12-03: Curve Sketching

Oblique asymptote

________, OA: when the degree of the numerator is greater than the degree of the denominator.

Critical numbers

________: value a in the domain of the function for which either f (a)= 0 or f (a)= DNE.

Hole

________ (open point): if a factor cancels out.

Absolute minimum

________: the lowest y coordinate on the function.

OA

________ y= the quotient of the numerator and denominator.

Local Maximum

________: if the y coordinate of all points are less than the y.

Rational function

________: a function in the form y= (f (x))/ (g (x))

Point of inflection

________: the point at which the graph changes concavity.

Second derivative test

________ helps classify critical points but also can identify points of inflection.

Absolute maximum

________: the highest y coordinate on the function.

Local minimum

________: if the y coordinate for all points in the vicinity are greater than the y coordinate of the point.

Local extrema

________: local maximum and minimum values of a function, also called turning points.

Critical numbers

________ occur when f (x)= 0 and local extrema occur when the sign of the derivative changes.

Critical numbers

________ are x values, to find a point, substitute the x values in and solve for y.

critical number

The first derivative test: indicates whether a(n) ________ yields a local maximum, a local minimum, or neither.

Local Maximum

if the y coordinate of all points are less than the y

Local minimum

if the y coordinate for all points in the vicinity are greater than the y coordinate of the point

Local extrema

local maximum and minimum values of a function, also called turning points

Absolute maximum

the highest y coordinate on the function

Absolute minimum

the lowest y coordinate on the function

Critical numbers

value a in the domain of the function for which either f(a) = 0 or f(a) = DNE

Critical points

(a, f(a))

The first derivative test

indicates whether a critical number yields a local maximum, a local minimum, or neither

If f(x) changes from positive to negative

Local max

If f(x) changes from negative to positive

Local min

If the sign of f(x) does not change

Not a local max or min, we have a horizontal tangent

A function increases over an interval if

it rises from left to right

A function decreases over an interval

it falls from right to left

When f(x) is greater than 0, positive and above the x axis

the function is increasing

When f(x) is less than 0, negative and below the x axis

the function is decreasing

Steps

to solve for increasing and decreasing intervals

Pick a number in the boundaries of the column headers (a number)

substitute this into the rows and record the sign overall

Concave up

all tangents on the interval are below the curve (slope increasing)

Concave down

all tangents on the interval are above the curve (slope decreasing)

Point of inflection

the point at which the graph changes concavity

If f"(x) is greater than 0

Concave up

If f"(x) is less than 0

Concave down

If f"(x) = 0

Possible point of inflection (f"(x) must change signs over the zero)

Rational function

a function in the form y=(f(x))/(g(x))

Vertical asymptote, VA

zeros of the denominator, solve the denominator

Oblique asymptote, OA

when the degree of the numerator is greater than the degree of the denominator

x int

sub y=0, zeros of the numerator (solve the numerator)

y int

sub x = 0

Hole (open point)

if a factor cancels out

HA

lim x→±∞

x int

sub y=0

y int

sub x=0