Unformatted text preview: d f ” ( x) = 6x
At x = 0 : f(0) = +1 and f ’(0) = 0, implying x=0 may be a maximum
or a minim um point. From the graph we can see it is neither. f(x) is
minimum
concav
continuously increasing. However, the curve is changing shape from concave
down
concav up.
do wn to conca v e up
At x=0 : f ”(0) = 0 and ore
positiv after.
neg tiv befor
f ”(x) is ne g a ti v e befor e and positi v e after. Consider the function f(x) =  x 3 + 1 . Please draw the graph.
f ’(x) =  3x 2 a nd f ”(x) =  6x
At x = 0 : f(0) = +1 and f ’(0) = 0, implying x = 0 may be a maximum
or a minim um point. From the graph we can see it is neither. f(x) is
minimum
continuously decreasing. However, the curve is changing shape from concave
concav
down.
up to concave down
At x = 0: f ” ( 0) = 0 and f ” ( x) is positi v e bef or e and ne g a ti v e after
positiv befor
ore
neg tiv after.
121 y
f ' (x) = 3x 2 f " (x) = 6x
f(x) = x3 + 1 4 3 2
11
x
 3  2  1 0
 1  2  3  4 122
122 1 2 3 In both cases, f(x) = x 3 + 1 a nd f(x) = − x 3 + 1 , we have f ’ (0) = 0.
But x = 0 is neither a maxim um nor a minim um How do we determine
maximum
minimum
um.
what kind of point x is ?
f ’ ( x) = 0 and f ” ( x) negative ⇒ maximum
maximum.
m aximum
f ’ ( x) = 0 and f ” ( x) positive ⇒ minimum
minimum.
m inimum
f ’ ( x) = 0 and f ” ( x) = 0 and changing sign ⇒ inflexion
inflexion.
inflexion
Let us now summarize in tabular form the results of these last few chapters. For
2
any wellbehaved function y = f(x), with dy/ dx = f ’ (x) and d y/ dx2 = f” (x),
we have:
Type of point dy tan θ = slope of tangent
s lope
=
dx
before
after d2y
dx2 Maximum 0 + − − or 0 Minimum 0 − + + or 0 Inflexion
(turning) 0 ± ± 0 and
changing sign 123 ...
Reviewing the example of the bouncing ball at the instants t1, t2, t3, .... w e
.
can analyse and say:
1. The function (height) is decreasing before and increasing
after . 2. The first derivatives are ne g a ti v e before and positi v e after
neg tiv
positiv
i.e. changing sign. 3. f ’ (x) = 0. W hat can you...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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