Must Know COLD Formulas- AP CALC AB

What is the definition of the derivative? and what does it give you?

f’(c)= lim h→0 for (f(x+h)-f(x))/h, gives you the derivative of the function.

What is the alternate derivative definition?

f’(a)= lim x→a for (f(x)-f(a))/x-a, evaluates the derivative at some point. so must have some value to be a.

derivatives:

sinx

cosx

tanx

cotx

secx

cscx

ln|u|

e^u

ANS:

cosx

-sinx

sec²x

-csc²x

secx tanx

-cscxcotx

1/u

e^u

differentiation rules:

[f(u)]

(uv)

(u/v)

ANS:

chain rule: f’(u)(du/dx)

product rule: uv’+vu’

Quotient rule: (vu’-uv’)/v²

Intermediate Value Theorem

If the func f(x) is cont on (a,b) and y is a number between f(a) and f(b), then there exists at least one number x=c in the open interval (a,b) such that f(c )=y

Mean Value Theorem

If the function is differentiable it is also continuous on (a,b) AND the first derivative exists on the interval (a,b) then there is at least one number x=c in (a,b) such that

f’(c )=(f(b)-f(a))/b-a

Rolle’s Theorem

If the func f(x) is cont on (a,b) AND the first deriv exists on the interval (a,b) AND f(a)=f(b), then there is at least one number x=c in (a,b) that f’(c )=0

Extreme Value Theorem

If the func f(x) is cont on (a,b) then the func is guaranteed to have a abs min and abs max on the interval

If f has an inverse function g then

g’(x)= 1/f’(g(x)) cause derivatives are reciprocal slopes

How do you do implicit differentiation

all y’s differentiate to dy/dx, next isolate dy/dx, if you take the secon deriv of that then you will prob plug in the expression for the first deriv to that.

AROC

msec=f(b)-f(a)/(b-a)

Instantaneous rate of change

m tan=def of derivative

critical point:

dy/dx=0 or undefined

Local min:

dy/dx goes from -,0,+

local max

goes from +,0,-

How to find points of inflection

where concavity changes, d²y/dx² goes from (+,0,-)(-,0,+)

What do these show us?

f’(x)>0

f’(x)<0

f’(x)=0 or DNE

ans:

increasing func

decreasing fun

critical values

what is the max value

y-value

What do these mean?

f”(x)>0

f”(x)<0

f”(x)=0 and sign of f”(x) changes

Ans;

func is concave up

func is concave down

then there is a point of inflection at x

When is relative max for f”(X)

When is relative min for f”(X)

f”(x)<0

f”(x)>0

Horizontal Asymptotes:

if the largest expo in the numerator is < largest expo in the denominator then

lim x→- + infinity f(x)=0

Horizontal Asymptotes:

if the largest expo in the numerator is > largest expo in the denominator then

lim x→- + infinity f(x)=DNE

Horizontal Asymptotes:

if the largest expo in the numerator is = largest expo in the denominator then

the quotient of the leading coeffficients is the asymptote. lim x→- +infinity f(x)=a/b

When does the particle move left or right

the particle moves to the left when velocity is negative and moves right when velocity is positive

equation for displacement

the integral from initial time to t v(t)dt

equation for total distance

the integ from initial time to final time |v(t)|dt

Average velocity equation

final position-initial position/ total time

accumulation equation

x(0)+inte from intiti t to t v(t)dt

ln N= P so

e^p=N

Ln e=

Ln 1=

ans

1

0

ln(MN)

ln(M/N)

p*lnM

ln M+ln N

lnM-lnN

lnM^p

Exponential growth and decay equation

y=Ce^kt

Fundamental Theorem of Calculus

the inte from a to b f(x)dt =F(b)-F(a) , where f’x)=f(x)

The average value of the function on the int a,b equation

f avg=1/b-a integr from a to b f(x)dx= integr a to b f(x)dx/b-a

sin pi/6

sin pi/3

sin pi/45

tan pi/6

tan pi/3

tan pi/45

½

rad 3/2

rad 2/ 2

rad3/3

rad 3

1

Trap rule equations

A trap =1/2h[b1+b2] (uneven intervals)

integral from a to b f(x)dx=b-a/2n(yinitial +2y1+2y2)

quotient identities:

tan x

cotx

cscx

secx

ans:

sinx/cosx

cosx/sinx

1/sinx

1/cosx

sinxcscx=

cosxsecx=

1

1

Pythag identify: There are 3

sin²x+cos²x=1

1+tan²x=sec2x

cit²x+1=csc²x