Foundations for Calculus reference sheet

This sheet follows the eight units of the course in order. Each unit starts with the big idea in a few sentences, then gives the definitions, formulas and rules you will look up most often, with the conditions they need. When you want the full explanation or more examples, follow the lesson links under each topic.

Every variable is a real number, and angles are in radians unless they are marked in degrees. A formula holds wherever both sides are defined unless a condition says otherwise.

Reading Quantities and Graphs

Calculus reads formulas, intervals and graphs constantly, usually without stopping to explain them. This unit makes that reading automatic, from substituting into a formula and describing allowed values on the number line to functions, graphs, lines, average rates of change and absolute value.

Formulas and number lines

To evaluate a formula, replace the variable by the input at every place it appears, with the same value each time and inside parentheses. Then follow the order of operations, which means grouping symbols first, then powers and roots, then multiplication and division from left to right, and last addition and subtraction from left to right. An exponent applies only to the symbol or parenthesized group just before it, so $(-3)^2 = 9$ while $-3^2 = -9$. For $q(x) = x^2 - 4x$ the input $a + h$ goes in as one package,

Terms that are added or subtracted must have the same unit, and an answer to an applied question is complete only with its unit. In the cost $C(n) = 40 + 12n$ dollars for $n$ hours of work, the $12$ is a rate in dollars per hour.

For real numbers $a < b$ the intervals are

A square bracket, drawn as a filled dot, means the endpoint belongs to the set, and a round parenthesis, drawn as a hollow dot, means it does not. The symbols $\infty$ and $-\infty$ are not real numbers, so they always get a parenthesis. The union $A \cup B$ holds every number in at least one of the two sets, so the condition $x < -2$ or $x \ge 5$ is $(-\infty, -2) \cup [5, \infty)$.

The allowed values of a formula are the real inputs at which every step of the calculation is defined. A denominator may not be $0$, and the radicand of an even root, such as a square root, must be at least $0$. A situation can restrict the inputs further, for example to times $t \ge 0$. The formula

needs both conditions at once, so its allowed values are $[2, 3) \cup (3, \infty)$.

Lessons Reading formulas, Number lines and intervals

Functions and graphs

A function $f$ from a set $D$ to a set $Y$ gives every input $x$ in $D$ exactly one output $f(x)$ in $Y$. Different inputs may share an output, but one input may never have two. The domain $D$ is part of the function. When only a formula is given, the domain is every real number for which the formula gives a real number, and a context such as time or length can restrict it further. The range is the set of outputs $\{f(x) \mid x \in D\}$.

The symbol $f(x)$ is the output at the input $x$, not $f$ times $x$, and evaluating at an expression means substituting the whole expression. For $f(x) = x^2 + 1$,

and $f(1 + 2) = f(3) = 10$ while $f(1) + f(2) = 7$.

The graph of $f$ is the set $\{(x, f(x)) \mid x \in D\}$, with the input on the horizontal axis and the output on the vertical axis. A point $(a, b)$ lies on it exactly when $a$ is in $D$ and $b = f(a)$, so every vertical line meets the graph of a function at most once. How steep a graph looks depends on the scales of the axes, so a slope is computed from coordinates, never read from the drawing. A table of measurements gives only the listed points, and a value between them is an estimate that rests on an added assumption.

Lessons Functions, Reading graphs

Lines, rates and distance

The slope of the line through $(x_1, y_1)$ and $(x_2, y_2)$, with $x_1 \ne x_2$, is

rise over run, with the points in the same order on top and bottom. Every line is the set of points with $Ax + By = C$, where $A$ and $B$ are not both zero.

For example the points $(-2, 5)$ and $(4, -1)$ give $m = \frac{-6}{6} = -1$, and then $5 = 2 + b$ gives the line $y = -x + 3$.

