high d-low, over low squared".

Watch out: the order in the numerator matters.

tan, sec, cot, csc

$(\tan x)' = \sec^2 x$ , $(\sec x)' = \sec x\tan x$ .
$(\cot x)' = -\csc^2 x$ , $(\csc x)' = -\csc x\cot x$ .

Watch out: co-functions take a minus sign. You may also write $\sec^2 x$ as $\dfrac{1}{\cos^2 x}$ .

Chain rule

$\big(f(g(x))\big)' = f'(g(x))\,g'(x)$ : outside's derivative (inside kept) times inside's derivative.

Watch out: do not forget the factor $g'(x)$ .

Inverse functions

$(\ln x)' = \dfrac{1}{x}$ , $(\ln u)' = \dfrac{u'}{u}$ .
$(\arcsin x)' = \dfrac{1}{\sqrt{1 - x^2}}$ , $(\arctan x)' = \dfrac{1}{1 + x^2}$ .

Watch out: the chain rule still applies to the inside.

Implicit differentiation

Differentiate both sides with respect to $x$ .
Each $y$ term gets a factor $y'$ ; use the product rule for $xy$ .
Solve for $y'$ .

Watch out: $(y^2)' = 2y\,y'$ , not $2y$ .

Tangents to implicit curves

Slope: substitute the point into $y'$ (both $x$ and $y$ ).
Line: $y = y_0 + m(x - x_0)$ .

Watch out: the tangent to a circle is perpendicular to the radius.

Derivative of an inverse

Find $a = f^{-1}(b)$ .
Compute $f'(a)$ .
$(f^{-1})'(b) = 1 / f'(a)$ .

Watch out: evaluate $f'$ at $a$ , the input of $f$ , not at $b$ .

Related rates

Equation linking the quantities.
Differentiate with respect to $t$ .
Substitute the values at that moment; solve for the unknown rate.

Watch out: substitute changing values only after differentiating.

Right-triangle rates

$x^2 + y^2 = D^2 \Rightarrow x\,x' + y\,y' = D\,D'$ .
Find the missing side at that moment first.

Watch out: rates at right angles do not add; decreasing lengths have negative rates.

Similar triangles

Use the fixed ratio to write one variable in terms of another.
Differentiate with respect to $t$ .
Substitute the moment's values.

Watch out: the big triangle's base runs from the lamp to the shadow's tip.

L'Hôpital's rule

Check the form is $0/0$ or $\infty/\infty$ .
Differentiate top and bottom separately.
Take the new limit (repeat if still $0/0$ ).

Watch out: no indeterminate form, no L'Hôpital.

First derivative test

Find the critical numbers.
Sign chart of $f'$ .
$+ \to -$ : max; $- \to +$ : min; no change: neither.

Watch out: $f'(c) = 0$ alone does not make an extremum.

Second derivative test

$f' = 0$ , $f'' > 0$ : min. $f' = 0$ , $f'' < 0$ : max. $f'' = 0$ : test fails.
Inflection: $f''$ changes sign.

Watch out: $f''(c) = 0$ is not enough for an inflection point.

Closed interval method

Critical numbers inside $(a, b)$ .
Evaluate $f$ there and at $a$ and $b$ .
Largest = maximum, smallest = minimum.

Watch out: the EVT needs continuity on a closed interval. Give the value, not just where it occurs, when asked.

Mean Value Theorem

Check: continuous on $[a, b]$ , differentiable on $(a, b)$ .
Average rate $\dfrac{f(b) - f(a)}{b - a}$ .
Solve $f'(c) =$ that rate, with $c$ in $(a, b)$ .

Watch out: $c$ is not always the midpoint.

Motion on a line

$v = s'$ , $a = v'$ ; at rest when $v = 0$ .
Same signs of $v$ and $a$ : speeding up.
Distance: add each leg between turns.

Watch out: displacement and distance differ when the particle turns back.

Optimization

Quantity and constraint.
One-variable function and its domain.
Critical numbers vs endpoints.
Answer the question: the value or the place, with units.

Watch out: not every critical number is the optimum.

Distance from velocity

Area under $v$ = distance (when $v \ge 0$ ).
Signed area (below the axis negative) = displacement.

Watch out: use the right area formula, ½ for triangles.

Riemann sums

$\Delta x = (b - a)/n$ ; add height × $\Delta x$ .
Left, right or midpoint heights.
Increasing $f$ : left under, right over.

Watch out: multiply by the width.

The definite integral

Signed area: below the axis is negative.
Reverse the limits: change the sign. Adjacent intervals add.
$\displaystyle\int_a^b c\,dx = c(b - a)$ .

Watch out: a constant is integrated over the whole length.

