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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
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Critical point
When f '(c) = 0 or undefined.
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Location of Extrema
When derivative is undefined, or 0. So the same as a critical point
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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
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What types of functions are always continuous and differentiable?
- Polynomials
- Rational Functions
- Trig Functions
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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).
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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
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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)] / (b-a)
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Rates of Change and Slopes
- Average Rate of Change = Instantaneous Rate of Change
- Slope of Secant = Slope of Tangent
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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
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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
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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
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Concavity
- Concave UP = f ' is INCREASING
- Concave DOWN = f ' is DECREASING
- Concave UP = f " > 0
- Concave DOWN = f " < 0
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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)
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2nd Derivative Test
- When f '(c) = 0
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f "(c) < 0 Relative MAX - f "(c) > 0 Relative MIN
- Steps: find f '(x)
- plug in results to f "(x)
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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
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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!
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Indeterminate Forms
(!= means does not equal)
- oo - oo != 0
- 0 x oo != 0
- 1^oo != 1
- oo^0 != 1
- 0^0 != 1
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Determinate Forms
- nonzero/0 = +/- oo
- 0^-oo = oo
- 0^oo = 0
- -oo - oo = - oo
- oo + oo = oo
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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)
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