AP® Calculus AB Beta

Formulas and theorems to know

The exam gives no formula sheet. These are the formulas and theorems to know by heart, what each is for and when to use it.

Three habits that save points

Name the theorem and check its conditions. Before you use the Intermediate Value Theorem, the Mean Value Theorem or the Extreme Value Theorem, say why the function is continuous (and differentiable, for the Mean Value Theorem) on the interval.
Three decimal places, setup first. On a calculator question, write the integral, derivative or equation you solve, then give the answer correct to three decimal places. Keep full precision until the end.
Radian mode. Every angle in calculus is in radians.

Limits and continuity

Formula	What it is	When you use it	Learn it in
$\lim_{x \to c} f(x)$ $= f(c)$	Continuity at $x = c$ : $f(c)$ is defined, the limit exists, and the two are equal.	To show a function is continuous at a point. For a piecewise function, find both one-sided limits first.	Unit 1 Continuity at a PointPractice it: Limit explorer
$\lim_{x \to 0} \frac{\sin x}{x}$ $= 1$	A special limit, with $x$ in radians, found with the squeeze theorem.	Limits such as $\lim_{x \to 0} \frac{\sin(3x)}{x}$ : rewrite as $3 \cdot \frac{\sin(3x)}{3x}$ .	Unit 1 The Squeeze TheoremPractice it: Limit explorer
$\lim_{x \to 0} \frac{1 - \cos x}{x}$ $= 0$	The companion special limit, also with $x$ in radians.	Limits with $1 - \cos x$ over $x$ ; together with the first, it gives the derivatives of sine and cosine.	Unit 1 The Squeeze Theorem
$g(x) \le f(x) \le h(x)$ $\Rightarrow \lim_{x \to c} f(x)$ $= L$	The squeeze theorem, when $g(x) \le f(x) \le h(x)$ near $c$ and $g$ and $h$ both have limit $L$ at $c$ .	When $f$ is trapped between two functions with the same limit, such as a bounded factor like $\sin\frac{1}{x}$ times something that goes to 0.	Unit 1 The Squeeze TheoremPractice it: Limit explorer
$f(c)$ $= d \ \text{for some}\ c \ \text{in}\ [a, b]$	The Intermediate Value Theorem, when $f$ is continuous on $[a, b]$ and $d$ is between $f(a)$ and $f(b)$ .	To show an equation $f(x) = d$ has a solution. Say first why $f$ is continuous on the closed interval.	Unit 1 The Intermediate Value Theorem
$\lim_{x \to c} \frac{f(x)}{g(x)}$ $= \lim_{x \to c} \frac{f'(x)}{g'(x)}$	L'Hospital's rule, when $f$ and $g$ both have limit 0, or both have infinite limits, and the limit on the right exists.	For $\frac{0}{0}$ and $\frac{\infty}{\infty}$ forms. Write the two limits separately first, then differentiate top and bottom on their own (not the quotient rule).	Unit 4 L'Hospital's Rule