Antiderivatives

$x^n \to \dfrac{x^{n+1}}{n+1}$ ; $\tfrac{1}{x} \to \ln|x|$ ; $e^x \to e^x$ ; $\cos \to \sin$ ; $\sin \to -\cos$ .
Use a given value to find $C$ .

Watch out: check by differentiating.

Fundamental Theorem (evaluation)

$\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)$ , with $F' = f$ .

Watch out: upper minus lower; check the antiderivative's signs.

Average value

$f_{\text{avg}} = \dfrac{1}{b - a}\displaystyle\int_a^b f(x)\,dx$ .

Watch out: divide by the length; do not just average the end values.

Accumulation functions

$\dfrac{d}{dx}\displaystyle\int_a^x f(t)\,dt = f(x)$ .
Upper limit $u(x)$ : multiply by $u'(x)$ .
$g$ rises where $f > 0$ , falls where $f < 0$ .

Watch out: the answer is the integrand, not its antiderivative.

Net change

$F(b) = F(a) + \displaystyle\int_a^b F'(t)\,dt$ : start + accumulated change.

Watch out: a rate at one moment is not an amount.

Substitution

Choose the inside $u = g(x)$ .
Replace $g'(x)\,dx$ by $du$ (adjust a constant).
Integrate in $u$ , then return to $x$ .

Watch out: keep track of constant factors from $du$ .

Definite integrals by substitution

New limits: $u(a)$ and $u(b)$ .
Integrate in $u$ between them.

Watch out: never mix $x$ -limits with a $u$ -integral.

Useful patterns

$\dfrac{g'}{g} \to \ln|g|$ .
$e^{kx} \to \dfrac{e^{kx}}{k}$ .
$\tan x \to -\ln|\cos x|$ .

Watch out: a minus sign from $du$ .

Algebra first

Divide term by term when the bottom is one term.
Complete the square for $\dfrac{1}{x^2 + bx + c}$ , then arctan.

Watch out: $\dfrac{1}{a + b} \ne \dfrac{1}{a} + \dfrac{1}{b}$ .

Differential equations

Check a solution: compute its derivatives, substitute, compare.
Equilibrium: set the rate equal to 0.

Watch out: equilibrium means $y' = 0$ , not $y = 0$ .

Slope fields

Slope at $(x, y)$ : substitute into $f(x, y)$ .
Only $x$ : parallel along verticals. Only $y$ : along horizontals.
$f = 0$ : horizontal segments (equilibria if constant in $y$ ).

Watch out: $(x, y)$ in that order.

Separation of variables

$\dfrac{dy}{h(y)} = g(x)\,dx$ .
Integrate both sides, one $+C$ .
Solve for $y$ ; use the initial value.

Watch out: $e^{x + C} = Ae^x$ , not $e^x + C$ .

Exponential models

$y' = ky$ , $y(0) = y_0 \Rightarrow y = y_0e^{kt}$ .
Decay: $k < 0$ ; $k = -\ln 2 / (\text{half-life})$ .

Watch out: equal times multiply by equal factors.

Newton's law of cooling

$T' = -k(T - A)$ .
$T = A + (T_0 - A)e^{-kt}$ , approaching $A$ .

Watch out: it is the difference $T - A$ that decays.

Motion with integrals

Position: start + $\int v$ .
Displacement $\int v$ ; distance $\int |v|$ .

Watch out: split where $v$ changes sign.

Area between curves

Find where they meet.
Decide which is on top.
$\int (\text{top} - \text{bottom})\,dx$ .

Watch out: a negative result means the order is reversed.

Area in $y$

Right minus left, $dy$ .
Limits: the $y$ -values where the curves meet.

Watch out: $x$ -values are not $y$ -limits.

Known cross-sections

Side $s(x)$ from the base.
Area $A(x)$ of the shape.
$V = \int A(x)\,dx$ .

Watch out: a diameter is twice the radius.

Disc method

$V = \pi\displaystyle\int_a^b (\text{radius})^2\,dx$ .

Watch out: square the radius; keep the $\pi$ .

Washer method

$V = \pi\displaystyle\int_a^b (R^2 - r^2)\,dx$ , radii from the axis.

Watch out: $R^2 - r^2$ , not $(R - r)^2$ .

Integration by parts

$\displaystyle\int u\,dv = uv - \int v\,du$ .

Watch out: the minus sign, and signs inside $\int v\,du$ .

Choosing $u$

LIATE: logs, inverse trig, powers, trig, exponentials.
$x^n e^x$ : parts $n$ times.

Watch out: if the new integral is harder, swap $u$ and $dv$ .

Partial fractions

Factor the denominator.
Find $A$ , $B$ by cover-up.
Integrate: $\displaystyle\int \dfrac{A}{x - r}\,dx = A\ln|x - r|$ .

