ap ab exam
critical point
f′(x) = 0 or f′(x) does not exist
point of inflection
the point where the graph changes concavity
graph of butterflies vs day. slope of this graph means
(increasing or decreasing) at a rate of butterflys/per day
to simply a limit
-multiple by difference of squares
things that are not differentiable
-sharp corners -not continuous -cusps -vertical tangents (straight lines) -a jump -straight line -hole(not continuous)
how f'(a) might fail to exist
1. corner ( absolute value) 2. a cusp (ex: x^2/3) (if 0 is in demoinator) 3. a vertical tangent (x^1/3) (if 0 is in demonator) 4. a discontinuity (piece wise functions)
Rules for finding Horizontal Asymptotes
1. if highest degree of the denominator is bigger than numerator the horizontal asymptote is y = 0. 2. if degrees are the same, ratio 3. if highest degree of the numerator is bigger than the denominator, there is no horizontal asymptote.
Intermediate Value Theorem
If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k
extreme value theorem
If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval. has to have a bound interval.
2nd derivative test
If f'(c) = 0 and f"(c)<0, there is a local max on f at x=c. If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.
Squeeze Theorem
If f(x) ≤ g(x) ≤ h(x) and limx→a f(x) = limx→a h(x) = L, then limx→a g(x) = L. if two functions are in between each other and the top 2 have the same limit, then the middle one will have the same limit too.
Linear Approximation
L(x) = f(a) + f'(a)(x-a) (point slope)
Mean Value Theorem
The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line. if the function is continuous on the closed interval and differentiable on the open interval.
range of X^e / ln
anything greater than 0
1/x-1
can use ln (x-1), cant split up terms, could spilt up terms if x-1 was on top.
differentiability implies
continuity, but vise versa does not work
pythagorean trig identities
cos²θ +sin²θ=1 1+tan²θ=sec²θ cot²θ+1=csc²θ
LRAM over approximation when
decreasing
not being continuous
does guarantee not being differentiable
Which of the following statements, if true, can be used to conclude that f(2) exists?
f is continuous at x=2. f is differentiable at x=2. (limits don't matter)
in order for f to be continuous at x=1, the following conditions must be satisfied:
f(1) exists, limx→1f(x) exists, and limx→1f(x)=f(1)
average rate of change
f(b)-f(a)/b-a
Let g be the inverse of f. means...
f(g(x))=x
A function is differentiable at
horizontal tangents
is the speed increasing
if acceleration and velocity the same sign, yes
One-sided limits
lim x--> 2^- (approaches from the left) lim x--> 2^+ (approaches from the right)
horizontal tangent
located where a function's derivative is zero.
How to find the vertical asymptote of a rational function
set demoimator to 0
derivative of ln x, cosx, sinx, and ex
simplifies to sin d/dx. if u see this form, recognize it and simplify!
If y=(sin−1)x then
siny= x
The approximation at x=2.3 is an underestimate of the corresponding function value, this means
the function is concave up
Let f be a function such that limx→5−f(x)=∞. Which of the following statements must be true?
there is a vertical asymptote at x = 5
vertical tangent
touches the curve at a point where the gradient (slope) of the curve is infinite and undefined. On a graph, it runs parallel to the y-axis.
"increases at a rate proportional to the amount present"
use pert
How many times does the particle change direction over the time interval?
when velocity crosses over the x axis (sign changes)
