sec 2.8: the derivative as a function replace a by x
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Sec 2.8: THE DERIVATIVE AS A FUNCTION
replace a by x
Sec 2.8: THE DERIVATIVE AS A FUNCTION
Slopes : 0 + -Slopes : 0 + -
Sec 2.8: THE DERIVATIVE AS A FUNCTION
)5.0('f
? 0)('when xf? negativeor positive )4('f
? positive )('when xf
Sec 2.8: THE DERIVATIVE AS A FUNCTION
? negativeor positive )0('f
? negative )('when xf
Sec 2.8: THE DERIVATIVE AS A FUNCTION
)(' xf
Sec 2.8: THE DERIVATIVE AS A FUNCTION
Sec 2.8: THE DERIVATIVE AS A FUNCTION
Sec 2.8: THE DERIVATIVE AS A FUNCTION
Definition: A function f is differentiable on an open interval (a, b) if it is differentiable at every number in the interval.
Definition: A function f is differentiable on an open interval (a, b) if it is differentiable at every number in the interval.
?0on abledifferenti Is
:Example
),(xf(x)
?0,on abledifferenti Is
:Example
)(xf(x)
0?at abledifferenti Is
:Example
xf(x)
Sec 2.8: THE DERIVATIVE AS A FUNCTION
2 properties2 propertiescontinuity continuity
differentiabilitydifferentiability
Proof: axax
afxfafxf
)()(
)()(
Remark: f cont. at a
f diff. at a
Remark: f discont. at a f not diff. at a
Remark: f discont. at a f not diff. at a
Sec 2.8: THE DERIVATIVE AS A FUNCTION
Example:
xxf )(
0if1
0ifexistnot does
0if1
)('
x
x
x
xf
f cont. at a
f diff. at a
f discont. at a f not diff. at a
f discont. at a f not diff. at a
Sec 2.8: THE DERIVATIVE AS A FUNCTION
Example:
xxf )(
0if1
0ifexistnot does
0if1
)('
x
x
x
xf
f cont. at a
f diff. at a
f discont. at a f not diff. at a
f discont. at a f not diff. at a
Sec 2.8: THE DERIVATIVE AS A FUNCTION
HOW CAN A FUNCTION FAIL TO BE DIFFERENTIABLE?
existnot does
Sec 2.8: THE DERIVATIVE AS A FUNCTION
Higher Derivativedx
dfxf )('
2
2')(')(''
dx
fd
dx
dfxf
dx
dxf
Note:
)(ts
)(')( tstv
)('')( tsta
jerk)(''')(' tsta
acceleration
velocity
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
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MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
R
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MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
H
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
H
H
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
H
R
R
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
H
H
H
R