Watch out: check by recombining the fractions.

Which technique?

Inside + its derivative: substitution.
Unrelated product: parts.
Factorable denominator: partial fractions.

Watch out: $x^2 + a^2$ (arctan) and $x^2 - a^2$ (fractions) differ.

Infinite intervals

Replace $\infty$ by $b$ .
Integrate.
Take $\lim_{b \to \infty}$ : a number means convergence.

Watch out: $f \to 0$ does not guarantee convergence.

Infinite discontinuities

Look for asymptotes inside $[a, b]$ before integrating.
Replace the bad end by $t$ ; take the one-sided limit.

Watch out: never apply the FTC across an asymptote.

The p-test

$\int_1^{\infty} x^{-p}\,dx$ : converges for $p > 1$ .
$\int_0^1 x^{-p}\,dx$ : converges for $p < 1$ .
$p = 1$ : both diverge.

Watch out: the condition flips between $\infty$ and $0$ .

Euler's method

$y_{n+1} = y_n + h\,f(x_n, y_n)$ : slope at the current point × step.

Watch out: update the slope at each new point.

Logistic growth

$P' = rP(1 - P/K)$ ; equilibria $0$ and $K$ .
$P \to K$ ; fastest growth at $K/2$ .

Watch out: $r$ is a rate, $K$ is a population.

The logistic solution

$P(t) = \dfrac{K}{1 + Ae^{-rt}}$ , $A = \dfrac{K - P_0}{P_0}$ , $P \to K$ .

Watch out: check $A$ by computing $P(0)$ .

Parametric slopes

$\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$ .
Horizontal: $y' = 0$ . Vertical: $x' = 0$ .

Watch out: $y$ 's rate on top.

Second derivative

$\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right) \Big/ \dfrac{dx}{dt}$ .

Watch out: the division by $dx/dt$ is needed again.

Arc length

$L = \displaystyle\int_a^b \sqrt{x'(t)^2 + y'(t)^2}\,dt$ .

Watch out: Pythagoras, not a sum.

Vector motion

$\vec v = \vec r\,'$ , $\vec a = \vec v\,'$ , speed $= |\vec v|$ .
Position: start + $\int \vec v$ .

Watch out: speed is a length, not a sum of components.

Polar slopes

$\dfrac{dy}{dx} = \dfrac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}$ .

Watch out: $dr/d\theta$ is not the slope.

Polar area

$A = \tfrac{1}{2}\displaystyle\int_{\alpha}^{\beta} r^2\,d\theta$ .

Watch out: the ½ and the square.

Convergence

Converges: the partial sums have a limit.
$a_n \not\to 0$ : diverges. $a_n \to 0$ : test another way.

Watch out: the harmonic series diverges.

Integral test, p-series

Positive decreasing terms: compare with $\int_1^{\infty} f$ .
$\sum 1/n^p$ : converges iff $p > 1$ .

Watch out: $p = 1$ diverges.

Comparison tests

Below convergent: converges. Above divergent: diverges.
Same leading behavior: limit comparison.

Watch out: below a divergent series tells you nothing.

Alternating series

$b_n$ decreasing to 0: converges.
Error after $N$ terms $\le b_{N+1}$ .
Absolute vs conditional: test $\sum |a_n|$ .

Watch out: the bound uses the first omitted term.

Ratio test

$L = \lim |a_{n+1}/a_n|$ : $L < 1$ converges, $L > 1$ diverges, $L = 1$ unknown.

Watch out: use it for factorials and powers; not for p-series.

Choosing a test

nth-term → known series → alternating → ratio → comparison → integral.

Watch out: a divergent absolute series does not make an alternating series diverge.

Taylor polynomials

$P_n(x) = \displaystyle\sum_{k=0}^{n} \dfrac{f^{(k)}(a)}{k!}(x - a)^k$ .

Watch out: divide by $k!$ ; use powers of $(x - a)$ .

Lagrange error bound

$|R_n| \le \dfrac{M|x - a|^{n+1}}{(n + 1)!}$ , $M \ge |f^{(n+1)}|$ .

Watch out: $n + 1$ everywhere.

Radius and interval

Ratio test → $|x - a| < R$ .
Test $x = a - R$ and $x = a + R$ separately.

Watch out: solve for $|x - a|$ carefully; never skip the endpoints.

Known Maclaurin series

$e^x$ , $\sin x$ (odd), $\cos x$ (even), $\dfrac{1}{1 - x}$ .

Watch out: signs alternate in sin and cos.

New series from old

Multiply by $x^k$ : shift the powers.
Substitute: replace $x$ by $u(x)$ .
Differentiate or integrate term by term.

Watch out: integrating divides by the new power.

Constructions

Compass for equal distances, straightedge for lines; justify with congruent triangles.

