APPLICATION OF DIFFERENTIATION

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3.4 APPLICATION OF DIFFERENTIATION
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Have you ever ride a roller coaster?
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Do you know that we can use differentiation to
find the highest point and the lowest point of
the roller coaster track?
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CRITICAL VALUE
important!!!
Critical value, c for a function f(x) is any
value of x in the domain of f at which
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GRAPHS
Tangent line (horizontal) at xc ?
y
y
y
y
x
x
x
x
c
c
c
c
Tangent line (vertical)?
y
y
y
y
x
x
x
x
c
c
c
c
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INCREASING OR DECREASING FUNCTIONS
y
x
c
b
e
d
a
f
Increasing? Decreasing? Constant?
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THEOREM 1
Let f be a continuous function on the interval
a, b a) f is increasing if b) f is
decreasing if c) f is constant if
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EXTREMUM RELATIVE
y
increasing
decreasing
increasing
x

• Refer to the maximum relative and minimum
  relative point

• Also known as local maximum and local minimum
  point

• Only at critical points or stationary points

• Not all critical points are extremum relative

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FIRST DERIVATIVE TEST
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CONCAVITY
y
f(x) increasing
f(x) decreasing
x
0
y
f(x) decreasing
f(x) increasing
0
x
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SECOND DERIVATIVE TEST
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POINT OF INFLECTION
If (c, f(c)) is a point of infection for f, then
For point of inflection, not all f(c) 0,
but f(c) 0
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POINT OF INFLECTION

• Let f be any function and (c,f(c)) is a point
  of inflection if
• f(x)gt0 for (a, c) and f(x)lt0 for (c, b) OR
• f(x)lt0 for (a, c) and f(x)gt0 for (c, b)

y
y
concave upwards
concave upwards
Concave downwards
concave downwards
x
x
0
0
b
c
a
a
c
b
(c, f(c)) point of inflection
(c, f(c)) point of inflection
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EXAMPLES