AB
Calculus Cram Sheet
[Courtesy of Will Felder. Edited/extended by EMH 5/10/00, 1/25/01, 8/3/02, 5/8/03, 4/22/04, 2/7/05.]
Formulas 



Trapezoid rule: A » ½ Dx (f(x_{0}) + 2f(x_{1}) + 2f(x_{2}) + . . . + 2f(x_{n}_{ – 1}) + f(x_{n})) 


Left endpoint rule: A » Dx (f(x_{0}) + f(x_{1}) + f(x_{2}) + . . . + f(x_{n}_{ – 1})) 


Right endpoint rule: A » Dx (f(x_{1}) + f(x_{2}) + f(x_{3}) + . . . + f(x_{n})) 


Chain rule: D_{x}(g(u(x))) = g¢(u(x)) u¢(x) 


Derivative of an inverse: D_{x}(f ^{–1}(x)) = 1 / f ¢(f ^{–1}(x)) 


Parametric chain rule: If y = y(t) and x = x(t), then dy/dx = (dy/dt) / (dx/dt). 


Product rule: (uv)¢ = uv¢ + vu¢ 


Quotient rule: (u/v)¢ = (vu¢ – uv¢)/v^{2} 


Change of base: log_{b} x = (ln x) / (ln b) 


Rewriting an exponential: b^{x} = e^{x}^{ ln b} 


Exponential growth: Diffeq. y' = ky has solution y = ce^{kx} 


Volume by disks: ò pr² dx if axis of rotation is parallel to xaxis (use dy if parallel to yaxis) 


Volume by washers: ò p(R^{2} – r²) dx if axis of rotation is parallel to xaxis (use dy if parallel to yaxis) 


Volume by plane slicing ò A(x) dx if A(x) is an area function for cross sections perpendicular to the xaxis (use dy if perpendicular to yaxis) 


Average value of f on [a, b] is (ò_{a}^{b} f (x) dx) / (b – a). 


f (x) » f (a) + f '(a)(x – a) 

Derivatives and Antiderivatives 



Should know derivatives and antiderivatives of all of these function families: 


polynomials 


Need to know derivatives (not antiderivatives) for these families: 


logarithmic 

u Substitution 



If a function (or some altered form of it) and the function’s derivative are both in the integrand, you can often do a “u substitution.” 


Example: 


ò x^{–1}(3 ln x + 7)^{11} dx 

IVT 



If f is continuous on [a, b], 


then " y Î (f(a), f(b))
[or, wlog, "
y Î (f(b), f(a))
if f(a) > f(b)] 


In words: For any intermediate value of a continuous function on a closed interval, there is at least one place in the interior of the open interval (a, b) where that intermediate value is actually attained. (Sometimes known as the “Cape of Good Hope Theorem.”) Interesting corollary: If f and g are both continuous on [a, b] and their difference is negative at one endpoint and positive at the other, then there is at least one place in (a, b) where f(x) = g(x). 

EVT 



If f is continuous on [a, b], 


then $ x_{1}, x_{2}
Î [a, b] ' 


Alternate (more cryptic) version of the theorem: If f is continuous
on [a, b], 


In words: A continuous function on a closed interval (the conditions are crucial) attains its maximum and minimum values somewhere on that closed interval. 

MVT 



If f is differentiable on (a, b) and continuous on [a, b], 


then $ c Î (a, b) ' f '(c) = (f(b) – f(a)) / (b – a). 


In words: There is at least one place where (slope of tangent line) equals (average slope between a and b). Conditions are crucial to know: f differentiable on (a, b) and continuous on [a, b]. 

FTC 



If f is integrable on [a, b] and g is any antiderivative of f, 


then ò_{a}^{b} f(x) dx = g(b) – g(a). 


Equivalent form (sometimes called FTC2): 

Diff. Þ Cont. Þ Integ. 


D Þ C Þ
I 

Definitions 



Derivative = limit of difference quotient = slope of tangent line = instantaneous rate of change. 


A function f is differentiable iff f has local linearity with finite slope. In other words, f is differentiable iff f “looks like a straight but nonvertical line when you zoom in closely enough.” 


Derivative at a point: f ' (c) = lim_{x}_{®c} [ (f(x) – f(c)) / (x – c) ] 


Derivative function: f ' (x) = lim_{h}_{®0} [ (f(x + h) – f(x)) / h ] 


Definite integral: lim_{Dx}_{®0} of Riemann sums = lim_{Dx}_{®0} S f(x_{i}) Dx = ò_{a}^{b} f(x) dx. Notice how, in the limit, the summation (S) is replaced by the stretchy S and the step size (Dx) is replaced by dx. In order for the definite integral to exist, the sufficient condition you are expected to know for the AP exam is that f is continuous on [a, b]. Although this condition can be weakened somewhat while still preserving Riemann integrability, do not worry about that for the AP exam. 


Critical point: f ¢ = 0 or DNE. 


Stationary point (a.k.a. plateau point): f ¢ = 0 but does not change sign. 


Local min.: f ¢ = 0 and is changing from negative to positive. 


Local max.: f ¢ = 0 and is changing from positive to negative. 

WARNING 
Although it is sometimes useful to check f ¢¢
when checking for max. or min., the second deriv.
test is not 100% reliable. 


Inflection point: A point of continuity for f where f ¢¢ changes sign. (Frequently, this is a point where f ¢ has a strict local max. or min. However, that is not necessarily the case; for example, think of the origin, which is an inflection point for the function y = f(x) = x^{1/3} even though f ¢(0) and f ¢¢(0) are undefined.) 

WARNING 
Some students think (wrongly) that “point of inflection” means f ¢¢ = 0. 


Concavity: A synonym for f ¢¢. 







Techniques for Multiple Choice 
Techniques for Free Response 

1. Pace yourself. Keep brainpower in reserve for free response. 
1. If you can’t get part (a), skip it and do the others. Part (a) may be
worth only a point. 