Watch out: do not change the compass opening between the two arcs.

Translations

$(x, y) \to (x + a, y + b)$ ; rigid: lengths and angles kept.

Watch out: left and down are negative.

Reflections

$x$ -axis: $(x, -y)$ ; $y$ -axis: $(-x, y)$ ; $y = x$ : $(y, x)$ .

Watch out: across the $x$ -axis, it is $y$ that changes sign.

Rotations about the origin

ccw 90°: $(-y, x)$ ; 180°: $(-x, -y)$ ; cw 90°: $(y, -x)$ .

Watch out: check the quadrant the image lands in.

Symmetry

Regular $n$ -gon: $n$ lines; rotations by $360^\circ/n$ .

Watch out: diagonals of a rectangle are not lines of symmetry.

Angle relationships

Vertical: equal. Straight line: 180°. Parallel lines: corresponding/alternate equal, same-side interior 180°. Triangle: 180°.

Watch out: equal or supplementary? Picture the rotation.

Corresponding parts

The order of letters in a congruence statement gives the matching.

Watch out: match by position, not by appearance.

Congruence criteria

SSS, SAS, ASA, AAS. Not SSA, not AAA.

Watch out: in SAS the angle is between the sides.

Isosceles triangles and bisectors

Base angles equal; perpendicular bisector = points equidistant from the ends.

Watch out: halve 180° − apex for each base angle.

Parallelograms

Opposite sides and angles equal; consecutive angles 180°; diagonals bisect each other.

Watch out: opposite (equal) vs consecutive (supplementary).

Polygon angles

Interior sum (n − 2) · 180°; exterior sum 360°.

Watch out: n − 2 triangles, not n.

Congruence proofs

Facts with reasons → criterion → conclusion → corresponding parts.

Watch out: never use what you are proving as a reason.

Dilations

$(x, y) \to (kx, ky)$ about the origin; lengths × $k$ , angles kept.

Watch out: multiply, do not add.

Similar figures

Lengths × $k$ , perimeters × $k$ , areas × $k^2$ .

Watch out: scale by multiplying, and square $k$ for areas.

Triangle similarity

AA (or SSS, SAS similarity) → corresponding sides proportional.

Watch out: AA works for triangles only.

Proportional segments

$DE \parallel BC$ : $AD/DB = AE/EC$ ; midsegment = half the third side.

Watch out: keep the parts in matching order.

Similar right triangles

$h^2 = pq$ , $a^2 = pc$ , $b^2 = qc$ ; hence $a^2 + b^2 = c^2$ .

Watch out: geometric mean, not average.

Special right triangles

45-45-90: $1 : 1 : \sqrt{2}$ . 30-60-90: $1 : \sqrt{3} : 2$ .

Watch out: √2 vs √3.

Trigonometric ratios

SOH-CAH-TOA, for an acute angle of a right triangle.

Watch out: opposite and adjacent depend on which angle you look from.

Finding sides

Label → choose the ratio → solve (multiply or divide).

Watch out: the hypotenuse is the longest side; check your answer.

Finding angles

$A = \sin^{-1}$ , $\cos^{-1}$ or $\tan^{-1}$ of the side ratio.

Watch out: inverse, not reciprocal; degree mode.

Complementary angles

$\sin x = \cos(90^\circ - x)$ .

Watch out: equal values need complementary angles, not the same angle.

Area of any triangle

½ab sin C, with C between a and b.

Watch out: use the included angle.

Law of sines

a / sin A = b / sin B = c / sin C.

Watch out: pair each side with its opposite angle.

Law of cosines

c² = a² + b² − 2ab cos C.

Watch out: Pythagoras alone works only for a right angle.

Cross sections and rotations

Parallel slices copy the base; rectangle → cylinder, right triangle → cone, semicircle → sphere.

Watch out: the angle of the slice decides the shape.

Scaling solids

Length $k$ , area $k^2$ , volume $k^3$ .

Watch out: cube roots undo volume ratios.

Prisms and cylinders

$V = Bh$ ; cylinder $V = \pi r^2 h$ .

Watch out: base area, not circumference.

Pyramids and cones

$V = \tfrac{1}{3}Bh$ .

Watch out: do not forget the ⅓.

Spheres and density

$V = \tfrac{4}{3}\pi r^3$ ; density = mass/volume.

Watch out: $r^3$ for volume, $r^2$ for area.

Distance and midpoint

$d = \sqrt{\Delta x^2 + \Delta y^2}$ ; midpoint = average of the coordinates.

Watch out: average, not half the difference.

Circles

$(x - h)^2 + (y - k)^2 = r^2$ : center $(h, k)$ , radius $r$ .
Expanded form: complete the square in $x$ and in $y$ .

