ap ab exam

to simply a limit

-multiple by difference of squares

pythagorean trig identities

cos�? +sin�?=1
1+tan�?=sec�?
cot�?+1=csc�?

write a limit f(x) can be made as close to 2 by taking x sufficiently close to 0, but not equal to 0

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.

removable discontinuity on graph

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)

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

One-sided limits

lim x--> 2^- (approaches from the left)
lim x--> 2^+ (approaches from the right)

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

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.

indeterminate form

average rate of change

f(b)-f(a)/b-a

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)

differentiability implies

continuity, but vise versa does not work

things that are not differentiable

-sharp corners
-not continuous
-cusps
-vertical tangents (straight lines)
-a jump
-straight line
-hole(not continuous)

A function is differentiable at

horizontal tangents

horizontal tangent

located where a function's derivative is zero.

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.

derivative of ln x, cosx, sinx, and ex

simplifies to sin d/dx. if u see this form, recognize it and simplify!

Let g be the inverse of f. means...

f(g(x))=x

If y=(sin?1)x then

siny= x

Linear Approximation

L(x) = f(a) + f'(a)(x-a) (point slope)

How many times does the particle change direction over the time interval?

when velocity crosses over the x axis (sign changes)

The approximation at x=2.3 is an underestimate of the corresponding function value, this means

the function is concave up

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.

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.

point of inflection

the point where the graph changes concavity

critical point

f?(x) = 0 or f?(x) does not exist

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.

LRAM over approximation when

decreasing

increases at a rate proportional to the amount present

use pert

not being continuous

does guarantee not being differentiable

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)

How to find the vertical asymptote of a rational function

set demoimator to 0

1/x-1

can use ln (x-1), cant split up terms, could spilt up terms if x-1 was on top.

is the speed increasing

if acceleration and velocity the same sign, yes

range of X^e / ln

anything greater than 0

graph of butterflies vs day. slope of this graph means

(increasing or decreasing) at a rate of butterflys/per day