Skills

Math you need: the one-page sheet

Functions and Their Graphs

A function sends each input to one output. Read it from a formula (substitute with parentheses), a graph (heights, filled and open dots) or a table. Know the domain, write lines in point-slope form, compose functions inside first, move graphs with f(x − h) + k , and find vertical asymptotes where a denominator is 0.

A function gives one output for each input. Substitute with parentheses: if $f(x) = x^2 - 3x$ , then $f(a + 1)$ $= (a + 1)^2 - 3(a + 1)$ .
The domain is every input the rule accepts; the range is every output it produces. Rule out zero denominators and square roots of negatives.
On a graph, a filled dot is on the graph and an open circle is not. $f(a)$ is the height of the graph at $x = a$ .
A line through $(x_1, y_1)$ with slope $m$ is $y - y_1 = m(x - x_1)$ (point-slope form). Slope is rise over run.
$f(g(x))$ means $g$ first, then $f$ . Order matters: usually $f(g(x)) \ne g(f(x))$ .
A rational function has a vertical asymptote where the denominator is 0 and the numerator is not.

f(3) names one number: the height of the graph above x = 3 those inputs are left out of the domain f(g(x)) means: do g first, then f moves the whole graph h units right and k units up

function: A rule that assigns exactly one output to each input. $f(3)$ is the output when the input is 3.
domain: The set of inputs a function accepts. The range is the set of outputs it produces.
graph of a function: All points $(x, f(x))$ for $x$ in the domain. The height of the graph above $x = a$ is $f(a)$ ; a filled dot is on the graph, an open circle is not.
slope: Rise over run: for a line through $(x_1, y_1)$ and $(x_2, y_2)$ , $m$ $= \frac{y_2 - y_1}{x_2 - x_1}$ .
point-slope form: The point-slope form of a line with slope $m$ through $(x_1, y_1)$ : $y - y_1 = m(x - x_1)$ . Slope-intercept form is $y = mx + b$ .
piecewise function: A function defined by different formulas on different parts of its domain, such as $f(x) = x^2$ for $x < 2$ and $f(x) = 3x - 1$ for $x \ge 2$ .
composite function: A composite function feeds one output into another function: $f(g(x))$ applies $g$ first, then $f$ .
graph transformation: A change that moves or reshapes a graph: $f(x - h) + k$ shifts it $h$ right and $k$ up; $-f(x)$ flips it over the $x$ -axis; $a f(x)$ stretches it vertically.
rational function: A quotient of two polynomials, such as $\frac{2x + 1}{x - 4}$ . A polynomial is a sum of terms $a x^n$ with whole-number powers $n$ .
asymptote: A line the graph approaches. A rational function has a vertical asymptote $x = a$ where its denominator is 0 and its numerator is not; a horizontal asymptote is a height the outputs settle toward as $|x|$ grows large.

Exponent Rules and Rewriting Powers

Exponent rules let you write every root and fraction as a power: 1/(x^n) = x^(−n) and n√(x^m) = x^(m/n) . Add exponents to multiply, subtract to divide, multiply for a power of a power, and remember an exponent applies only to the base it touches and never splits over a sum.

Multiply powers of the same base: add exponents. Divide: subtract. Power of a power: multiply.
$x^{-n} = \frac{1}{x^n}$ . A negative exponent means reciprocal, never a negative number.
$x^{m/n} = \left(\sqrt[n]{x}\right)^m$ . Take the root first to keep numbers small: $8^{2/3} = 2^2 = 4$ .
An exponent applies only to what it touches: $3x^{-2} = \frac{3}{x^2}$ , but $(3x)^{-2} = \frac{1}{9x^2}$ .
Exponents do not split over sums: $(x + 3)^{-1} \ne x^{-1} + 3^{-1}$ and $\sqrt{x^2 + 9} \ne x + 3$ .

so x^m⋅x^n = x^(m + n) , and (x^m)/(x^n) = x^(m − n) a negative exponent is a reciprocal: x^(−n) = 1/(x^n) a fractional exponent is a root: x^(1/2) = √x and x^(m/n) = (n√x)^m expressions like 3/x² + 2√x become 3x⁻² + 2x^(1/2) , ready for calculus

exponent rules: The laws for powers of one base: $x^m x^n = x^{m+n}$ , $\frac{x^m}{x^n} = x^{m-n}$ , $(x^m)^n = x^{mn}$ , $x^0 = 1$ . A negative exponent is a reciprocal ( $x^{-n} = \frac{1}{x^n}$ ); a fractional exponent is a root ( $x^{m/n} = (\sqrt[n]{x})^m$ ).