Watch out: $(x + 3)$ means $h = -3$ , and the right side is $r^2$ .

Parabolas

$x^2 = 4py$ : focus $(0, p)$ , directrix $y = -p$ .
$y^2 = 4px$ : focus $(p, 0)$ , directrix $x = -p$ .

Watch out: divide by 4 to get $p$ .

Parallel and perpendicular slopes

Parallel: equal. Perpendicular: $m_1 m_2 = -1$ .

Watch out: flip and change the sign.

Equations of lines

$y = y_1 + m(x - x_1)$ ; find $b$ by substituting the point.

Watch out: parallel lines share the slope, not the intercept.

Partitioning a segment

$A + t(B - A)$ ; ratio $m : n$ gives $t = \dfrac{m}{m + n}$ .

Watch out: count the parts of the ratio.

Inscribed angles

Inscribed = ½ central; on a diameter: 90°.

Watch out: halve, do not double.

Tangent lines

Tangent ⊥ radius; tangent segments from a point are equal.

Watch out: a right triangle, not a subtraction.

Cyclic quadrilaterals

Opposite angles add to $180^\circ$ .

Watch out: supplementary, not equal.

Arcs and sectors

Arc $= \tfrac{\theta}{360} 2\pi r$ , sector $= \tfrac{\theta}{360}\pi r^2$ .

Watch out: circumference for arcs, area for sectors.

Basics

Equally likely: favorable ÷ total.
$P(\text{not } A) = 1 - P(A)$ ; "at least one" via the complement.

Watch out: make sure the outcomes really are equally likely.

Addition rule

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ ; for disjoint events the last term is 0.

Watch out: disjoint is not the same as independent.

Conditional probability

$P(A \mid B) = P(A \text{ and } B) / P(B)$ .

Watch out: $P(A \mid B)$ and $P(B \mid A)$ are different.

Independence

Independent: $P(A \text{ and } B) = P(A)P(B)$ .
In general: $P(A)P(B \mid A)$ .

Watch out: without replacement, update the second probability.

Permutations and combinations

Order matters: nPr. Order does not: nCr = nPr ÷ r!.

Watch out: ask whether swapping two choices changes the result.

Area of parallelograms

$A = b \cdot h$ , with the height at a right angle to the base.

Watch out: the slanted side is not the height.

Area of triangles

$A = \tfrac{1}{2} b h$ .

Watch out: do not forget the ½.

Surface area

Add the areas of all faces (square units).

Watch out: covering is area; filling is volume.

Equivalent ratios

Multiply both parts by the same number.

Watch out: adding the same amount changes the ratio.

Part-part-whole

Whole = sum of parts; one part = total ÷ number of parts.

Watch out: a : b is part to part, not part to whole.

Unit rates

Amount for one: divide by the number of units.

Watch out: which quantity is per what?

Mass and volume

Grams per cm³ = mass ÷ volume; mass = grams per cm³ × volume.

Watch out: divide mass by volume, not volume by mass.

Percent of a quantity

$p\% = p/100$ ; percent = part ÷ whole × 100.

Watch out: divide the part by the whole, not the reverse.

Dividing fractions

Multiply by the reciprocal of the divisor.

Watch out: flip the second fraction, not the first.

Fractional lengths

Multiply lengths for areas and volumes, fractions included.

Watch out: a 1/k cube is 1/k³ of a unit cube.

Adding and subtracting decimals

Line up the points; regroup as with whole numbers.

Watch out: never subtract the smaller digit from the larger by habit.

Multiplying decimals

Multiply, then count the decimal places.

Watch out: estimate to place the point.

Dividing decimals

Scale both numbers to make the divisor whole.

Watch out: dividing by 0.something makes it bigger.

One-step equations

Do the opposite operation to both sides.

Watch out: undo × with ÷ and ÷ with ×.

Equivalent expressions

Distribute to every term; combine like terms with their signs.

Watch out: a minus before parentheses changes every sign inside.

Expressions with exponents

Parentheses, exponents, × ÷, + −.

Watch out: in kx² only x is squared.

Proportional relationships

$y = kx$ , $k = y \div x$ ; the graph passes through $(0, 0)$ .

Watch out: scale by multiplying, not by adding.

Ordering and absolute value

Right is greater; |a| is the distance to 0.

Watch out: −8 is less than −3.

The coordinate plane

x right/left, y up/down; same coordinate: subtract or add across 0.

Watch out: x comes first.

Common factors and multiples

GCF: largest common divisor. LCM: smallest common multiple.

Watch out: do not swap them.

Circle graphs

Count = percent × total; angle = percent × 360°.

Watch out: a percent is not a count.

Mean absolute deviation

Average distance from the mean.

Watch out: distances, not signed differences.

Scale drawings

Drawing → real: multiply by the scale; real → drawing: divide.