For $a \ne b$ in the domain of $f$, the change in input is $\Delta x = b - a$, the change in output is $\Delta y = f(b) - f(a)$, and the average rate of change is

It is the slope of the secant line through $(a, f(a))$ and $(b, f(b))$, measured in output units per input unit, and it describes the interval as a whole, not any single moment. A tank holding $W(t) = 200 + 3t^2$ liters after $t$ minutes gains on average $\frac{308 - 200}{6} = 18$ liters per minute over $[0, 6]$, but only $6$ liters per minute over $[0, 2]$. For a position $s(t)$ on a line the average rate is the average velocity. Average speed is the total distance divided by the elapsed time, which is never negative and is larger than the size of the average velocity when the object turns around.

The absolute value of a real number $x$ is

For every real $x$ we have $\lvert x \rvert \ge 0$, with $\lvert x \rvert = 0$ only at $x = 0$, and $\sqrt{x^2} = \lvert x \rvert$. The distance between $a$ and $b$ on the number line is $\lvert a - b \rvert = \lvert b - a \rvert$. The equation $\lvert x - a \rvert = d$ has the two solutions $a - d$ and $a + d$ when $d > 0$, only $x = a$ when $d = 0$, and none when $d < 0$. So $\lvert x - 2 \rvert = 5$ gives $x = -3$ or $x = 7$.

The graph of $y = \lvert x - a \rvert$ is a V with vertex $(a, 0)$, slope $-1$ to the left of it and slope $1$ to the right, and symmetry about the line $x = a$. On all real inputs its range is $[0, \infty)$. The vertex is a corner, where no single slope describes the graph.

Lessons Lines from slope and intercepts, Average rate of change, Absolute value

Choosing Useful Algebra

The same expression can be written in many equivalent forms, and each form is good at something different. This unit picks the form that fits the job and keeps track of every excluded value along the way. It then solves equations and inequalities by steps that neither add nor lose solutions, or checks the candidates when a step might.

Expanding and factoring

Terms are joined by addition at the top level and factors by multiplication. Only a factor shared by the whole numerator and the whole denominator can be canceled, and only where it is not zero. For all real $a$ and $b$,

Squaring a sum is not squaring each piece, so $(x + 5)^2 = x^2 + 10x + 25$, not $x^2 + 25$. A sum of squares $x^2 + c^2$ with $c \ne 0$ does not factor over the reals. A polynomial is factored completely over the reals when every factor is linear or a quadratic with no real zero, and a quadratic $px^2 + qx + r$ with $p \ne 0$ splits further exactly when $q^2 - 4pr \ge 0$. For example

Two expressions are equivalent on a set when they have the same value at every input of that set. Expanding a factored form back confirms a rewrite. Agreement at a few numbers proves nothing, though one disagreement shows that two expressions are not equivalent.

Lessons Expanding and factoring

Cancellation and fractions

The excluded values of a rational expression are the real zeros of its denominator, found before any simplification.

A simplified formula agrees with the original only on the original domain, so excluded values stay excluded. For example $\frac{x^2 - 4}{x - 2} = x + 2$ for $x \ne 2$, and its graph is the line $y = x + 2$ with an open circle at $(2, 4)$. Cancel factors, never terms, since $\frac{x + 6}{x + 2}$ is not $3$. To simplify a complex fraction, multiply its numerator and denominator by the least common denominator of the inner fractions, which is nonzero on the original domain. For $x \ne 0$ and $x \ne 2$,

Lessons Cancellation and excluded values, Common denominators

Powers, roots and signs

For a positive integer $n$ the power $a^n$ is the product of $n$ factors equal to $a$. A negative exponent means a reciprocal, never a negative number.

An even root needs a radicand of at least $0$ and is never negative, while an odd root accepts every real number and keeps its sign, as in $\sqrt[3]{-8} = -2$. The equation $x^2 = 9$ has the two solutions $\pm 3$, but $\sqrt{9} = 3$ only. In this course a rational exponent has a base $a > 0$, or the base $0$ with a positive exponent, where $0^{m/n} = 0$. An odd root of a negative number is written with the radical sign, because the exponent rules break for negative bases. The expression $0^{-n}$ is undefined. In algebra $0^0 = 1$ by convention, so a polynomial still gives $a_0$ at $x = 0$, but in calculus a limit of the form $0^0$ is indeterminate. For example $8^{2/3} = 4$, $16^{-3/4} = \frac{1}{8}$ and $(-2)^{-3} = -\frac{1}{8}$.