Algebra for Limits: Factoring and Rationalizing

Limits that give zero over zero are settled with precalculus algebra: factor and cancel a common factor, multiply by a conjugate to clear a square root, or combine a complex fraction into one. Cancel factors, never terms, and expand powers in full. The simplified expression agrees with the original everywhere except the troublesome point.

Factoring toolkit: common factor; difference of squares $a^2 - b^2 = (a - b)(a + b)$ ; trinomials (with the $ac$ method when the leading coefficient is not 1); cubes $a^3 \pm b^3$ $= (a \pm b)(a^2 \mp ab + b^2)$ .
Cancel factors, never terms: $\frac{x(x - 4)}{(x - 4)(x + 4)}$ $= \frac{x}{x + 4}$ , but $\frac{x^2 + 4}{x + 2}$ does not simplify.
Rationalizing: multiply top and bottom by the conjugate ( $\sqrt{A} - B$ and $\sqrt{A} + B$ form a pair). Their product is $A - B^2$ . Leave the other factor unmultiplied.
A complex fraction becomes one fraction with a common denominator; then divide.
Expand powers fully: $(a + h)^2 = a^2 + 2ah + h^2$ . The middle term is the one that matters.

factor top and bottom so that factor is visible gives a simpler expression equal to the original for every x ≠ a multiplying by the conjugate √x + 2 turns it into x − 4 , which can cancel a common denominator turns the complex fraction into one fraction you can simplify

factoring: Writing an expression as a product, such as $x^2 - 9 = (x - 3)(x + 3)$ (a difference of squares). A factored form shows which factors can cancel.
rationalizing: Removing a square root from part of a fraction by multiplying top and bottom by the conjugate: $\sqrt{A} - B$ and $\sqrt{A} + B$ are conjugates, and $(\sqrt{A} - B)(\sqrt{A} + B)$ $= A - B^2$ .
complex fraction: A fraction whose top or bottom contains fractions, such as $\frac{\frac{1}{x + 2} - \frac{1}{2}}{x}$ . Combine the inner fractions over a common denominator to simplify it.

Trigonometry and the Unit Circle

Radians measure angles by arc length on the unit circle, where the point at angle θ is (cos θ, sin θ) . Exact values come from the first-quadrant special angles plus the quadrant signs. Tangent, secant, cosecant and cotangent are ratios of sine and cosine, and sin² x + cos² x = 1 links them. Keep calculators in radian mode.

Radians: $\pi$ radians $= 180^{\circ}$ . To convert degrees to radians, multiply by $\frac{\pi}{180}$ . Calculus always uses radians, including on the calculator.
On the unit circle, the point at angle $\theta$ is $(\cos\theta, \sin\theta)$ : cosine is the $x$ -coordinate, sine the $y$ -coordinate.
Know the first-quadrant values at $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$ : $\frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}$ . Other quadrants use a reference angle and a sign.
$\sec = \frac{1}{\cos}$ , $\csc = \frac{1}{\sin}$ , $\cot = \frac{\cos}{\sin}$ . $\sin^2 x$ means $(\sin x)^2$ .
Pythagorean identity: $\sin^2 x + \cos^2 x = 1$ ; also $1 + \tan^2 x = \sec^2 x$ and $\sin 2x = 2\sin x\cos x$ .

