Math Chapter 3

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craziieegurl13
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45578
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Math Chapter 3
Updated:
2010-10-28 00:51:05
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applications differentiation rolles theorem extrema math
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Chapter 3: Applications of Differentiation
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  1. 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
  2. Critical point
    When f '(c) = 0 or undefined.
  3. Location of Extrema
    When derivative is undefined, or 0. So the same as a critical point
  4. 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
  5. What types of functions are always continuous and differentiable?
    • Polynomials
    • Rational Functions
    • Trig Functions
  6. 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).
  7. 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
  8. 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)
  9. Rates of Change and Slopes
    • Average Rate of Change = Instantaneous Rate of Change
    • Slope of Secant = Slope of Tangent
  10. 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
  11. 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
  12. 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
  13. Concavity
    • Concave UP = f ' is INCREASING
    • Concave DOWN = f ' is DECREASING

    • Concave UP = f " > 0
    • Concave DOWN = f " < 0
  14. 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)
  15. 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)
  16. 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
  17. 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!
  18. Indeterminate Forms
    (!= means does not equal)

    • oo - oo != 0
    • 0 x oo != 0
    • 1^oo != 1
    • oo^0 != 1
    • 0^0 != 1
  19. Determinate Forms
    • nonzero/0 = +/- oo
    • 0^-oo = oo
    • 0^oo = 0
    • -oo - oo = - oo
    • oo + oo = oo

    • 1/0 = oo
    • 1/oo = 0
  20. 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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