Watch out: convert units at the end.

Scaling and area

Lengths × $k$ , areas × $k^2$ .

Watch out: scale both sides before multiplying.

Tables, graphs, equations

Constant ratio; line through the origin; $(1, k)$ ; $y = kx$ .

Watch out: a constant difference is not a constant ratio.

Circumference

$C = \pi d = 2\pi r$ .

Watch out: radius or diameter? Check before using π.

Area of a circle

$A = \pi r^2$ ; around is $C = 2\pi r$ .

Watch out: square the radius, not the diameter.

Percent increase and decrease

New = $(100 \pm p)\%$ of the original; change ÷ original for the percent.

Watch out: the base is the original amount.

Adding and subtracting signed numbers

Move on the number line; subtract = add the opposite.

Watch out: minus a negative is plus.

Multiplying and dividing signed numbers

Same signs: positive. Different signs: negative.

Watch out: negative times negative is positive.

Angle relationships

Complementary: 90°. Supplementary: 180°. Vertical: equal.

Watch out: complement 90, supplement 180.

Building triangles

Two shorter sides must add to more than the longest.

Watch out: equal is not enough.

Slope

Rise over run, the same for every slope triangle on the line.

Watch out: down from left to right means negative.

Slope-intercept form

$y = mx + b$ : rate $m$ , start $b$ .

Watch out: the rate multiplies $x$ ; the starting amount stands alone.

How many solutions?

$3 = 7$ : none. $5 = 5$ : all. $x =$ a number: one.

Watch out: a true statement like $2 = 2$ means infinitely many.

Functions

Exactly one output for each input.

Watch out: shared outputs are fine; two outputs for one input are not.

Exponent rules

Multiply: add. Divide: subtract. Power of a power: multiply.

Watch out: count the factors when in doubt.

Zero and negative exponents

$a^0 = 1$ ; $a^{-n} = 1/a^n$ .

Watch out: negative exponent, small positive number.

Scientific notation

$a \times 10^n$ with $1 \le a < 10$ .

Watch out: count the places the point moves, not the digits.

Computing in scientific notation

Multiply/divide: separately. Add/subtract: same power first.

Watch out: adding does not add the exponents.

Square and cube roots

$\sqrt{a}$ : side of a square of area $a$ ; $\sqrt[3]{a}$ : edge of a cube of volume $a$ .

Watch out: a root is not half or a third.

Pythagorean theorem

$a^2 + b^2 = c^2$ , with $c$ the hypotenuse.

Watch out: for a leg, subtract the squares.

Converse and applications

$a^2 + b^2 = c^2$ means right angle; distances are hypotenuses.

Watch out: always test with the longest side as $c$ .

All zeros from one zero

Check $p(r) = 0$ .
Divide by $x - r$ : $p(x) = (x - r)\,q(x)$ .
Solve $q(x) = 0$ ; list $r$ with those solutions.

Watch out: $x^2 = 9$ gives $3$ and $-3$ .

Tangent and reciprocals

$\tan = \dfrac{\sin}{\cos}$ , $\cot = \dfrac{\cos}{\sin}$ .
$\sec = \dfrac{1}{\cos}$ , $\csc = \dfrac{1}{\sin}$ .

Watch out: sec goes with cos, csc with sin.

Right-triangle trigonometry

SOH CAH TOA.
Side from an angle: multiply ( $\text{opposite} = \text{hyp} \cdot \sin\theta$ ).
Angle from two sides: $\sin^{-1}$ , $\cos^{-1}$ or $\tan^{-1}$ .

Watch out: name the sides from the angle you use.

Inverse trig functions

$\sin^{-1}$ : angle in $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ . $\tan^{-1}$ : in $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ .
$\cos^{-1}$ : angle in $[0, \pi]$ .

Watch out: $\sin^{-1}x$ is an angle, not $\dfrac{1}{\sin x}$ .

Solving sin x = c, cos x = c, tan x = c

Reference angle from $|c|$ .
The two quadrants with the right sign.
Write both solutions in $[0, 2\pi)$ .

Watch out: there are usually two solutions, not one.

Solving by factoring

One side 0, then factor (or set $u = \sin x$ ).
Each factor = 0.
Solve each $\sin x = u$ in $[0, 2\pi)$ .

Watch out: never divide by $\sin x$ or $\cos x$ .

Multiple angles

Let $\theta = nx$ : it runs over $[0, 2n\pi)$ .
List every $\theta$ there.
Divide each by $n$ .

Watch out: $\sin 2x$ has twice as many solutions as $\sin x$ .

Sum and difference identities

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$ .
$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$ .

Watch out: $\cos(A + B) \ne \cos A + \cos B$ ; cosine flips the sign.