Derivative rules

Formula	What it is	When you use it	Learn it in
$f'(a)$ $= \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$	The definition of the derivative: the limit of the slopes of secant lines.	To recognize a limit as a derivative in disguise, or to check differentiability where two pieces meet.	Unit 2 The Derivative and Its NotationPractice it: Tangent tracer
$\frac{f(b) - f(a)}{b - a}$	The average rate of change of $f$ on $[a, b]$ , the slope of a secant line.	Rates over an interval from a formula or a table; to estimate $f'(c)$ from a table, use the two values closest to $c$ on either side.	Unit 2 Average and Instantaneous Rates of ChangePractice it: Tangent tracer
$\frac{d}{dx} x^n = n x^{n - 1}$	The power rule, for any real number $n$ .	Powers, roots and reciprocals. Rewrite $\sqrt{x}$ as $x^{1/2}$ and $\frac{1}{x^2}$ as $x^{-2}$ first.	Unit 2 The Power RulePractice it: Rule drills
$\frac{d}{dx}\left[f(x)g(x)\right]$ $= f'(x)g(x)$ $+ f(x)g'(x)$	The product rule, for a product of two functions.	A product of two functions of $x$ ; for three factors, use it twice.	Unit 2 The Product RulePractice it: Rule drills
$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]$ $= \frac{f'(x)g(x) - f(x)g'(x)}{\left[g(x)\right]^2}$	The quotient rule, for a quotient of two functions.	A quotient that does not simplify by dividing first. The order on top matters.	Unit 2 The Quotient RulePractice it: Rule drills
$\frac{d}{dx}\sin x$ $= \cos x,$ $\quad \frac{d}{dx}\cos x$ $= -\sin x$	Derivatives of sine and cosine, with $x$ in radians.	Every trigonometric derivative starts here. The minus sign belongs to cosine.	Unit 2 Derivatives of sin x, cos x, e to the x and ln xPractice it: Rule drills
$\frac{d}{dx}e^x$ $= e^x,$ $\quad \frac{d}{dx}\ln x$ $= \frac{1}{x}$	Derivatives of the natural exponential and the natural logarithm.	Exponentials and logarithms. With the chain rule, $\frac{d}{dx}e^{u} = e^{u}u'$ and $\frac{d}{dx}\ln u = \frac{u'}{u}$ .	Unit 2 Derivatives of sin x, cos x, e to the x and ln xPractice it: Rule drills
$\frac{d}{dx}\tan x$ $= \sec^2 x,$ $\quad \frac{d}{dx}\sec x$ $= \sec x \tan x$	Derivatives of tangent and secant, from the quotient rule.	Tangent and secant directly, without rewriting them as quotients each time.	Unit 2 Derivatives of tan, cot, sec and csc
$\frac{d}{dx}\cot x$ $= -\csc^2 x,$ $\quad \frac{d}{dx}\csc x$ $= -\csc x \cot x$	Derivatives of cotangent and cosecant.	The co-functions follow the same pattern as tangent and secant, with minus signs.	Unit 2 Derivatives of tan, cot, sec and csc
$\frac{d}{dx}f(g(x))$ $= f'(g(x)) \cdot g'(x)$	The chain rule, for a composite function.	Any function inside another, such as $e^{3x}$ , $\sin(x^2)$ or $(2x + 1)^5$ : multiply by the derivative of the inside.	Unit 3 The Chain RulePractice it: Rule drills
$\frac{d}{dx}\left[y^n\right]$ $= n y^{n - 1}\frac{dy}{dx}$	Implicit differentiation: $y$ is a function of $x$ , so the chain rule brings out $\frac{dy}{dx}$ .	Curves such as $x^2 + xy + y^2 = 7$ : differentiate both sides, keep the $= 0$ , then solve for $\frac{dy}{dx}$ .	Unit 3 Implicit DifferentiationPractice it: Rule drills
$\left(f^{-1}\right)'(b)$ $= \frac{1}{f'(a)} \ \text{where}\ f(a)$ $= b$	The derivative of an inverse function at a point.	When you know $f$ but not its inverse: find the input $a$ with $f(a) = b$ first.	Unit 3 Derivatives of Inverse Functions
$\frac{d}{dx}\arcsin x$ $= \frac{1}{\sqrt{1 - x^2}},$ $\quad \frac{d}{dx}\arctan x$ $= \frac{1}{1 + x^2}$	Derivatives of inverse sine and inverse tangent.	Inverse trigonometric functions, with the chain rule for $\arcsin u$ and $\arctan u$ .	Unit 3 Derivatives of Inverse Trigonometric Functions
$y = f(a) + f'(a)(x - a)$	The tangent line at $x = a$ , in point-slope form.	Tangent lines and local linear approximation. Whether the estimate is too big or too small comes from concavity.	Unit 4 Local Linearity and LinearizationPractice it: Tangent tracer

