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4.3

Using Derivatives

for Curve Sketching

Greg Kelly, Hanford High School, Richland, ashington!hoto "y #ickie Kelly, $%%&

'ld (aithful Geyser, )ello*stone +ational !ark

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4.3

Using Derivatives

for Curve Sketching

Greg Kelly, Hanford High School, Richland, ashington!hoto "y #ickie Kelly, --

)ello*stone (alls, )ello*stone +ational !ark

7/21/2019 Calc03_3 Uso de Las Derivadas

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4.3

Using Derivatives

for Curve Sketching

Greg Kelly, Hanford High School, Richland, ashington!hoto "y #ickie Kelly, --

/a00oth Hot S1rings, )ello*stone +ational !ark

7/21/2019 Calc03_3 Uso de Las Derivadas

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2n the 1ast, one of the i01ortant uses of derivatives *as as

an aid in curve sketching. ven though *e usually use a

calculator or co01uter to dra* co01licated gra1hs, it is still

i01ortant to understand the relationshi1s "et*een

derivatives and gra1hs.

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(irst derivative

is 1ositive Curve is rising.

is negative Curve is falling.

is 5ero !ossi"le local 0a6i0u0 or

0ini0u0.

Second derivative

is 1ositive Curve is concave u1.

is negative Curve is concave do*n.

is 5ero !ossi"le inflection 1oint

7*here concavity changes8.

y

y

y

y

y

y

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6a01leGra1h

9here are roots at and .

Set (irst derivative test

negative

1ositive

1ositive

!ossi"le e6tre0e at .

e can use a chart to organi5e our thoughts.

( ) ( )23 2

3 4 1 2y x x x x= + = +

1x = 2x =

23 6y x x =

0y =

20 3 6x x=

20 2x x=

( )0 2x x=

0, 2x =

y

0 2

0 0 ++

( ) 21 3 1 6 1 3y = =

( ) ( ) ( )2

1 3 1 6 1 9y = =

( ) 2

3 3 3 6 3 9y = =

0, 2x =

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6a01leGra1h

(irst derivative test

NOTE:'n the :! 6a0, it is notsufficient to si01ly dra*the chart and *rite the ans*er. )ou 0ust give a writtenexplanation!

There is a local maximum at (0,4)because for all x in

and for all x in (0,2).

There is a local minimum at (2,0)because for all x in

(0,2)and for all x in .

( ) ( )23 2

3 4 1 2y x x x x= + = +

23 6y x x =

y

0 2

0 0 ++

0y >

( , 0) 0y