Limits The laws below are valid so long as we don't have undefined expressions such as 0/0, ∞/∞, ∞ · 0, ∞ - ∞ and so on. Constant law: lim C = C  x→a Sum law: lim (f(x) + g(x)) =  lim f(x) + lim g(x)  x→a  x→a  x→a Difference law: lim (f(x) - g(x)) =  lim f(x) - lim g(x)  x→a  x→a  x→a Product law: lim (f(x) · g(x)) =  lim f(x) · lim g(x)  x→a  x→a  x→a Quotient law: lim (f(x)/g(x)) =  lim f(x)/ lim g(x) if g(x) ≠ 0  x→a  x→a  x→a Multiple law: lim Cf(x) = C lim f(x)  x→a  x→a Root law: lim n√x =  lim n√a if x is positive if n is even x→a x→a Composition law: lim f(g(x)) = f( lim g(x))  x→a  x→a Fundamental trig limit: lim sinx  = 1  x→0 x

Derivatives Constant rule: (cf(x))' = cf '(x) Sum rule: (f(x) + g(x))' = f '(x) + g'(x) Difference rule: (f(x) - g(x))' = f '(x) - g'(x) Product rule: (f(x) · g(x))' = f '(x)g(x) + f(x)g'(x) Reciprocal rule: ((f(x))-1)' = -f(x)-2f '(x) Quotient rule: (f(x)/g(x))' =  f '(x)g(x) - f(x)g'(x) (g(x))2 Power rule: (xn)' = nx(n - 1) if n is an integer General power rule: ((f(x))n)' = nf(x)(n - 1)f '(x) if n is rational Chain rule: (f(g(x))' = (f '(g(x)))(g'(x)) Multiple rule: (f(cx))' = cf '(cx) (sin x)' = cos x (cos x)' = -sin x (tan x)' = sec2x (ex)' = ex (bx)' = (logeb)(bx) (logex)' = 1/x (logbx)' = 1/(x logeb)

Integrals Constant rule: ∫cf(x)dx = c∫f(x)dx Sum rule: ∫(f(x) + g(x))dx = ∫f(x)dx + ∫g(x)dx Difference rule: ∫(f(x) - g(x))dx = ∫f(x)dx - ∫g(x)dx Product rule: ∫f(x)g(x)dx = f(x)g(x) - ∫f(x)g'(x)dx ∫sin x dx = cos x + C ∫cos x dx = -sin x + C ∫ex dx = ex + C ∫xn dx = (x(n + 1)) / (n + 1) + C if n ≠ 1 and ln(x) + C if n = 1

Intermediate value theorem: Assume f is continuous on [a, b] Let C be any number between f(a) and f(b) Then there is some c between a and b so that f(c) = C Rolle's theorem: Assume f is continuous on [a, b] and differentiable on (a, b) If f(a) = f(b), then there is some c in (a, b) such that f '(c) = 0 Mean value theorem: Assume f is continuous on [a, b] and differentiable on (a, b) Then there is at least one number c in (a, b) such that f '(c) =  f(b) - f(a) b - a Fundamental theorem of calculus: Assume f is continuous on [a, b] Let F(x) be any antiderivative for f(x) so that f '(x) = f(x) Then ∫ b  f(x)dx = F(b) - F(a) a where the LHS denotes the area under the curve y = f(x) from x = a to b.

Continuity: A function is continuous at x = a if	lim	f(x) = f(a)
	x→a

First derivative: If the first derivative is positive, the function is increasing; if the first derivative is negative, the function is decreasing.
If f '(a) = 0 then a is called a critical point of f.

Second derivative: If f ''(a) = 0 then a is called an inflection point of f.

Finding extrema: Check the values of f at the endpoints of the interval and at all critical points.

First derivative test: Let f '(a) = 0. Then f has a local maximum if the first derivative is positive slightly to the left of a and negative slightly to the right of a (so the signs of f ' look like +++0---); f has a local minimum if the first derivative is negative slightly to the left of a and positive slightly to the right of a (so the signs of f ' look like ---0+++).

Second derivative test: Let f ''(a) = 0. If f ''(a) > 0 then f has a local minimum at a; if f ''(a) < 0 then f has a local maximum at a.

Implicit differentiation: y = y(x), then differentiate.

L'Hôpital's Rule: If f(a) = g(a) = 0 or ∞, then	lim	f(x)	=	lim	f '(x)
	x→a	g(x)		x→a	g'(x)

Linear approximation: To estimate the value of f(x) for x near a, find the tangent line to f(x) at a. The tangent line is given by y - f(a) = f '(a) (x - a).

Squeeze theorem:

f(x) ≤ g(x) ≤ h(x)

If	lim	f(x) =	lim	h(x) = L
	x→a		x→a

then	lim	g(x) = L
	x→a

For x = any real number:

∞ + x = ∞
-∞ + x = -∞
(∞)(∞) = ∞
(-∞)(-∞) = ∞
(∞)(-∞) = -∞
x(∞) = ∞
(-x)(∞) = -∞
(∞)(0) = -1

If f(-x) = f(x) then say f is an even function
If f(-x) = -f(x) then say f is an odd function

Definition of the derivative: f '(x) =	lim	f(x + h) – f(x)
	h→0	h

Definite integral has b, a on integral sign, and represents area under the curve y = f(x) from x = a to b; indefinite integral does not, and represents all anti-derivatives.