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