Theorems and curve analysis

Formula	What it is	When you use it	Learn it in
$\text{differentiable at}\ c$ $\Rightarrow \text{continuous at}\ c$	Differentiability implies continuity. Not the other way: $\|x\|$ is continuous at 0 but not differentiable there.	To supply the continuity a theorem needs when you are told only that $f$ is differentiable.	Unit 2 Differentiability and Continuity
$f'(c)$ $= \frac{f(b) - f(a)}{b - a},$ $\ a < c < b$	The Mean Value Theorem, when $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ .	To show an instantaneous rate equals an average rate. State both hypotheses in words first.	Unit 5 The Mean Value TheoremPractice it: Curve analyzer
$\max_{[a, b]} f \ \text{and}\ \min_{[a, b]} f \ \text{exist}$	The Extreme Value Theorem, when $f$ is continuous on the closed interval $[a, b]$ .	To know absolute extrema exist before you look for them. It needs a closed interval and continuity.	Unit 5 Extreme Value Theorem, Global and Local Extrema, Critical Points
$f' > 0$ $\Rightarrow f \ \text{increasing},$ $\quad f' < 0$ $\Rightarrow f \ \text{decreasing}$	Increasing and decreasing from the sign of $f'$ .	Where $f$ rises or falls, from a formula, a table, or a graph of $f'$ .	Unit 5 Where a Function Increases or DecreasesPractice it: Curve analyzer
$f' \ \text{from}\ + \ \text{to}\ - \ \text{at}\ c$ $\Rightarrow \text{relative max}$	The first derivative test. From negative to positive: a relative minimum.	Any relative extremum. Name the function that changes sign and say from what to what; $f'(c) = 0$ alone is not enough.	Unit 5 The First Derivative TestPractice it: Curve analyzer
$\text{compare}\ f(a),$ $\ f(b),$ $\ f(\text{each critical point})$	The candidates test for absolute extrema on $[a, b]$ , also called the closed interval method.	An absolute (global) maximum or minimum on a closed interval. A sign change only shows a relative extremum.	Unit 5 The Candidates TestPractice it: Curve analyzer
$f'' > 0$ $\Rightarrow \text{concave up},$ $\quad f'' < 0$ $\Rightarrow \text{concave down}$	Concavity. A point of inflection is where $f''$ changes sign (where $f'$ changes from increasing to decreasing, or back).	Concavity and inflection points, and whether a tangent line estimate is an overestimate (concave down) or an underestimate (concave up).	Unit 5 Concavity and Points of InflectionPractice it: Curve analyzer
$f'(c)$ $= 0,$ $\ f''(c) < 0$ $\Rightarrow \text{relative max}$	The second derivative test. If $f''(c) > 0$ , a relative minimum; if $f''(c) = 0$ , the test says nothing.	A relative extremum at a critical point where $f''$ is easy to evaluate.	Unit 5 The Second Derivative TestPractice it: Curve analyzer

