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Requirements for relative max/min?
Must be continuous and if its on a closed interval, it cannot be an endpoint.
 f " > 0 relative MIN
 f " < 0 relative MAX

Critical point
When f '(c) = 0 or undefined.

Location of Extrema
When derivative is undefined, or 0. So the same as a critical point

Finding Extrema on Closed Interval
 Find Critical Numbers (f '(c))
 Evaluate f(c) for each critical number
 Evaluate f(c) for endpoints
 High is MAX, Low is MIN
 NOTE: If it says a polynomial to the 3rd degree, there can only be 2 relative numbers.
 ex: 4th degree = 3 relative numbers etc

What types of functions are always continuous and differentiable?
 Polynomials
 Rational Functions
 Trig Functions

Rolle's Theorem
 Is f(x) continuous on [a,b]? why?
 Is f(x) differentiable on (a,b)? Why?
 Does f(a) = f(b)?
If all of these apply: find the critical points at f '(x).

Does Rolle's Theorem work for Absolute Values?
Only if the closed interval does NOT include the vertex. Because it is not differentiable at a sharp point

Mean Value Theorem (MVT)
If f is continuous on [a,b] and differentiable on (a,b) then there exists c on (a,b) such that f '(c)= [f(b)  f(a)] / (ba)

Rates of Change and Slopes
 Average Rate of Change = Instantaneous Rate of Change
 Slope of Secant = Slope of Tangent

Increasing/Decreasing
 f '(c) > 0 f is increasing
 f '(c) < 0 f is decreasing
 f '(c) = 0 f is constant
 When f ' changes from  to +, relative MINimum
 When f ' changes from + to , relative MAXimum

Velocity and Acceleration General Info
Speeding Up/Speeding Down?
 s(t) = position > 0 right of origin / < 0 left of origin
 s'(t) = velocity > 0 moving right / < 0 moving left
 s"(t) = acceleration
 Find when s '(t) = 0 and s "(t) = 0.
 Put on number line.
 Plug numbers in for BOTH velocity and acceleration.
 When signs are same it is speeding UP
 When signs are different it is slowing DOWN

Find polynomial of least degree such that
(given relative min/max)
f(x) = ax^4 + bx^3 + cx^2 + dx + e
 Plug in (x,y) in each f equation.
 Find derivative for equation and plug in (x,y)
 Solve for a,b,c,d,e

Concavity
 Concave UP = f ' is INCREASING
 Concave DOWN = f ' is DECREASING
 Concave UP = f " > 0
 Concave DOWN = f " < 0

Point of Inflection
When concavity changes from + to  or  to + at a point
 f " = 0 or undefined
 TANGENT MUST EXIST
 (vertical/horizontal line is usually the case)

2nd Derivative Test
 When f '(c) = 0
 f "(c) < 0 Relative MAX
 f "(c) > 0 Relative MIN
 Steps:
 find f '(x)
 plug in results to f "(x)

Limits of Infinity (oo)
Divide by highest power of the DENOMINATOR
 Denominator > Numerator f(x) = 0
 Denominator = Numerator f(x) = coefficients
 Denominator < Numerator f(x) = +/ infinity (oo)
lim (x = > oo) (sinx)/x = 0

L'Hopital's Rule
ONLY USE WHEN IT IS 0/0 OR oo/oo !!!
 Always substitute first to find out ^
 Find derivative, and substitute again.
 Keep going until it works
NOTE: do NOT use quotient rule!

Indeterminate Forms
(!= means does not equal)
 oo  oo != 0
 0 x oo != 0
 1^oo != 1
 oo^0 != 1
 0^0 != 1

Determinate Forms
 nonzero/0 = +/ oo
 0^oo = oo
 0^oo = 0
 oo  oo =  oo
 oo + oo = oo

Graphing
 If f is increasing, f ' > 0 (above axis)
 If f is decreasing, f ' < 0 (below axis)
If f '(x) = 0, there is a HORIZONTAL TANGENT
If f '(x) does not exist, there is a SHARP TURN or VERTICAL TANGENT
 If f " > 0, concave UP
 If f " < 0, concave DOWN
If f ' does not exist, then it is a VERTICAL TANGENT
When f " = 0, there is an inflection point (CONCAVITY CHANGES)

