calc exam 3 - deriv tests for curve sketching & graph behavior

steps for curve sketching using 1st deriv test

• deriv of f(x)

• critical points by setting f’(x)=0

• choose critical points inside interval

• plug in a point to the L & R of critical points into f’(x)

• determine if each value is > or < 0 to determine if f(x) is increasing or decreasing @ the critical point

if the L of the crit. pt. is negative then positive to the R, the crit. pt. is a

local min (f(x) decreasing on L & increasing on R)

if the L of the crit. pt. is positive then negative to the R, the crit. pt. is a

local max (f(x) increasing on L then decreasing on R)

when f(x) is increasing, f’(x) is

positive/greater than 0

when f(x) is decreasing, f’(x) is

negative/less than 0

if value of f’(x) is negative/less than 0, f(x) is

decreasing

if value of f’(x) is positive/greater than 0, f(x) is

increasing

horizontal & constant lines don’t have

local extrema

crit. pt. of fractions for deriv tests are found when

• f’(x) numerator is set = 0

• f’(x) denom. is set = 0 (aka undefined in domain)

• MUST FIND BOTH

steps for curve sketching using 2nd deriv test

• find first deriv

• find second deriv

• factor if possible

• find critical points of f’’(x)

• determine where f’’(x) > 0 or < 0 by testing points in f’’(x) to find concave up or down

if value of f’’(x) is greater than 0, f(x) is

concave up

if value of f’’(x) is less than 0, f(x) is

concave down

if f(x) is concave up, f’’(x) is

positive

if f(x) is concave down, f’’(x) is

negative

point of inflection

point where concavity changes