2.2 derivatives of polynomial functions
DESCRIPTION
2.2 Derivatives of Polynomial Functions. Differentiate means “find the derivative” A function is said to be differentiable if he derivative exists at a point x=a. NOT Differentiable at x=a means that you cannot find the slope of the tangent at x=a. Examples (not differentiable at x=a) - PowerPoint PPT PresentationTRANSCRIPT
2.2 Derivatives of Polynomial Functions• Differentiate means “find the derivative”• A function is said to be differentiable if he derivative exists at a point x=a.• NOT Differentiable at x=a means that you cannot find the slope of the tangent at x=a.• Examples (not differentiable at x=a)CUSP VERTICAL TANGENT DISCONTINUITY
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2.2 Derivatives of Polynomial Functions
Constant rule and Power ruleConstant Rule: If where k is a constant then
(Prime notation)
OR
(Leibniz notation)
( )f x k
'( ) 0f x
( ) 0d kdx
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2.2 Derivatives of Polynomial Functions
Proof of Constant Rule:
'
0
0
0
( ) ( )( ) lim
lim
lim 0
0
h
h
h
f x h f xf xh
k kh
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2.2 Derivatives of Polynomial Functions
Power Rule:If then: where x is one term
where n is a real #OR
( ) nf x x' 1( ) nf x nx
1( )n nd x nxdx
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0limh
k kh
0limh
k kh
2.2 Derivatives of Polynomial Functions
Proof of Power Rule:
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'
0
0
1 2 2 1
0
1 2 2 1
0
1 2 2
( ) ( )( ) lim , ( )
( )lim
( ) ( ) ( ) ... ( )lim
lim[( ) ( ) ... ( ) ]
... ( )
n
h
n n
h
n n n n
h
n n n n
h
n n n
f x h f xf x where f x xh
x h xh
x h x x h x h x x h x x
h
x h x h x x h x x
x x x x x
1
1
( )n
n
x there are n terms
nx
2.2 Derivatives of Polynomial Functions
Ex. 1: Differentiate with respect to x:a) 33 2( ) 2 5 5 3.95f x x x x
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2.2 Derivatives of Polynomial Functions
b) ( ) (3 4)(7 2)g x x x
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2.2 Derivatives of Polynomial Functions
c) 1 xyx
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2.2 Derivatives of Polynomial Functions
Ex. 2: Find the slope of the tangent line to the curve at x=13 1( 1)y x x
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2.2 Derivatives of Polynomial Functions
Ex. 3: Find the co-ordinates of the point(s) on the graph of
at which the slope of the tangent is 12.
3 2y x
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2.2 Derivatives of Polynomial Functions
Ex. 4: Tangents are drawn from point (0,-8) to the curve
. Find the co-ordinates of the point(s) at which these tangents touch the curve.
2 4y x
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2.2 Derivatives of Polynomial Functions
Vocabulary:Derivative: Also known as instantaneous rate of change with
respect to the variable.Displacement, Change in position.Velocity, Rate of change of position with respect to time. Acceleration, Rate of change of velocity with respect to time.
( )v t
( )s t
( )a t
'( ) ( ) dva t v t
dt
'( ) ( ) d sv t s t
dt
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