Ap Calc exam

What is the definition of the limit of a function at a given x-value

the height the function intends to reach there

can a function have a limit at an x value if the function has a hole there?

yes

when does a limit exist

if the limits from the left and right exist and are equal for a value of x=c

Does a limit exist in cases of infinite growth?

no

what do limits at infinity imply

a vertical or horizontal asymptote

what do you do if a limit comes out to be 0/0 after direct substitution

l’hospital

what is l’hopital

when taking the limit through direct substitution has an outcome of 0/0, Take the derivitive of the top over the derivitive of the bottom.

if the degree of the numerator is less than the degree of the denominator

y=0 is a horizontal asymptote

if the degree of the numerator = the degree of the denominator

y = leading coefficient a/leading coefficient b is the horizontal asymptote

if the degree of the numerator is greater than the degree of the denominator

no horizontal asymptote

critical point

must be continuous

y’=0 or undefined

rel min

y’ goes from - to +

y’’ > 0

rel max

y’ goes from + to -

y’’<0

point of inflection

concavity changes

y’’ changes sign

d/dx x^(n)

nx^(n-1)

d/dx (sin x)

cos x

d/dx (cos x)

-sin x

d/dx (tan x)

sec²x

d/dx (cot x)

-csc²x

d/dx (sec x)

sec x tan x

d/dx (csc x)

-csc x cot x

d/dx (ln x)

1/x

d/dx (e^x)

e^x

d/dx arcsin

1/√(1-x²)

d/dx arccos

-1/√(1-x²)

d/dx arctan

1/(1+x²)

d/dx arccot

-1/(1+x²)

d/dx arcsec

1/(|x|√(x²-1))

d/dx arccsc

-1/(|x|√(x²-1))

d/dx a^u

a^(u) ln a

d/dx log(a)x

1/(x ln a)

IVT

if the function f(x) is continuous on (a,b), for any number c between f(a) and f(b) there exists a number d in the open interval (a,b) such that f(d)=c

MVT

If the function f(x) is continuous on (a,b), and the first dirivative exists then there exists a number x=c on (a,b) such that f’( c )= [f(b)-f(a)]/b-a

Volume

pi( the integral from a to b of [R(x)²-r(x)²] )

area of a trapezoid

(1/2)h(b1+b2)

speed

|v(t)|

total distance traveled

the integral from a to b of |v(t)|

average velocity

(final position-initial position)/total time