final exam Flashcards
what rule do we use when we try to evaluate and it gives 0/0 or inf/inf
L’hopital rule
what happens when denominator is bigger than numerator
it will be 0
ln(x)
1/x
cos(0)
1
d/dx of cosx
-sinx
sin(0)
0
d/dx of sinx
cosx
in limits, sinx/x
its 1
what to do with LH
keep find doing derivative until it does not give 0/0 or inf/inf
a^3 - b^3
(a-b)(a^2+ab+b^2)
in limits, tanx
its sinx/cosx
in limits, if polynomials
ignores insignificant terms
in limits, if we look for HA
we use limit x->inf
in limits, if we look for VA
we use limit x->0
any power of 0
= 1
loga (b) = c
= a^c b
ln(1)
0
squeeze theorem
h(x) <_ f(x) <_ g(x)
d/dx of x
= 1
d/dx of e
= 0
d/dx of ln(u)
= u’/u
log,diff and inv trig: d/dx of a^u
= a^u . u’ . ln (a)
d/dx of e^u
= e^u . u’
in piecewise functions, if there’s a letter except x and y
have to make it the same and that letter = ?
3 steps continuity test
- f(a) is defined
- lim x->a f(x) exist
- lim x->a f(x) = f(a)
if step 2 fails in “3 steps continuity test”
it will be a jump discontinuity
if step 3 fails in “3 steps continuity test”
it will be a hole discontinuity
definition of derivatives
lim h->0 f(x+h) - f(x) / h
d/dx of tanx
sec^2x
d/dx of cotx
-csc^2x
d/dx of secx
secxtanx
d/dx of cscx
-cscxcotx
eqn of tangent line
y-y1 = m (x-x1)
when look for eqn of tangent line, how to find y
f(x) will be y and plug x in the f(x) function.
when look for eqn of tangent line, how to find m
m is f’(x) of f(x)
quotient rule
(hi)(d’lo) - (lo)(d’hi) / lo^2
product rule
uv’ +vu’
chain rule
f(g(x)) = f’(g(x)) . x’
d/dx of cot(g(x))
-csc^2(g(x)) . (g’(x))
d/dx of arctan g(x)
(1/ 1 + g(x)^2) . g’(x)
d/dx of arcsin g(x)
(1/ sqrt( 1- g(x)^2) . g’(x)
d/dx of arccos g(x)
(-1/ sqrt( 1- g(x)^2) . g’(x)
steps in Logarithmic Differentiation
- put ln both sides
- ln multiply with the eqn
- put d/dx
- solve
steps of implicit difference
- deriv both sides
- rearrange and factor
if looking for slope of an eqn: plug numbers in
(whenever y is affected, there will d/dx next to it)
inverses functions
exchange with y and x