radians: a half turn is π , a full turn 2π the point reached is (cos θ, sin θ) , and tan θ = (sin θ)/(cos θ) sin² θ + cos² θ = 1 , the Pythagorean identity gives the exact value at any special angle

radian: A measure of angle: the length of the arc the angle cuts on a circle of radius 1. A full turn is $2\pi$ radians, so $\pi$ radians $= 180^{\circ}$ .
unit circle: The circle of radius 1 centered at the origin, $x^2 + y^2 = 1$ . The point at angle $\theta$ (measured counterclockwise from $(1, 0)$ ) is $(\cos\theta, \sin\theta)$ .
sine: Sine and cosine give the coordinates of the point at angle $\theta$ on the unit circle: $(\cos\theta, \sin\theta)$ . The others are ratios: $\tan\theta = \frac{\sin\theta}{\cos\theta}$ , $\sec\theta = \frac{1}{\cos\theta}$ , $\csc\theta = \frac{1}{\sin\theta}$ , $\cot\theta = \frac{\cos\theta}{\sin\theta}$ .
Pythagorean identity: An equation true for every angle. The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ comes from the unit circle; dividing by $\cos^2\theta$ gives $\tan^2\theta + 1 = \sec^2\theta$ .

Exponentials and Logarithms

e^x is the exponential function calculus uses; it is always positive and passes through (0, 1) . ln x , defined for x > 0 , undoes it: e^(ln x) = x and ln (e^x) = x . Logs turn products into sums, quotients into differences and powers into multipliers, which is how you solve for a variable in an exponent.

$e^x > 0$ for every $x$ ; $e^0 = 1$ . The graph rises steeply to the right and approaches the $x$ -axis to the left.
$\ln x$ is defined only for $x > 0$ ; $\ln 1 = 0$ , $\ln e = 1$ . Its graph is the graph of $e^x$ reflected across $y = x$ .
Log rules: $\ln(ab) = \ln a + \ln b$ , $\ln\frac{a}{b} = \ln a - \ln b$ , $\ln(a^p) = p\ln a$ . There is no rule for $\ln(a + b)$ .
$e$ and $\ln$ cancel: $e^{\ln x} = x$ ( $x > 0$ ) and $\ln(e^x) = x$ . Use them to solve equations.
$A e^{kt}$ grows when $k > 0$ and decays when $k < 0$ ; the time to double is $\frac{\ln 2}{k}$ .

it models growth by a fixed percentage; the base e ≈ 2.718 is the one calculus prefers e^(ln x) = x for x > 0 and ln (e^x) = x for every x give the log rules: ln (ab) = ln a + ln b , ln a/b = ln a − ln b , ln a^p = pln a take ln of both sides to bring it down: 2x = ln 9

exponential function: A function $b^x$ with a fixed positive base $b \ne 1$ ; it changes by the same factor over equal steps in $x$ . The natural exponential $e^x$ uses $e \approx 2.71828$ . $A e^{kt}$ shows exponential growth when $k > 0$ and exponential decay when $k < 0$ .
natural logarithm: $\ln x$ is the power of $e$ that gives $x$ : $\ln x = y$ means $e^y = x$ . It is defined for $x > 0$ . Log rules: $\ln(ab) = \ln a + \ln b$ , $\ln\frac{a}{b} = \ln a - \ln b$ , $\ln a^p = p\ln a$ .

Inverse Functions

An inverse function runs a one-to-one function backward: f⁻¹(b) = a exactly when f(a) = b . Its graph is the reflection of f across y = x , so points swap coordinates and slopes become reciprocals. Find a formula by solving x = f(y) for y . The inverse trig functions use restricted ranges, and f⁻¹ never means 1/f .