Double-angle identities

$\sin 2x = 2\sin x\cos x$ .
$\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$ .

Watch out: $\sin 2x \ne 2\sin x$ (a sine is never above 1, but $2\sin x$ can be).

Equations with identities

Rewrite with one angle (double-angle identities).
One side 0, factor (or $u = \sin x$ , $u = \cos x$ ).
Solve each factor in $[0, 2\pi)$ .

Watch out: factor, never divide by $\sin x$ or $\cos x$ .

Magnitude and direction

$|\langle a, b \rangle| = \sqrt{a^2 + b^2}$ .
$\tan\theta = \dfrac{b}{a}$ ; sketch the vector to pick the quadrant.
Pointing left: add $180^\circ$ to the calculator's angle.

Watch out: the magnitude is not $a + b$ .

Vector operations

Add, subtract, scale: component by component.
$\vec{u} + \vec{v}$ : tip to tail.
$|\vec{u} + \vec{v}|$ : add first, then take the length.

Watch out: $|\vec{u} + \vec{v}| \ne |\vec{u}| + |\vec{v}|$ unless they point the same way.

From magnitude and direction

$\langle r\cos\theta,\ r\sin\theta \rangle$ .
Unit vector: $\dfrac{\vec{v}}{|\vec{v}|}$ , length 1.

Watch out: horizontal goes with cosine, vertical with sine.

The dot product

$\langle a, b \rangle \cdot \langle c, d \rangle = ac + bd$ , a number.
$\cos\theta = \dfrac{\vec{u} \cdot \vec{v}}{|\vec{u}|\,|\vec{v}|}$ .
$\vec{u} \cdot \vec{v} = 0$ : perpendicular.

Watch out: multiply matching components, not crossed ones.

Determinants and inverses

det = ad − bc; inverse = (1/det)(d, −b; −c, a) when det ≠ 0.

Watch out: the inverse is not the matrix of reciprocals.

Systems with matrices

AX = B, so X = A⁻¹B when det A ≠ 0.

Watch out: one row per equation; multiply by A⁻¹ on the left.

Parametric equations

The point at time $t$ is $\big(x(t), y(t)\big)$ .
"Where at time $t$ ": evaluate. "When at $x = c$ ": solve $x(t) = c$ .

Watch out: $x$ first, then $y$ .

Eliminating the parameter

Solve one equation for $t$ .
Substitute into the other.
With $\cos t$ and $\sin t$ : use $\cos^2 t + \sin^2 t = 1$ .

Watch out: solve for $t$ ; do not just replace $t$ by $x$ .

Circles and ellipses

$x = a\cos t$ , $y = b\sin t$ : half-axes $a$ (horizontal) and $b$ (vertical).
Counterclockwise from $(a, 0)$ ; swapping $\sin$ and $\cos$ reverses it.
$\cos(kt)$ , $\sin(kt)$ : one turn in $\dfrac{2\pi}{k}$ .

Watch out: the number with $\cos$ in $x$ is the horizontal half-axis.

Polar coordinates

$x = r\cos\theta$ , $y = r\sin\theta$ .
$r = \sqrt{x^2 + y^2}$ , $\tan\theta = \dfrac{y}{x}$ .

Watch out: choose $\theta$ in the quadrant of the point.

Polar equations

Convert with $x = r\cos\theta$ , $y = r\sin\theta$ , $r^2 = x^2 + y^2$ .
$r = a$ : circle; $\theta = \alpha$ : line; $r = 2a\sin\theta$ : circle of radius $|a|$ through the origin.

Watch out: in $r = 2a\sin\theta$ , $2a$ is the diameter.

Polar form of a complex number

$z = r(\cos\theta + i\sin\theta)$ .
$r = \sqrt{a^2 + b^2}$ ; $\tan\theta = \dfrac{b}{a}$ , quadrant of $(a, b)$ .
$a = r\cos\theta$ , $b = r\sin\theta$ .

Watch out: plot the point before choosing $\theta$ .

Products and powers in polar form

Product: multiply moduli, add arguments.
De Moivre: $(r\,\mathrm{cis}\,\theta)^n = r^n\,\mathrm{cis}(n\theta)$ .

Watch out: do not raise the real and imaginary parts separately.

Distance and midpoint in the complex plane

|z − w| and (z + w)/2.

Watch out: distance uses √(x² + y²), not x + y.

Ellipses

$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ : vertices at distance $a$ , co-vertices at $b$ .
Foci: $c^2 = a^2 - b^2$ (larger minus smaller).
The larger denominator gives the major axis.

Watch out: minus for ellipses, plus for hyperbolas.

Hyperbolas

Vertices on the axis of the positive term, at distance $a$ .
Foci: $c^2 = a^2 + b^2$ .
$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ : asymptotes $y = \pm\dfrac{b}{a}x$ .