Integrals and the Fundamental Theorem

Formula	What it is	When you use it	Learn it in
$\int x^n\,dx$ $= \frac{x^{n + 1}}{n + 1}$ $+ C$ $\quad (n \ne -1)$	The power rule for antiderivatives.	Powers and roots, rewritten as powers first. Check by differentiating.	Unit 6 Antiderivatives and Indefinite Integrals
$\int \frac{1}{x}\,dx = \ln\|x\| + C$	The case $n = -1$ , with the absolute value.	An integrand $\frac{1}{x}$ , or $\frac{u'}{u}$ after a substitution.	Unit 6 Antiderivatives and Indefinite Integrals
$\int e^x\,dx = e^x + C$	The antiderivative of the natural exponential.	Exponentials; with a substitution, $\int e^{kx}\,dx = \frac{1}{k}e^{kx} + C$ .	Unit 6 Antiderivatives and Indefinite Integrals
$\int \sin x\,dx$ $= -\cos x$ $+ C,$ $\quad \int \cos x\,dx$ $= \sin x$ $+ C$	Antiderivatives of sine and cosine.	Trigonometric integrands. The minus sign is on the antiderivative of sine.	Unit 6 Antiderivatives and Indefinite Integrals
$\int \sec^2 x\,dx$ $= \tan x$ $+ C,$ $\quad \int \sec x \tan x\,dx$ $= \sec x$ $+ C$	The derivatives of tangent and secant, read backward.	Integrands that are exactly one of these derivatives.	Unit 6 Antiderivatives and Indefinite Integrals
$\int \frac{1}{\sqrt{1 - x^2}}\,dx$ $= \arcsin x$ $+ C,$ $\quad \int \frac{1}{1 + x^2}\,dx$ $= \arctan x$ $+ C$	The inverse trigonometric derivatives, read backward.	Integrands of these forms, sometimes after a substitution or after completing the square.	Unit 6 Antiderivatives and Indefinite Integrals
$\int_a^b f(x)\,dx$ $= \lim_{n \to \infty}\sum_{i=1}^{n} f(x_i)\Delta x,$ $\quad \Delta x$ $= \frac{b - a}{n}$	The definite integral as a limit of Riemann sums.	To recognize a limit of sums as an integral: read $\Delta x$ and $x_i$ to find $a$ , $b$ and $f$ .	Unit 6 Riemann Sums, Sigma Notation and the Definite IntegralPractice it: Riemann sums
$\frac{\Delta x_1}{2}\left[f(x_0) + f(x_1)\right]$ $+ \frac{\Delta x_2}{2}\left[f(x_1) + f(x_2)\right]$ $+ \cdots$	The trapezoidal sum, each trapezoid with its own width.	Approximating an integral from a table. Widths may differ; never assume they are equal.	Unit 6 Estimating Area with Riemann SumsPractice it: Riemann sums
$\frac{d}{dx}\int_a^x f(t)\,dt$ $= f(x)$	The Fundamental Theorem of Calculus, first part, for continuous $f$ .	Derivatives of accumulation functions. With an upper limit $g(x)$ , multiply by $g'(x)$ : $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt$ $= f(g(x))g'(x)$ .	Unit 6 The Fundamental Theorem and Accumulation FunctionsPractice it: Accumulation
$\int_a^b f(x)\,dx$ $= F(b) - F(a),$ $\ \text{where}\ F'$ $= f$	The Fundamental Theorem of Calculus, second part: evaluate with any antiderivative.	Definite integrals without a calculator.	Unit 6 The Fundamental Theorem and Definite IntegralsPractice it: Accumulation
$\int_b^a f(x)\,dx$ $= -\int_a^b f(x)\,dx$	Reversing the limits changes the sign.	An accumulation function evaluated below its lower limit, such as $\int_4^1 f(t)\,dt$ .	Unit 6 Properties of Definite IntegralsPractice it: Accumulation
$\int_a^b f(x)\,dx$ $+ \int_b^c f(x)\,dx$ $= \int_a^c f(x)\,dx$	Integrals over adjacent intervals add.	Integrals given in pieces, or a graph made of pieces whose areas you add.	Unit 6 Properties of Definite IntegralsPractice it: Accumulation
$\int f(g(x))g'(x)\,dx$ $= \int f(u)\,du,$ $\quad u$ $= g(x)$	Substitution: the chain rule run backward.	An inside function whose derivative, up to a constant, is also a factor. For a definite integral, change the limits too.	Unit 6 Integrating by Substitution

Motion, rates and accumulation

Formula	What it is	When you use it	Learn it in
$v(t)$ $= x'(t),$ $\quad a(t)$ $= v'(t)$ $= x''(t)$	Velocity and acceleration from position.	Particle motion on a line.	Unit 4 Straight-Line Motion: Position, Velocity, AccelerationPractice it: Motion lab
$v, a \ \text{same sign}$ $\Rightarrow \text{speeding up}$	Speeding up or slowing down. Opposite signs: slowing down. Speed is $\|v(t)\|$ .	Any speeding-up question: give the signs of both $v$ and $a$ at that time.	Unit 4 Straight-Line Motion: Position, Velocity, AccelerationPractice it: Motion lab
$\frac{d}{dt}\left[x^2 + y^2\right]$ $= 2x\frac{dx}{dt} + 2y\frac{dy}{dt}$	Related rates: differentiate the equation that links the quantities with respect to time.	Ladders, cones, shadows: differentiate first, then substitute the values at the moment asked about.	Unit 4 Introduction to Related RatesPractice it: Related rates
$f_{\text{avg}}$ $= \frac{1}{b - a}\int_a^b f(x)\,dx$	The average value of $f$ on $[a, b]$ .	"Average temperature" or "average velocity" asks for the average value of the function, not its average rate of change.	Unit 8 Average Value of a Function
$x(t)$ $= x(t_0)$ $+ \int_{t_0}^{t} v(s)\,ds$	Position from velocity and a known position.	Position at a time: start from the time where the position is known, not always $t = 0$ .	Unit 8 Motion with IntegralsPractice it: Motion lab
$\int_a^b v(t)\,dt \ \text{and}\ \int_a^b \|v(t)\|\,dt$	Displacement (the first) and total distance traveled (the second).	Total distance: integrate the speed, or, without a calculator, split where $v$ changes sign.	Unit 8 Motion with IntegralsPractice it: Motion lab
$A(t)$ $= A(0)$ $+ \int_0^t r(s)\,ds$	An amount from a rate and a starting amount (net change).	Water in a tank, people in a park: keep the starting amount, and subtract the rate out from the rate in.	Unit 8 Accumulation in ContextPractice it: Accumulation