$f^{-1}(b) = a$ means $f(a) = b$ . So $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ on the right domains.
$f^{-1}(x)$ is not $\frac{1}{f(x)}$ . The $-1$ means "undo", not "reciprocal".
Only a one-to-one function (each output from one input; passes the horizontal line test) has an inverse. Restrict the domain if needed.
To find a formula: write $x = f(y)$ and solve for $y$ .
$\arcsin x$ and $\arctan x$ return angles in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ ; $\arccos x$ returns angles in $[0, \pi]$ .

the inverse function f⁻¹ sends b back to a : f⁻¹(b) = a exactly when f(a) = b swaps the coordinates of every point, reflecting the graph across y = x slopes at matching points are reciprocals their inverses use restricted ranges: arcsine and arctangent give angles from −π/2 to π/2, arccosine from 0 to π

inverse function: For a one-to-one function $f$ , the function $f^{-1}$ that undoes it: $f^{-1}(b) = a$ exactly when $f(a) = b$ . Its graph is the reflection of the graph of $f$ across $y = x$ . A function is one-to-one when no two inputs share an output.
inverse trigonometric function: $\arcsin x$ , $\arccos x$ and $\arctan x$ (also written $\sin^{-1} x$ , ...) give the angle with a given sine, cosine or tangent, from a restricted range: $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ for arcsine, $[0, \pi]$ for arccosine, $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ for arctangent.

Geometry for Calculus

Calculus word problems start with a geometric equation. Right angles call for the Pythagorean theorem; nested triangles call for similar triangles and a ratio; shapes call for their area or volume formulas. Label changing quantities with letters, fixed ones with numbers, and use a ratio to reduce a formula to one variable.

Pythagorean theorem: in a right triangle with legs $a$ , $b$ and hypotenuse $c$ , $a^2 + b^2 = c^2$ . The hypotenuse is opposite the right angle.
Similar triangles have equal angles, so their sides are in proportion: $\frac{a}{b} = \frac{A}{B}$ . Look for a small triangle inside a big one.
Know the area and volume formulas: circle $\pi r^2$ , semicircle $\frac{1}{2}\pi r^2$ , trapezoid $\frac{b_1 + b_2}{2}h$ , cylinder $\pi r^2 h$ , cone $\frac{1}{3}\pi r^2 h$ , sphere $\frac{4}{3}\pi r^3$ .
Draw the picture, label what changes with letters and what is fixed with numbers, then write one equation.

the Pythagorean theorem links the sides: x² + y² = z² similar triangles: corresponding sides are in the same ratio, such as r/h = 4/10 a formula like V = 1/3 πr²h becomes a function of one variable, V = 4π/75 h³

similar triangles: Triangles with the same three angles. Their corresponding sides are in the same ratio, so $\frac{r}{h} = \frac{R}{H}$ for matching sides.
Pythagorean theorem: In a right triangle with legs $a$ and $b$ and hypotenuse $c$ : $a^2 + b^2 = c^2$ .
volume of a cone: Volume of a cone $\frac{1}{3}\pi r^2 h$ , of a cylinder $\pi r^2 h$ , of a sphere $\frac{4}{3}\pi r^3$ ; area of a trapezoid $\frac{b_1 + b_2}{2}h$ , of a semicircle $\frac{1}{2}\pi r^2$ .

Parametric Equations and Vectors

Parametric equations describe a path by giving x and y as functions of a parameter t ; increasing t traces the path in a direction. Eliminating t shows the shape but loses the direction. A vector ⟨a, b⟩ records a change in position, from start to end; its magnitude is √(a² + b²) , and vectors add and scale component by component.

Parametric equations $x = x(t)$ , $y = y(t)$ give a point for each $t$ . Make a table of $t$ , $x$ , $y$ , plot the points in order of $t$ , and draw arrows for the direction.
Eliminate the parameter to see the shape: solve a linear equation for $t$ and substitute, or square and add sines and cosines. The direction and the timing are lost when you do.
$x = r\cos t$ , $y = r\sin t$ is a circle of radius $r$ traced counterclockwise from $(r, 0)$ ; swapping sine and cosine starts at the top and goes clockwise.
A vector $\left\langle a, b \right\rangle$ is a change: $a$ across and $b$ up. From $P$ to $Q$ it is end minus start; its magnitude is $\sqrt{a^2 + b^2}$ . Add vectors and multiply by numbers one component at a time.