Watch out: plus for hyperbolas, minus for ellipses.

Which conic?

One variable squared: parabola.
Both, same sign: ellipse (equal coefficients: circle).
Both, opposite signs: hyperbola.

Watch out: look at the squared terms, not the constant.

Infinite geometric series

$|r| < 1$ : $S = \dfrac{a}{1 - r}$ .
$|r| \ge 1$ : no sum.
Repeating decimals are such series.

Watch out: check $|r| < 1$ before using the formula.

Transforming data

$a + bx$ : center → $a + b\cdot$ center; spread → $|b|\cdot$ spread.

Watch out: adding a constant never changes the spread.

Unusual points

Outlier: large residual. Leverage: extreme $x$ .
Influential: removing it changes the line.

Watch out: leverage, not residual size, drives influence.

Sampling methods

SRS · stratified (some from every group) · cluster (all from some groups) · systematic (every $k$ th).

Watch out: proportional allocation in stratified samples.

Scope of inference

Random sampling → generalize.
Random assignment → cause.

Watch out: you need both for a causal claim about a population.

Combining random variables

Means add (and subtract).
Independent: variances add, for sums and differences.

Watch out: never add SDs directly.

Binomial probabilities

$P(X = k) = \binom{n}{k}p^k(1 - p)^{n - k}$ in a binomial setting.

Watch out: multiply by the number of arrangements.

Binomial mean and SD

$\mu = np$ , $\sigma = \sqrt{np(1 - p)}$ .

Watch out: the SD is the square root of $np(1 - p)$ .

Geometric distribution

$P(X = k) = (1 - p)^{k - 1}p$ , $\mu = 1/p$ .

Watch out: $k - 1$ failures before the first success.

Sampling distribution of p̂

$\mu = p$ , $\sigma = \sqrt{p(1 - p)/n}$ ; normal if $np, n(1 - p) \ge 10$ .

Watch out: proportion, not count.

Sampling distribution of x̄

$\mu_{\bar{x}} = \mu$ , $\sigma_{\bar{x}} = \sigma/\sqrt{n}$ .

Watch out: divide by $\sqrt{n}$ , not $n$ .

Central limit theorem

Large $n$ : $\bar{x}$ approximately normal, whatever the population.

Watch out: it is about means, not about the data.

Differences

Independent samples: means subtract, variances add.

Watch out: paired data are not independent.

Bias and variability

Unbiased: centered on the parameter.
Variability: shrinks with $n$ .
10% condition for independence.

Watch out: large does not mean unbiased.

Confidence level

$z^*$ : 1.645, 1.96, 2.576. Higher confidence, wider interval.

Watch out: the level describes the method over many samples.

Sample size

$n \ge (z^*/ME)^2 p(1 - p)$ , $p = 0.5$ if unknown; round up.

Watch out: halving ME needs 4 times the sample.

t interval for μ

$\bar{x} \pm t^* s/\sqrt{n}$ , df $= n - 1$ .

Watch out: t*, not z*, when σ is estimated.

Intervals for differences

Estimate difference ± critical value × √(sum of variances).

Watch out: does the interval contain 0?

Hypotheses

H₀: parameter = value. Hₐ: <, >, or ≠.

Watch out: parameters, never statistics.

z test for p

$z = (\hat{p} - p_0)/\sqrt{p_0(1 - p_0)/n}$ ; P-value assumes H₀.

Watch out: use $p_0$ in the SE.

Conclusions and errors

P ≤ α: reject H₀. Otherwise: fail to reject.
Type I: reject true H₀. Type II: keep false H₀.
Power rises with $n$ .

Watch out: never "accept" or "prove" H₀.

t test for μ

$t = (\bar{x} - \mu_0)/(s/\sqrt{n})$ , df $= n - 1$ ; two-sided: double the tail.

Watch out: failing to reject is not proving H₀.

Chi-square tests

$\chi^2 = \sum (O - E)^2/E$ ; GOF: $E = np$ ; two-way: $E =$ row × column / total.

Watch out: divide by $E$ ; use counts, not percentages.

The regression model

$\hat{y} = a + bx$ estimates $\mu_y = \beta_0 + \beta_1 x$ . Conditions: LINER.

Watch out: no condition on the size of $r$ .

t for the slope

$t = b/SE_b$ , df $= n - 2$ .

Watch out: divide by $SE_b$ ; two parameters, so $n - 2$ .

Interval for β₁

$b \pm t^* SE_b$ (df $n - 2$ ); 0 outside → evidence of a relationship.

Watch out: association, not causation.

Testing the slope

H₀: β₁ = 0; $t = b/SE_b$ , df $= n - 2$ .

Watch out: a significant slope is not proof of causation.