Calc unit 2 derivatives Flashcards
find slope of tangent using limits
f(x+h) - f(x)
h
find slope of tangent using limits at point x=a
f(x)-f(a)
x-a
sum rule
f’(x+y) = f’(x) + f’(y)
scalar rule
f’(c . y) = c . f’(x)
horizontal tangent
numerator= 0
vertical tangent
denomerator=0
continuity
continous: make sure orig function is continous. its differentiable is derivitive is contious
absolute value
set limit of both side for orig. if theyre the same, then continous at that point
then take derivative
plug in x to find the two ys. if they’re the same, then the function is differentiable at that point
IVT
if function if differentiable then f’ has very y value btw y1 and y2
with variable and continuity n absolute value
set both orig equation to eachother solve (A)
set derivative of the equation to eachother solve (B)
solve for intersection of A n B- elimination/substitution
critical number
when derivative = o or undefined
veritcal tangent
orig increasing
f’ positive
orig decreasing
f’ negative
orig max
f’ posit to negat
orig mi
f’ neg to posit
how to find relative max/min and intervals of ncrease/decrease
find critical numbers
plot on sign chart
points of inflection
orig graph changes concavity
relative max/min of f’
f” changes sign
when f”=0/undefined
can’t b a point of discontinuity on orig
how to find points of inflection/interval of concavity (always open)
find f”
find points of inflection
make sign chart
orig concave up
f” is postive
orig concave down
f” is negative
2nd derivative
relative min at x
f’ at x is zero
f” at x is positive
2nd derivative
relative max at x
f’ at x is zero
f” at x is negative
chain rule
derivative at outside
derivative on inside
product rule
find derivative of two function multiplying eachother
f(x)g(x)
f.g’ + g.f’
quotient rule
lo dHi - Hi dlo
lo.lo
sinx
cosx
cosx
-sinx
tanx
sec^x
secx
tanxsecx
cotx
-csc^x
cscx
-cotxcscx
how many times can you take the derivative until its zero
highest degree +1 = when derivative is zero
4+1=5
5th derivative=0
mean value theorem
f(x) continuous on [a,b]
differentiable on (a,b) and c is btw a n b
then
slope of secant= derivative at c
slope of tangent=secant slope
how to find c in mvt
verify that mtv applies on (x1,x2)
find secant slope: y2-y1 / x2,x1
set secant slope= derivative equation
rolle’s theorem
horizontal tangent exist
f(x) continuous on [a,b] and differentiable on (a,b) and y value of endpoints are the same.
then there’s a point called C btw a n b where tangent at c is 0 and secant slope is 0
L Hopital’s rule
only when limit is indeterminate
lim of f(x)/g(x) is f’(x)/g’(x)
take derivative of top and bottom
if answer still indeterminate then take derivative again
types of indeterminate
0/0
infinity/infinty
extreme value theorem
if f is continuous at closed intereval then it must have absolute max and min
limit of
constant/ 0
infinity
limit of
0/ contant
0
limit of 1/infinty
0
f’ (e^x)
e^x
f’ (e^u)
u’ . e^u
f’ (a^x)
x’ . a^x . lna
f’ (lnx)
x’/x
f’ (loga X)
x’ / x . lna
how high does something go
velocity=0
plu back in position function
velocity tells direction
at rest: velocity=0
optimalization
EVT
to find absolute max
find critical numbers on (a,b)
make a sign chart with endpoints too
see where relative maxes are
then plug values into original function, see which relative is highest
applied optimalization
set up contrainst equation and optimization equation(which would equal what is asked for). plu constainst into optimization equation
