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critical point
or undefined

or undefined
critical point





concavity inc slope
f''>0

f''>0
concavity inc slope

concavity dec slope
f''<0

f''<0
concavity dec slope

local minimum
goes (,0,+) or (,DNE,+) or

goes (,0,+) or (,DNE,+) or
local minimum

local maximum
goes (+,0,) or (+,DNE,) or

goes (+,0,) or (+,DNE,) or
local maximum

point of inflection
 f''=0 or DNE and concavity changes
 goes from (+ to ), ( to +)

f''=0 or DNE and concavity changes
goes (+ to ),( to +)
point of inflection

abs. max/min
eval. crit# & endpoints or discuss "always inc or always dec"

eval. crit# & endpoints or discuss "always inc or always dec"
abs. max/min

intermediate value theorem
 if the function f(x) is continuous on [a,b], for all k between f(a) and f(b), there exists at least one number x=c in the open interval:
 (a,b) such that f(c)=k

if the function f(x) is continuous on [a,b], for all k between f(a) and f(b), there exists at least one number x=c in the open interval:(a,b) such that f(c)=k
intermediate value theorem

extreme value theorem
if the function f(x) is continuous on [a,b], then there exists an absolute max and min on that interval

if the function f(x) is continuous on [a,b], then there exists an absolute max and min on that interval
extreme value theorem

Rolle's theorem
 if the function f(x) is continuous on [a,b], AND
 differential on the interval (a,b), AND
 f(a)=f(b), then there is at least on number
 x=c in (a,b) such that f'(c)=0

if the function f(x) is continuous on [a,b], ANDdifferential on the interval (a,b), ANDf(a)=f(b), then there is at least on numberx=c in (a,b) such that f'(c)=0
Rolle's theorem

mean value theorem
 If the function f(x) is continuous on [a,b], AND
 differential on the interval (a,b), then
 there is at least one number x=c in (a,b)
 such that

If the function f(x) is continuous on [a,b], AND
differential on the interval (a,b), then
there is at least one number x=c in (a,b)
such that
mean value theorem

MVT of Integrals i.e. AVERAGE VALUE

If the function f(x) is continuous on [a,b] and differential on the interval (a,b), then there exists at least one number x=c on (a,b) such that
This value f(c) is the "average value" of the function on the interval [a,b].
MVT of Integrals i.e. AVERAGE VALUE

Limit Strategies
 Factor and cancel, rationalize numerator, usub HA rules:

Factor and cancel, rationalize numerator, usub HA rules:
Limit Strategies

To find all HA
Take limit as

Take limit as
To find all HA

Approximation methods for integration
Rectangles
Left, right and middle Riemann sums A=bh

Left, right and middle Riemann sums A=bh
 Approximation methods for integration
 Rectangles

Approximation methods for integration
Trapezoids

 Approximation methods for integration
 Trapezoids

Effects of inc/dec & concavity on approximation
Concave up
M under estimate, T over estimate

Effects of inc/dec & concavity on approximation Concave down
M over estimate, T under estimate

Effects of inc/dec & concavity on approximation
inc
L=under, R=over

Effects of inc/dec & concavity on approximation
dec
L=over,R=under

First fundamental theory of calculus
\

First fundamental theory of calculus

Second fundamental theory of calculus

Second fundamental theory of calculus





volume by cross section (not rotated)

Volume by cross section (not rotated)











