Formula sheetHigh school
AP Calculus AB and BC Formula Sheet
Everything you need to have memorised for AP Calculus AB, with the BC-only topics at the end. There is no formula sheet in the exam, so this is the list to know cold.
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01Limits and continuity
- Definition of the derivative f'(x) = limh→0 [f(x + h) − f(x)]/h
- Continuity at a f(a) exists and limx→a f(x) = f(a)
- L'Hôpital's rule for 0/0 or ∞/∞, lim f/g = lim f'/g'
- Key limits sin x/x → 1 and (1 − cos x)/x → 0 as x → 0
02Derivatives
- Basic xn → nxn−1; ex → ex; ax → ax ln a; ln x → 1/x; logax → 1/(x ln a)
- Trigonometric sin → cos; cos → −sin; tan → sec2; sec → sec tan; csc → −csc cot; cot → −csc2
- Inverse trig arcsin x → 1/√(1 − x2); arctan x → 1/(1 + x2)
- Rules product, quotient, chain; implicit differentiation for relations in x and y
- Inverse function (f−1)'(b) = 1/f'(a) where f(a) = b
03The big theorems
- Intermediate Value Theorem f continuous on [a, b] takes every value between f(a) and f(b)
- Extreme Value Theorem f continuous on [a, b] has an absolute max and min there
- Mean Value Theorem f'(c) = [f(b) − f(a)]/(b − a) for some c in (a, b)
- Fundamental Theorem d/dx ∫ax f(t)dt = f(x); ∫ab f(x)dx = F(b) − F(a)
04Using derivatives
- Shape f' > 0 increasing; f'' > 0 concave up; inflection where f'' changes sign
- Extrema candidates where f' = 0 or is undefined, plus the endpoints
- Linearisation L(x) = f(a) + f'(a)(x − a)
- Motion v = x'(t), a = v'(t); speeding up when v and a have the same sign; speed = |v|
05Integrals and applications
- Average value [1/(b − a)]∫ab f(x)dx
- Area between curves ∫(top − bottom)dx
- Volumes disc π∫R2dx; washer π∫(R2 − r2)dx; known cross-sections ∫A(x)dx
- Riemann sums left, right, midpoint; trapezoid ½Δx[f(x0) + 2f(x1) + … + 2f(xn−1) + f(xn)]
- Motion displacement ∫v dt; total distance ∫|v| dt
- u-substitution change the limits when you change the variable
06Differential equations
- Separation of variables put all y terms with dy, all x terms with dx, integrate
- Exponential growth dy/dt = ky gives y = Cekt
- Slope fields draw short segments with slope dy/dx at each point
07BC only
- Integration by parts ∫u dv = uv − ∫v du; partial fractions; improper integrals as limits
- Euler's method yn+1 = yn + h·f(xn, yn)
- Logistic growth dP/dt = kP(1 − P/L); fastest growth at P = L/2
- Arc length ∫√[1 + (dy/dx)2]dx
- Parametric dy/dx = (dy/dt)/(dx/dt); speed √[(dx/dt)2 + (dy/dt)2]
- Polar area ½∫r2dθ; x = r cosθ, y = r sinθ
- Series tests geometric a/(1 − r) for |r| < 1; nth-term test; p-series converges for p > 1; ratio test; alternating series test
- Alternating series error |error| ≤ first omitted term
- Maclaurin ex = Σxn/n!; sin x = x − x3/3! + x5/5! − …; cos x = 1 − x2/2! + x4/4! − …; 1/(1 − x) = Σxn
- Lagrange error bound |Rn(x)| ≤ M|x − a|n+1/(n + 1)!
How to use this sheet
- In free-response answers, write the integral before you evaluate it: setup earns points.
- Justify every conclusion with the theorem's name and its conditions.
- Check the College Board site each year for the current exam format and timing.
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