MAth

average speed

divding distance covered by elapsed time

jerk

third derivative

cos^2x

(1+cos2x)/2

sin^2x

(1-cos2x)/2

secant line (or limit) slope formula

(f(x1+h)-f(x1))/h

csc^2 formula

1+cot^2=csc^2

sec^2 formula

1+tan^2=sec^2

ratio for 45-45-90 triagnel

1,1 sqrt2

sin2x

2sinxcosx

unit circle for pi over 3

1/2, sqrt3/2

exponent rules

a^n ⋅ a^m = a^n+m a^n ⋅ b^n = (a ⋅ b)^n a^n / a^m = a^n-m a^n / b^n = (a / b)^n (a^n)^m = a nxm

speed

absolute value of velocity

algebraic function

all rational functions. can have ys too

central angle formula

angle=s/r where s is length of outside of circle and r is the radius

how can we write the absolute value functoin

as a piecewise function where it is X for greater than or equal to zero and -x for less than 0

polnomial

ax^n+bx^n-1

limit laws

breath apart under sum, difference, cosntants go out front, products, quotients, power comes in, limits pass through functions

propertie of logarithms

can multiply, quotient, power comes out front, and ln1/x=-lnx

derivative rules

constant comes out front, splits through addition and subtraction

cos(A+B)

cosAcosB-sinAsinB

cos2x

cos^2x-sin^2x

sinx

crosses at 0

cosx

crosses at 1

taan graph

crosses at o and asmptotoes at pi over 2 and -pi over 2

law of cosines

c²=a²+b²-2abcosC

derivative quotient rule

d/dx (u/v)=vdu/dx-udv/dx/v^2

derivative product rules

d/dx(uv)= ux dv/dx + v du/dx

what is the area under the curve

definite integral

fundamental theorem of calc part 1

derivative of definite integral from a to x is just the function evaluated at the upper limit

instantaneous velocity

derivative of the function

how to find the limit as going to infinity

divide every term by the highest power of x in the denominator and find it that way

how to find area between two curves

do the definite integral of the upper curve minus the lower

how do estimate area between curves that is vertical

do the most righ curve is first and then the left is the seconds

summetry of even and odd functoins

even functions are the same if flip over y axis and odds are same if flip over y and x, (about origin)

sin transofrmations

f(x)= Asin(2pi/b (x-C))+D a is amplitude, b is period, c is horizontal shift, d i vertical shift

linear function

f(x)= mx+b.

exponential function

f(x)=a^x

identity function

f(x)=x

power function

f(x)=x^n

how to do applied optimization

find a formula for what you are looking for like area and then find the derivative to find the minimum or max value

riemann sum formula

for doing area under the curve. episilon f(a+k(b-a)/n) times (b-a/n). where b is the upper value and a the lower

cot graph

going down crosses at pi over 2 and asymptotes at pi and o

greatest integer function or integer floor functoin

has bars at bottom, y is greatestest integer less than or equal to x

leaster interger function/ integer ceiling function

has little bar at top. y is the smallest integer greater than or equal to x

newtons method

has n, x of n, f(xofn) f'(xofn) , x of n+1 = x of n- (f(xofn)/f'(xofn). then take the last value and use as x ofn . start with n of 0

mean value theorem

if a function if continuus from a to b and differentiable on the interior, then there is at least one point where the slope of the secant line is a tangent line of the function

Rolle's theorem

if a function is continuous from a to be and differentiable and f(a)=f(b), then there is a least one point where the derivative is equal to 0

sandwich theorem

if g(x) is less than f(x) is less than h(x). or equal to for all, thenif other two are equal middle must be too

intermediate value theorem

if is a value between F(a0 and F(b) and the function is continuous, the the function will equal y at some point

differentiable and continous relatedness

if it is differential at a point it is continous, doesnt apply other way

horizontal asymptotes

if num is less tha nden, the it is y=o, if they are equal it is a/b, if it is greater then maybe oblique

properties of definite integrals

if put negative it will swtich a and b. if from a to a then it equals o. can take constant out. can add and substract.

corollary 2 of mean value

if the derivaitves of two functions are equal then they only differ by a cosntant

corollary one of mean value

if the derivative is 0 on every point then it is a cosntant function

fundamental theorem of calc part 2

if the function is contuous from a to b then the integral is just the antiderivate at b - anti at a

three things to be continuous

if the limit approaches the value at x, if the right and left limits approach same value

how to add inegrals with different a and b

if the upper limit and lower limit of other are the same then can do lower limit of one to upper of other

L Hopitals rule

if you have the limit of quotient if take limit of top over the limit of bottom and its like 0 over 0 or infinity over infiintiy then you can do the derivative of each over each other and find the limit of that instead

limit of function with sindelta over delta

is 1

how to do change of base formula

is have log base a of x, can do lnx/lna

definition of definite integral

is the limit as n goes to infinity of the riemann sum

half life

ln2/k

what is the inverse of the exponential function y=A^x

log base a of x

how to do u substitution

make something u and then find the deriviate and set that equal to du so you can then find the antiderivate of what is left and then plug u back in

how to find oblique asymptote

only if num is bgger than den, just divide them and the non remainder part is the line

log function

only positive x,

rational function

quotient

finite sum rules

same as sigma, can add, subtract, pull csonatn out and if a constant function and just multiple constant by the top n value

derivative

same as slope of tangent line

indefinite integral

same as the antiderivative basically

acceleration

second derivative of function

how to do related rates problems

set up equation and differentiate it all using them as functions of x. can help you find how fast something is changing

function

set where there is only one y value or F(x) for each x value

ratio for 90-30-60 triangle

short is 1, hypo is 2 and other side is sqrt3

sin(a+B)

sinAcosB+cosAsinB

average rate of change

slope of secant line

instantenous change

slope of tangent line which is same as secant but if take lim as h goes to 0

unit circle for pi over 6

sqrt3/2, 1/2

implicit differentiation

when you have y in the formula too, you can differentiate that as a function of x. so if you have like sinxy you will have to use chain rule. can solve for dy/dx