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AP Calculus AB formula sheet
These are the formulas AP Calculus AB students are expected to know by heart. On the calculator version, ∫ from a to b is written int[a,b].
Limits and continuity
| Special limits | limx→0 sin x / x = 1; limx→0 (1 − cos x)/x = 0 |
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| L’Hospital’s rule | if a limit is 0/0 or ∞/∞, lim f/g = lim f′/g′ |
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| Continuity at c | f(c) exists, the limit exists, and they’re equal |
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| Asymptotes | horizontal: limit as x → ±∞; vertical: f → ±∞ as x → c |
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Definition of the derivative
| At x | f′(x) = limh→0 [f(x + h) − f(x)]/h |
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| At a | f′(a) = limx→a [f(x) − f(a)]/(x − a) |
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| Differentiability | differentiable means continuous, not the other way round (corners, cusps, vertical tangents) |
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Derivative rules
| Power | d/dx xⁿ = nxⁿ⁻¹ |
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| Product | (fg)′ = f′g + fg′ |
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| Quotient | (f/g)′ = (f′g − fg′)/g2 |
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| Chain | d/dx f(g(x)) = f′(g(x))·g′(x) |
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| Inverse function | (f−1)′(a) = 1/f′(f−1(a)) |
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| Implicit | differentiate both sides, attach dy/dx to each y term, solve |
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Common derivatives
| Trig | (sin x)′ = cos x; (cos x)′ = −sin x; (tan x)′ = sec2 x; (cot x)′ = −csc2 x; (sec x)′ = sec x tan x; (csc x)′ = −csc x cot x |
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| Exponential and log | (eˣ)′ = eˣ; (aˣ)′ = aˣ ln a; (ln x)′ = 1/x; (loga x)′ = 1/(x ln a) |
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| Inverse trig | (arcsin x)′ = 1/√(1 − x2); (arccos x)′ = −1/√(1 − x2); (arctan x)′ = 1/(1 + x2) |
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Using derivatives
| Tangent line | y = f(a) + f′(a)(x − a), also the linear approximation |
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| Critical points | f′ = 0 or f′ undefined |
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| Increasing and concavity | f′ > 0 increasing; f″ > 0 concave up; inflection where f″ changes sign |
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| First derivative test | f′ changes + to −: relative max; − to +: relative min |
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| Second derivative test | f′(c) = 0 and f″(c) < 0: relative max; f″(c) > 0: relative min |
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| Absolute extrema on [a, b] | check critical points and both endpoints |
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The big theorems
| Mean Value Theorem | f continuous on [a, b], differentiable on (a, b) ⇒ some c has f′(c) = [f(b) − f(a)]/(b − a) |
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| Intermediate Value Theorem | f continuous on [a, b], k between f(a) and f(b) ⇒ some c has f(c) = k |
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| Extreme Value Theorem | f continuous on [a, b] ⇒ f has an absolute max and min there |
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Motion
| Velocity and acceleration | v(t) = x′(t); a(t) = v′(t) = x″(t) |
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| Speed | |v(t)|; speeding up when v and a have the same sign |
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| Displacement | ∫ab v(t) dt |
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| Total distance | ∫ab |v(t)| dt |
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Integrals
| Power | ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1 |
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| Log and exponential | ∫ 1/x dx = ln|x| + C; ∫ eˣ dx = eˣ + C; ∫ aˣ dx = aˣ/ln a + C |
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| Trig | ∫ cos x dx = sin x + C; ∫ sin x dx = −cos x + C; ∫ sec2 x dx = tan x + C; ∫ csc2 x dx = −cot x + C; ∫ sec x tan x dx = sec x + C; ∫ csc x cot x dx = −csc x + C; ∫ tan x dx = −ln|cos x| + C |
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| Inverse trig | ∫ 1/√(1 − x2) dx = arcsin x + C; ∫ 1/(1 + x2) dx = arctan x + C |
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Fundamental Theorem of Calculus
| Part 1 | d/dx ∫ag(x) f(t) dt = f(g(x))·g′(x) |
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| Part 2 | ∫ab f(x) dx = F(b) − F(a), where F′ = f |
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| Accumulation | f(b) = f(a) + ∫ab f′(x) dx |
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| Average value | [1/(b − a)] ∫ab f(x) dx |
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Riemann sums and differential equations
| Trapezoid rule (equal widths) | (Δx/2)[y0 + 2y1 + … + 2yn−1 + yn] |
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| Left and right sums | f increasing: left sum underestimates, right sum overestimates |
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| Separation of variables | y’s with dy, x’s with dx, integrate, add C, use the initial condition |
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| Exponential growth | dy/dt = ky ⇒ y = y0ekt |
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Area and volume
| Area between curves | ∫ab (top − bottom) dx |
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| Disc | V = π∫ab R2 dx |
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| Washer | V = π∫ab (R2 − r2) dx |
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| Known cross sections | V = ∫ab A(x) dx; square s2; semicircle on diameter s: (π/8)s2; equilateral triangle (√3/4)s2 |
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On the TI-84 Plus CE
| Mode | use RADIAN (press MODE) |
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| Derivative at a point | MATH, nDeriv( |
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| Definite integral | MATH, fnInt( |
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| From a graph | 2ND TRACE (CALC): zero, minimum, maximum, intersect, dy/dx, ∫f(x)dx |
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What it looks like on the calculator
The calculator version is the same sheet in plain text with short lines, so it fits the screen. Square roots are written as sqrt( ), powers with ^ and Greek letters as words. On the calculator, search a word like “chain” or “washer” to jump straight to it. The notes are stored in Archive, so a RAM clear won’t erase them.
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