Lessons Powers, roots and signs

Solving equations and inequalities

A solution of an equation is a number at which both sides are defined and equal. Write the restrictions of the original equation first, such as nonzero denominators and nonnegative radicands of even roots. Adding the same defined quantity to both sides, or multiplying or dividing by a nonzero constant, gives an equivalent equation. Dividing by a variable expression can lose solutions, so move everything to one side and factor instead. The zero product property says that $AB = 0$ exactly when $A = 0$ or $B = 0$, and it needs a $0$ on one side.

Squaring both sides, or multiplying by an expression that is defined at every allowed input and may be zero there, keeps every solution but can add extra candidates. Check every candidate in the original equation. For example $\sqrt{x + 7} = x - 5$ squares to $(x - 2)(x - 9) = 0$, and only $x = 9$ works, since $x = 2$ gives $3$ on the left and $-3$ on the right.

Adding a number to both sides of an inequality, or multiplying both sides by a positive number, keeps its direction. Multiplying or dividing by a negative number reverses it. Never multiply by an expression such as $x$ whose sign is unknown, and split into cases or use a sign chart instead. For example $\frac{1}{x} < 2$ has the solution $(-\infty, 0) \cup \left(\frac{1}{2}, \infty\right)$. A compound inequality is solved on all its parts at once, so $-1 \le 2x + 3 < 9$ gives $-2 \le x < 3$, the interval $[-2, 3)$.

Lessons Solving equations, Solving inequalities

Conjugates, completing the square and quadratics

For $u \ge 0$ and any real $c$,

Multiplying the numerator and denominator by a conjugate rationalizes a fraction without changing its value wherever the conjugate is nonzero. For example $\frac{4}{3 - \sqrt{5}} = 3 + \sqrt{5}$, and for $x \ge -4$ with $x \ne 0$

The new form gives $\frac{1}{4}$ at $x = 0$, where the original is still undefined. A conjugate that is zero at some input creates a new exclusion there.

Completing the square rests on the identity

With $a \ne 0$ it gives the vertex form

Factor $a$ out of the $x$ terms first, and subtract $a$ times whatever you add inside the parentheses. If $a > 0$ the minimum value is $k$, attained only at $x = h$, and if $a < 0$ the maximum value is $k$. For example $2x^2 + 12x + 5 = 2(x + 3)^2 - 13$ has minimum $-13$ at $x = -3$.

If $a \ne 0$ and $b^2 - 4ac \ge 0$, the real solutions of $ax^2 + bx + c = 0$ are

with the whole numerator divided by $2a$. The discriminant $D = b^2 - 4ac$ tells how many there are.

This course works over the real numbers, so a negative discriminant means that there is no solution. For example $x^2 - 4x + 1 = 0$ has $D = 12$ and the solutions $2 \pm \sqrt{3}$.

Lessons Conjugates, Completing the square, The quadratic formula

Rearranging formulas

To make a variable the subject, undo the operations applied to it in the reverse order. Adding or subtracting the same quantity and multiplying or dividing by a nonzero quantity keep the solutions. Dividing by an expression that contains a parameter needs that expression to be nonzero, and the zero case is examined separately in the original equation. So $y = mx + b$ gives

while for $m = 0$ the line is horizontal and $x$ cannot be made the subject. State the admissible values with the formula and carry them over. For a rectangle with perimeter $P = 2L + 2W$ the width is $W = \frac{P}{2} - L$, and $W > 0$ requires $0 < L < \frac{P}{2}$. Units follow the algebra, so $t = \frac{d}{v}$ comes out in hours when $d$ is in kilometers and $v$ in kilometers per hour.

Lessons Making a variable the subject

Understanding Individual Graphs

A small set of basic graphs comes up again and again in calculus. This unit sketches each one from facts that can be justified, its domain and range, symmetry, direction, intercepts and asymptotes, and then moves and scales those graphs point by point.

