AP Calculus Chapter 3

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  1. Critical Numbers
    Values of x in the domain of f (x) where f 'x= 0 or f 'x is undefined
  2. How do you find absolute minimums and maximums on a closed interval?
    • 1. Find critical numbers
    • 2. Find values for the function at each endpoint and critical number
    • 3. Determine absolute max and absolute min for the interval by your results from step 2.
  3. Rolle’s Theorem
    • If f is differentiable on the interval (a , b) and f(a) = f(b) then there is at least one place in the interval
    • where f 'x= 0
  4. Mean Value Theorem (MVT)
    • If f (x) is continuous and differentiable on the interval [a , b], there is someplace in the interval where f ' x
    • equals the slope of the line going through the endpoints of the interval.
  5. First Derivative ITSC
    1) Use critical numbers to determine intervals.

    • Interval
    • Test Value
    • Sign of f ' x
    • Conclusion
  6. If sign of f ' x is positive what is happening to f(x)?
    f(x) is increasing
  7. If sign of f ' x is negative then f(x) is _______.
    f(x) is decreasing
  8. First Derivative Test
    Do a first derivative ITSC,

    • If f ' x is changing from pos to neg, then relative max
    • If f ' x is changing from neg to pos, then relative min
  9. Concave Up
    f'(x) is increasing, f''(x)is positive, f(x) will hold water
  10. Concave Down
    f ' x is decreasing, f '' x is negative, f (x) will shed water
  11. Second Derivative ITSC
    Find where the 2nd Derivative is 0 or undefined. Use those numbers to make intervals

    • Intervals
    • Test Value
    • Sign of f '' x
    • Conclusio
  12. If f '' x is positive
    interval is concave up
  13. If f '' x is negative,
    interval is concave down
  14. Inflection Point
    • Where the function is switching from concave up to concave down or
    • from concave down to concave up.
  15. 2nd Derivative Test
    • 1. Plug your critical numbers into the 2nd Derivative
    • 2. If the result is positive, then the there is a relative min at the critical
    • number
    • 3. If the result is negative, then there is a relative max at the critical
    • number.
    • 4. If the result is 0, the test fails
  16. Image Upload
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  18. Image Upload
  19. Definition of a Differential
    dy= f(x)dx
  20. Definition of Δy
    Δy= f(x+Δx) − f(x)
  21. Optimization Problem
    • 1. Write your two equations. The equation that you are minimizing or maximizing is the Primary Equation.
    • 2. Get your Primary equation in terms of one variable by substituting in the other equation.
    • 3. Differentiate your primary equation and find the critical numbers.
    • 4. Show that the critical numbers are a max or min by the 2nd Derivative Test or a First Derivative ITSC
    • 5. Answer the question.
Card Set:
AP Calculus Chapter 3
2011-10-21 05:56:02
AP Calculus BC Chapter

These are my flashcards for the third chapter of my book, Calculus of a single variable AP edition
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