Differential equations

Formula	What it is	When you use it	Learn it in
$\frac{dy}{dx}$ $= f(x)g(y)$ $\Rightarrow \int \frac{1}{g(y)}\,dy$ $= \int f(x)\,dx$	Separation of variables.	A particular solution: add $+ C$ as you antidifferentiate, use the initial condition right away, then solve for $y$ .	Unit 7 General Solutions by Separation of VariablesPractice it: Slope fields
$\frac{dy}{dt} = ky \Rightarrow y = y_0 e^{kt}$	Exponential growth ( $k > 0$ ) and decay ( $k < 0$ ); $y_0$ is the value at $t = 0$ .	A rate proportional to the amount present.	Unit 7 Exponential ModelsPractice it: Slope fields

Area and volume

Formula	What it is	When you use it	Learn it in
$\int_a^b \left[f(x) - g(x)\right]dx$	Area between curves, $f$ above $g$ , slicing along $x$ .	Area between two graphs. Split at crossings if the top curve changes.	Unit 8 Area Between Curves (Functions of x)Practice it: Area and volume
$\int_c^d \left[p(y) - q(y)\right]dy$	Area slicing along $y$ : the right curve minus the left curve.	When the region is easier to describe with $x$ as a function of $y$ .	Unit 8 Area Between Curves (Functions of y)Practice it: Area and volume
$\int_a^b A(x)\,dx,$ $\quad A$ $= s^2 \ \text{for squares}$	Volume from cross sections of area $A(x)$ ; a square of side $s$ has area $s^2$ .	Known cross sections on a base region: the side $s$ is the length across the base at $x$ .	Unit 8 Volumes with Square and Rectangle Cross SectionsPractice it: Area and volume
$A$ $= \frac{\sqrt{3}}{4}s^2,$ $\quad A$ $= \frac{\pi}{8}s^2,$ $\quad A$ $= \frac{1}{2}s^2$	Cross-section areas: an equilateral triangle of side $s$ , a semicircle of diameter $s$ , an isosceles right triangle with a leg $s$ in the base.	Triangle and semicircle cross sections; with the hypotenuse $s$ in the base, the right triangle has area $\frac{1}{4}s^2$ .	Unit 8 Volumes with Triangle and Semicircle Cross SectionsPractice it: Area and volume
$\pi\int_a^b \left[R(x)\right]^2 dx$	The disc method, with the radius $R$ measured from the axis of revolution.	A region that touches the axis. About a line $y = k$ , the radius is $\|f(x) - k\|$ .	Unit 8 Disc Method About the AxesPractice it: Area and volume
$\pi\int_a^b \left(\left[R(x)\right]^2 - \left[r(x)\right]^2\right)dx$	The washer method: outer radius $R$ and inner radius $r$ , both from the axis.	A region with a gap from the axis. Square each radius on its own; $(R - r)^2$ is wrong.	Unit 8 Washer Method About the AxesPractice it: Area and volume

Easy to forget

$+ C$ . Every indefinite integral ends with $+ C$ . In a separable differential equation the constant appears as you antidifferentiate, before you solve for $y$ . General Solutions by Separation of Variables
Units of a rate of a rate. If $r(t)$ is in liters per minute, $r'(t)$ is in liters per minute per minute, and $\int_a^b r(t)\,dt$ is in liters. The Derivative in Context
The chain factor on an accumulation function. $\frac{d}{dx}\int_0^{x^2} f(t)\,dt$ $= f(x^2) \cdot 2x$ , not $f(x^2)$ . The Fundamental Theorem and Accumulation Functions
Average value is not average rate. $\frac{1}{b - a}\int_a^b f(x)\,dx$ is the average value of $f$ ; $\frac{f(b) - f(a)}{b - a}$ is its average rate of change. Average Value of a Function
Vertical tangents. Where the denominator of $\frac{dy}{dx}$ is 0 and its numerator is not, at a point that is on the curve. $\frac{dy}{dx} = 0$ gives horizontal tangents. Behavior of Implicit Relations
$\pi$ in discs and washers. Every disc or washer volume has a factor of $\pi$ ; a square or triangle cross section has none. Disc Method About the Axes

Practice with it

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