Domain and range

The natural domain of a formula is every real $x$ for which the formula, exactly as written, gives a real value. Canceling a factor does not add back the inputs where it was zero. The domain is the intersection of the conditions set by every root, denominator and other restricted piece.

For example $\frac{\sqrt{x + 2}}{x - 3}$ has domain $[-2, 3) \cup (3, \infty)$, while $\frac{x - 3}{\sqrt{x + 2}}$ has domain $(-2, \infty)$. A stated domain, or one forced by the context, replaces the natural domain.

The range is the set of outputs that actually occur. To find it, bound the outputs, then show that every value within the bound is attained by naming an input in the domain. An endpoint belongs to the range only when some input produces it. For example $x^2$ has range $[0, \infty)$ on all real inputs, range $[0, 4]$ on $[-1, 2]$ and range $(0, 4]$ on $(0, 2]$. A minimum or maximum value must be attained, so $x^2$ on $(0, 2]$ has the maximum $4$ and no minimum.

Lessons Finding the domain, Finding the range

Quadratic and power functions

A quadratic function $f(x) = ax^2 + bx + c$ with $a \ne 0$ has a parabola as its graph, opening up when $a > 0$ and down when $a < 0$. Its three common forms agree for every real $x$.

The range is $[k, \infty)$ when $a > 0$ and $(-\infty, k]$ when $a < 0$. Inputs at equal distances from the axis give equal heights, since $f(h + t) = f(h - t)$. For example $x^2 - 4x - 5 = (x - 2)^2 - 9 = (x - 5)(x + 1)$ has vertex $(2, -9)$, intercepts $(0, -5)$, $(-1, 0)$ and $(5, 0)$, and range $[-9, \infty)$.

A function is even when $f(-x) = f(x)$ and odd when $f(-x) = -f(x)$ for every $x$ in its domain, which must contain $-x$ whenever it contains $x$. An even graph is symmetric about the $y$-axis and an odd graph is symmetric about the origin. Most functions are neither. For a positive integer $n$ the power function $x^n$ has domain $\mathbb{R}$.

Higher powers are flatter between $-1$ and $1$ and steeper beyond, since $x^4 < x^2$ for $0 < \lvert x \rvert < 1$ and $x^4 > x^2$ for $\lvert x \rvert > 1$. A cubic $ax^3 + bx^2 + cx + d$ with $a \ne 0$ has its two ends going in opposite directions. Factoring locates its zeros, as $x^3 - 4x = x(x - 2)(x + 2)$, but finding its turning points exactly needs derivatives.

Lessons Quadratic functions, Integer powers

Root functions

The graph of $y = \sqrt{x}$ is the upper half of the sideways parabola $x = y^2$, and the graph of $y = \sqrt[3]{x}$ is the reflection of $y = x^3$ across the line $y = x$. Every $y \ge 0$ is the output of $\sqrt{x}$ at $x = y^2$, and every real $y$ is the output of $\sqrt[3]{x}$ at $x = y^3$.

For $\sqrt{mx + b} + k$ with $m \ne 0$ the only endpoint is at $x = -\frac{b}{m}$, at height $k$. The domain is $\left[-\frac{b}{m}, \infty\right)$ when $m > 0$ and $\left(-\infty, -\frac{b}{m}\right]$ when $m < 0$. So $\sqrt{6 - x}$ has domain $(-\infty, 6]$ and decreases to its endpoint $(6, 0)$. The cube root of a negative number is negative, as in $\sqrt[3]{-27} = -3$. Many calculators read $(-8)^{1/3}$ as a complex number, so graph with a real cube root.

Lessons The square root function, The cube root function

Reciprocal functions

The line $x = a$ is a vertical asymptote when $f(x) \to \infty$ or $f(x) \to -\infty$ as $x$ approaches $a$ from at least one side. The line $y = L$ is a horizontal asymptote when $f(x) \to L$ as $x \to \infty$ or as $x \to -\infty$. Neither reciprocal graph ever reaches $y = 0$.

