calc exam 3 stuff

marginal cost function (cost of making 1 more item)

[C(x+1) - C(x)] / [(x+1) - x]

or C’(x)

exponential growth/decay formula

y=Ce^kt

[ k=relative growth rate=(dy/dt)/y ]

[C=initial value]

radioactive decay formula

…?

newton’s law of cooling

…?

definition of e

lim_n→infinity (1+ (1/n) )ⁿ

continuous compound interest formula

A(t)=Pe^rt

simple interest formula

(1+ (k/n) )^nt

volume of sphere formula

4/3 * pi * r³

pythagorean triples & special triangles

(1, 1, 2^1/2), (3, 4, 5), (5, 12, 13),

x, x√3, 2x (30-60-90 triangle)

area of a circle

pi * r²

linear approximation (*only works when “a” is close to x)

L(x)= [f’(a) * (x-a) ] + f(a)

differentials

d( f(x) )= f’(x)*dx

percentage error

~(dA)/A

max/min names

extrema= collective term

absolute / global

local / relative

conditions for absolute extrema

absolute extrema at x=c if f( c) is (>/< + equal to) f(x) for all x in domain of f

conditions for local extrema

local extrema at x=c if f( c) is (>/< + equal to) f(x) for all x near c, in open interval around c

extreme value theorem

if f is continuous on a closed interval [a,b] then f has an absolute maximum & minimum value on that interval

fermat’s theorem

if f has a local max or min at c, then f’(c)=0 or =DNE

critical number

number C where f’(c)=0 or =undefined

rolle’s theorem

(I) f is continuous on [a,b]

(II) f is differentiable (“well defined”) on (a,b)

(III) f(a)=f(b)

there exists at least one number C on (a,b) such that f’(c)=0

mean value theorem

(I) f is continuous on [a,b]

(II) f is differentiable (“well defined”) on (a,b)

there exists at least one number C on (a,b) such that f’(c)= [ f(b)—f(a) ] / [ b—a ]