each t gives one point (x(t), y(t)) , and together the points trace a path the path gets a direction (its orientation), which an equation in x and y cannot show eliminating t gives the shape of the path as an equation in x and y a vector, whose length √((Δx)² + (Δy)²) comes from the Pythagorean theorem

parametric equations: A pair of equations $x = x(t)$ , $y = y(t)$ that give both coordinates of a point in terms of a third variable $t$ , the parameter (often time). As $t$ increases, the point traces a path in a direction; eliminating $t$ can give an equation in $x$ and $y$ , but it loses the direction and the timing.
vector: A quantity with a size and a direction, written by its components as $\left\langle a, b \right\rangle$ : $a$ across and $b$ up. The vector from $(x_1, y_1)$ to $(x_2, y_2)$ is $\left\langle x_2 - x_1, y_2 - y_1 \right\rangle$ ; its magnitude (length) is $\sqrt{a^2 + b^2}$ . Vectors add and scale component by component.

Polar Coordinates and Polar Graphs

Polar coordinates name a point by an angle θ and a signed distance r . Convert with x = rcos θ and y = rsin θ . A polar curve r = f(θ) is plotted from a table of θ and r ; watch for r = 0 (the curve meets the pole) and r < 0 (the point is on the opposite ray).

A polar point $(r, \theta)$ : turn to the ray at angle $\theta$ , then go $r$ units. A negative $r$ goes backward, through the pole, onto the opposite ray.
$x = r\cos\theta$ , $y = r\sin\theta$ , $r^2 = x^2 + y^2$ . These connect every polar question to rectangular ones.
A polar curve $r = f(\theta)$ is a parametric curve with parameter $\theta$ : $x = f(\theta)\cos\theta$ , $y = f(\theta)\sin\theta$ .
Shapes to know: $r = a$ (circle), $\theta = c$ (line through the pole), $r = a\cos\theta$ and $r = a\sin\theta$ (circles through the pole), $r = a + b\cos\theta$ (limaçons, cardioid when $a = b$ ), $r = a\sin(n\theta)$ (roses).

you reach the point (r, θ) x = rcos θ and y = rsin θ as θ turns, the point traces a polar curve you see where r = 0 (the pole) and where r < 0 (the opposite ray)

polar coordinates: A way to name a point by $(r, \theta)$ : turn to the ray at angle $\theta$ from the positive $x$ -axis, then go a signed distance $r$ along it. They convert by $x = r\cos\theta$ , $y = r\sin\theta$ and $r^2 = x^2 + y^2$ ; a curve $r = f(\theta)$ is a polar curve.

Sequences, Sigma Notation and Factorials

An infinite sequence lists a term a_n for each n . It converges if its terms approach one number and diverges otherwise; find the limit as for functions at infinity. Simplify factorial ratios by expanding the larger factorial, and read sigma notation from its starting index.

A sequence is a function whose inputs are $n = 1, 2, 3, \ldots$ Its limit is found with the tools for $\lim_{x \to \infty} f(x)$ : leading terms, L'Hospital's rule, the squeeze theorem.
$n! = n(n - 1)(n - 2) \cdots 2 \cdot 1$ , with $0! = 1$ . Simplify ratios by writing the larger factorial down to the smaller: $(n + 1)! = (n + 1)\,n!$ .
Sigma notation $\sum_{k=1}^{m} a_k$ adds the terms from $k = 1$ to $k = m$ . Check the starting index: it decides the first term.

an infinite sequence a_1, a_2, a_3, … , a function of n the terms settle to one number (the sequence converges) or they do not (it diverges) lim as n → ∞, a_n comes from lim as x → ∞, f(x) when that limit exists the common factorial cancels, leaving a few factors

infinite sequence: A list of numbers $a_1, a_2, a_3, \ldots$ with one term for each counting number $n$ ; a function of $n$ . It converges to $L$ if $\lim_{n \to \infty} a_n = L$ , and diverges if its terms do not approach one number.
factorial: For a positive integer $n$ , $n! = n(n - 1)\cdots 2 \cdot 1$ ; also $0! = 1$ . Ratios simplify by expanding the larger factorial: $(n + 1)! = (n + 1)\,n!$ .