A quantity $y$ is inversely proportional to $x$ when $y = \frac{k}{x}$ with a constant $k \ne 0$, so the product $xy = k$ stays fixed. An inverse-square law $g(x) = \frac{c}{x^2}$ with $c \ne 0$ satisfies

so doubling the input divides the output by $4$.

Lessons The reciprocal function, The reciprocal square function

Piecewise functions

A piecewise function uses different formulas on different parts of its domain. Find the condition that the input meets first, then use that formula and only that one. An input that meets no condition is not in the domain, and conditions that overlap must give equal values on the overlap. Where a branch ends at $c$, a filled dot marks an endpoint that its condition includes and a hollow dot one that it excludes.

At a switching input $c$, compare the value the left formula approaches, the value the right formula approaches and the actual value $f(c)$. If the left branch approaches $L$ and the right branch approaches $R \ne L$, the graph has a jump of size $\lvert R - L \rvert$. If all three numbers agree the graph has no break there, though it may still have a corner. For example the function

has $f(1) = 3$, a hollow dot at $(1, 1)$, a filled dot at $(1, 3)$ and a jump of size $2$.

Lessons Piecewise functions

Polynomials, sign charts and rational functions

A polynomial of degree $n$ has the form $p(x) = a_n x^n + \cdots + a_1 x + a_0$ with $a_n \ne 0$. It is defined for every real $x$, its graph has no breaks, and for $n \ge 1$ it has at most $n$ real zeros. The number $c$ is a zero exactly when $x - c$ is a factor. A zero $c$ has multiplicity $m$ when $p(x) = (x - c)^m q(x)$ with $q(c) \ne 0$. At a real zero of odd multiplicity the graph crosses the axis, flattening against it when $m \ge 3$, and at one of even multiplicity it touches the axis and turns back. For $n \ge 1$ the end behavior follows the leading term $a_n x^n$.

For example $p(x) = (x + 2)(x - 1)^2(x - 3)$ crosses the axis at $-2$ and $3$, touches it from below at $1$ and rises at both ends.

A sign chart marks the critical inputs, which are the real zeros of every factor of the numerator and of the denominator of the original expression. Between neighboring critical inputs the sign cannot change, so one test value per interval decides it. To solve an inequality, move everything to one side, combine into one fraction and factor. Never multiply by a denominator whose sign is unknown. Include the zeros of the numerator for $\le$ or $\ge$, and never include a zero of a denominator. For example $(x - 1)(x + 3) \le 0$ gives $[-3, 1]$, and

gives $(1, 2]$.

A rational function $\frac{p(x)}{q(x)}$ is undefined at every real zero of the original denominator. Write it in lowest terms $\frac{P(x)}{Q(x)}$ by canceling every common factor.

• A zero $a$ of the original denominator with $Q(a) \ne 0$ leaves a hole at $\left(a, \frac{P(a)}{Q(a)}\right)$.
• A zero $a$ of $Q$ gives a vertical asymptote $x = a$, where the outputs grow without bound in size.

With leading terms $a_n x^n$ on top and $b_m x^m$ below, and $p$ not the zero polynomial, the far ends behave as follows.

The graph can cross a horizontal asymptote, but never a vertical one. Its $x$-intercepts are the zeros of the numerator that are not excluded. For example the function

has a hole at $\left(2, \frac{4}{3}\right)$, the vertical asymptote $x = -1$, the horizontal asymptote $y = 1$, the $x$-intercept $(-2, 0)$ and the $y$-intercept $(0, 2)$. A function $\frac{a}{x - h} + k$ with $a \ne 0$ is the graph of $\frac{1}{x}$ moved so that its asymptotes are $x = h$ and $y = k$, as in $\frac{3x - 5}{x - 2} = 3 + \frac{1}{x - 2}$.

Lessons Polynomial zeros and signs, Sign charts, Rational functions

Moving and scaling graphs

To move a graph, follow its points. For each new function ask which input makes the inside equal to an old input $u$, then apply whatever is done to the output. Here $h$ and $k$ are real numbers and $A$ and $B$ are nonzero.

