Historical Note
Maria Gaetana Agnesi
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1718-1799 - Milan, Habsburg Empire (now Italy)
Oldest of 21children (3 mothers)
Wealthy (and busy) father
She spoke three languages
Writer/Debater of philosophy and natural science
Studied religious books and mathematics
Admired Newton
Learned Calculus from a monk - Ramiro Rampinelli
He encouraged her to write a book on Calculus - She
is famous for writing that book in 1748
Increasing and Decreasing
Functions
(page 290)
Closed and Open Interval
Notation
(page 291)
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There is not universal agreement on
derivative interval notation.
The theorem below shows this author’s
preference.
AP Exam will accept open or closed notation.
First and Second Derivative
Summary Concepts
G raph A bov e x-axis
f  x   0 f  x  is positive
f   x   0 f  x  is increasing
f   x   0 f  x  is concave up
G raph at x- a xis
f  x   0 x is a root/solution/x-intercept
f   x   0 f  x  is a m axim um /m inim um /inflection point
f   x   0 f  x  behavior is incon clusi ve
G raph B elow x-axi s
f  x   0 f  x  is negative
f   x   0 f  x  is decrea sing
f   x   0 f  x  is concave d ow n
5.2 Extrema
(page 300-305)
Relative (Local) Maximum or Minimum Values
Absolute (Global) Maximum or Minimum Value
Critical Numbers
(page 302)
Critical Numbers Summary
(page 300)
D efinition of C ritical N um bers - T he num bers a w hich either f   x   0
or f is not differentiable are called critical num bers of f .
P oints w here f   x   0 are called stationary po ints and occur w here
f  x  has a m axim um , m inim um or an inflection point.
Inflection points m ark the places on the curve y  f  x  w here
the rate of change of y , f   x  , w ith respect to x changes form
increasing to decreasing, or visa versa.
Points w here f   x  D N E occur w here f  x  has
a cusp, corner point, vertical tangent line or point of discontinuity.
Critical Numbers
(page 302)
Critical Numbers and Relative
Extrema
(page 302)
Relative Extrema occur at critical numbers.
However, not every critical number has a relative extrema.
First Derivative Test
(page 302)
If f   x  changes from increasing to decreasin g at a
critical num ber, then there is relative m axim um .
If f   x  changes from decreasing to increasing at a
critical num ber, then there is relative m inim um .
If f   x  has the sam e sign on either side of
a critical num ber, then there is no extrem e value.
Second Derivative Test
(page 303)
If f   x  = 0 an d f   x  > 0 , th en f h as a m in im u m .
If f   x  = 0 an d f   x  < 0 , th en f h as a m ax im u m .
Homework Problem #3
(page 298)
Homework Problem #4
(page 296)
Homework Problem #5
(page 298)
Homework Problem #6
(page 298)
Homework Problem #7
(page 298)
Example 5a,5b,5c
(page 295)
6b f
6a f
x 
xe
x
 x   sin x ,
0  x  2
6c f
x 
tan
1
x
Example 2
(page 303)
Example 3
(page 303)
Example 4
(page 304)
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