Test for symmetry:

to x (x,y)-->

to y (x, y)-->

to origin (x,y)-->

to x (x,y)--> (x,-y)

to y (x, y)--> (-x, y)

to origin (x,y)-->(-x,-y)

average rate of change

f(x+Δx) - f(x) / Δx

y2 - y1 / x2 - x1

instantaneous rate of change

lim as Δx-->0 of f(x+Δx) - f(x) / Δx

i. f(c) is defined

ii. lim x->c of f(x) exists

iii. lim x->c of f(x) = f(c)

1) jump

2)removable--hole--

3) infinite (non-removable)

If f is continuous on [a,b] and k is any number between f(a) and f(b), then thre is at least one number c on (a,b) such that f(c)=k

soooo....

1) continuous, 2)between f(a) and f(b), 3) f(a) does not equal f(b)-----> then, f(c) = k

the derivative is:

(2 things)

-the instantaneous rate of change

-the "m" slope of the tangent line at a point

product rule for derivatives

f(x)*g(x) =

f(x)g'(x) + f'(x)g(x)

f(x) / g(x) =

g(x)*f'(x) - f(x)*g'(x) / [g(x)]²

chain rule

d[f(g(x))]

dx

d[f(g(x))]

dx

=[f'(g(x))]*g'(x)

double angle formula

sin2x=

If f is continuous on [a,b] and is differenitable on (a,b) and f(a) = f(b), then there exists at least one c on (a,b) such that f'(c) = 0

-to find extreme max or min

If f is continuous on [a,b] and is differentiable on (a,b), then there is a c on (a,b) such that

f'(c)=f(b) - f(a) / b - a

-to find instantaneous rate of change, tangent line

n(n+1)

2

∑i²

n(n+1)(2n+1)

6

[n(n+1)]²

2²

If f is continuous on [a,b], then f has both a min and max on the interval

--to find relative max and min

amount of change

propgated error

exact change in tan

increasing or decreasing

max or min

critical numbers

concavity

points of inflection

Average Value of the function f

f(c) =

f(c)= 1 ∫_a^b f(x)dx

b-a      

=-ln¦cosx¦ + C

or

ln¦secx¦ + C

ln¦sin x¦ + C

or

-ln¦cscx¦ + C

the derivative of an inverse function

g'(x)=

T=Ce^kt + room temp.

C=difference between object temp and room temp

V=∏∫_a^br²dx

-if about x-axis, dx

-if about y-axis, dy

V=2∏∫_a^brh(thickness)<--dx or dy

-dx if about y axis

-dy if about x axis