A negative $A$ or $B$ adds a reflection. The new domain is the set of $x$ whose inside lies in the domain of $f$. In general the graph of $A f\big(B(x - h)\big) + k$ takes each point $(u, f(u))$ to

For example $\sqrt{3 - x} - 1 = f\big(-(x - 3)\big) - 1$ with $f(x) = \sqrt{x}$ has domain $(-\infty, 3]$ and takes the point $(4, 2)$ to $(-1, 1)$. An input scaling and an output scaling are different operations, although for a few functions they coincide, as in $\sqrt{4x} = 2\sqrt{x}$ for $x \ge 0$.

Lessons Shifts and reflections, Stretches and compressions

Connecting Functions and Curves

New functions are built from old ones by combining outputs at the same input, by feeding one output into another function, and by running a function backwards. Comparing two curves then comes down to the sign of their difference, and a circle splits into two functions.

Combining and composing functions

A composition applies $g$ first, and the order matters. With $f(x) = \sqrt{x}$ and $g(x) = x - 4$, the composite $f(g(x)) = \sqrt{x - 4}$ has domain $[4, \infty)$, while $g(f(x)) = \sqrt{x} - 4$ has domain $[0, \infty)$. The product $fg$ multiplies outputs and is a different function from $f \circ g$. Substitute the whole inner expression in parentheses, so $f(x) = x^2 - 2x$ and $g(x) = x + 1$ give

The domain comes from the original functions, not from the simplified formula. The quotient $\frac{x - 1}{\sqrt{x} - 1}$ equals $\sqrt{x} + 1$ only on $[0, 1) \cup (1, \infty)$, and with $f(x) = x^2$ and $g(x) = \sqrt{x}$ the composite $f(g(x))$ equals $x$ only for $x \ge 0$. Along a chain of functions from a real situation, the output units of the inner function must be the input units the outer one expects.

Lessons Adding, multiplying and dividing, Composing functions

Inverse functions

A function is one-to-one on its domain when $f(a) = f(b)$ forces $a = b$, which means that every horizontal line meets its graph at most once. A one-to-one function $f$ with domain $A$ and range $B$ has an inverse $f^{-1}$ with domain $B$ and range $A$, defined by

Then $f^{-1}(f(x)) = x$ for every $x$ in $A$ and $f(f^{-1}(y)) = y$ for every $y$ in $B$. The point $(a, b)$ is on the graph of $f$ exactly when $(b, a)$ is on the graph of $f^{-1}$, so the two graphs are mirror images across the line $y = x$ when the axes share a scale. To find a formula, solve $y = f(x)$ for $x$ and state the domain of the inverse. For example $f(x) = \sqrt{x + 1}$ has the inverse $x^2 - 1$ for $x \ge 0$.

A function that is not one-to-one gets an inverse once its domain is restricted to a piece where it is one-to-one, and the chosen piece must be stated. On $[2, \infty)$ the function $f(x) = (x - 2)^2 + 1$ has the inverse

The symbol $f^{-1}$ means the inverse function, not the reciprocal $\frac{1}{f(x)}$.

Lessons Inverse functions

Intersections and circles

The graphs of $y = f(x)$ and $y = g(x)$ meet at $(a, b)$ when $a$ is in both domains and $f(a) = g(a) = b$, the same input with the same output. Solve $f(x) = g(x)$ on the common domain, and when a step such as squaring was used, check every candidate in the original equation. So $\sqrt{x} = x - 2$ gives the candidates $1$ and $4$, and only $(4, 2)$ is an intersection.

The difference $d(x) = f(x) - g(x)$ is the signed vertical gap, positive where the graph of $f$ is higher. When $f$ and $g$ are continuous, $d$ keeps one sign between its zeros and the breaks of the common domain, so one test input per piece decides which graph is higher. Samples alone can miss two crossings. The values of $x^2 - (3x - 2.16)$ at $x = 0$, $1$, $2$ and $3$ are all positive, yet the line is higher on $(1.2, 1.8)$.

The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is given by the first formula below, and the circle with center $(h, k)$ and radius $r > 0$ by the second.