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CONCAVITY AND THE SECOND DERIVATIVE TEST

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Locate the critical numbers of Use these numbers to determine the test intervals. Determine the sign of at one test value in each of the intervals. ... – PowerPoint PPT presentation

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Title: CONCAVITY AND THE SECOND DERIVATIVE TEST


1
CONCAVITY AND THE SECOND DERIVATIVE TEST
  • Section 3.4

2
When you are done with your homework, you should
be able to
  • Determine intervals on which a function is
    concave upward or concave downward
  • Find any points of inflection of the graph of a
    function
  • Apply the Second Derivative Test to find relative
    extrema of a function

3
I lived from 1642-1715. I developed Calculus. I
called Calculus Fluents and Fluxions. I
discovered the law of gravity. I generalized the
Binomial Theorem. Who am I?
  • Fermat
  • Newton
  • Pythagoras
  • Pascal

4
Definition of Concavity
  • Let f be differentiable on an open interval I.
    The graph of f is concave upward on I if
    is increasing on the interval and concave
    downward on I if is decreasing on the
    interval.
  •  
  • concave down concave up

5
Theorem Test for Concavity
  • Let f be a function second derivative exists on
    an open interval I.
  • If for all x in I, then f is
    concave upward in I.
  • If for all x in I, then f is
    concave downward in I.

6
Definition Point of Inflection
  • Let f be a function that is continuous on an open
    interval and let be a point in the
    interval. If the graph of I has a tangent line
    at this point , then this point is a
    point of inflection of the graph of f if the
    concavity of f changes from upward to downward
    (or downward to upward) at the point .

7
Theorem Point of Inflection
  • If is a point of inflection of the
    graph of f, then either or
    does not exist at
  • Hmmm.so this means that c is a ________
  • __________ of the ___________.

8
Guidelines for Determining Concavity on an
Interval I and Finding Points of Inflection
  • Locate the critical numbers of Use
    these numbers to determine the test intervals.
  • Determine the sign of at one test value
    in each of the intervals.
  • Use the theorem regarding the test for concavity
    to determine whether is concave upward
    or concave downward on each interval.
  • Examine the results of the test for a change in
    concavity to determine if there are any
    inflection points.

9
Find the points of inflection and discuss the
concavity of the graph of the function
  • The function is concave upwards over its entire
    domain. There is a point of inflection at (0 ,
    3)
  • The function is concave upwards on and
    concave downwards on
  • The function is concave upwards over its entire
    domain and there are no points of inflection.
  • None of the above

10
The most famous algebraist of the 1600s was
Fermat. Along with Pascal, he founded the
subject of
  • Probability
  • Number Theory
  • Abstract Algebra
  • All of the above.

11
5 cards are selected without replacement from a
standard 52 card deck. Find the probability that
all the cards are spades. This is called a
straight flush in poker.
  • Both B and D

12
Theorem Second Derivative Test
  • Let f be a function such that and
    the second derivative of f exists on an open
    interval containing c.
  • If then f has a relative minimum
    at
  • If then f has a relative maximum
    at
  • If the test fails. That is, f may
    have a relative maximum, a relative minimum, or
    neither. In such cases, you can use the First
    Derivative Test.

13
The following function has a relative minimum at
  • 6856.0
  • 0.0
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