Sec 2.7: DERIVATIVES AND RATES OF CHANGE

Example:Find the derivative of the function at x = 2. Find

982)( xxxf)2('f

Example:Find the derivative of the function at a. Find

982)( xxxf)(' af

Sec 2.7: DERIVATIVES AND RATES OF CHANGE

Example:Find the derivative of the function at a. Find )(' af

21)( )29

x

xf

Sec 2.7: DERIVATIVES AND RATES OF CHANGE

axhhax If

axh approaches 0 approaches

an equivalent way of stating the definition of the derivative

Sec 2.7: DERIVATIVES AND RATES OF CHANGE

)(' afThe slope of the tangent line at the point (a,f(a))

Tangent line at x=3

Slope of this line )(' af

Sec 2.7: DERIVATIVES AND RATES OF CHANGE

Sec 2.7: DERIVATIVES AND RATES OF CHANGE

RATES OF CHANGE

xy on depends suppose

f(x)y

21

21

yyxx

12

12

yyyxxx

Chane in x =

Chane in y =

increment in x

average rate of change of y with respect to x over the interval [x1,x2] x

y

12

12

12

12 )()(xxxfxf

xxyy

xy

x= time, y = distanceaverage velocity

Sec 2.7: DERIVATIVES AND RATES OF CHANGE

RATES OF CHANGE

average rate of change of y with respect to x over the interval [x1,x2] x

y

12

12

12

12 )()(xxxfxf

xxyy

xy

12 xx 0x

instantaneous rate of change =

xy

x

0

lim

12

12 )()(lim12 xx

xfxfxx

x= time, y = distanceinstantaneous velocity

Sec 2.7: DERIVATIVES AND RATES OF CHANGE

RATES OF CHANGE

average rate of change of y with respect to x over the interval [x1,x2] x

y

12

12

12

12 )()(xxxfxf

xxyy

xy

12 xx 0x

instantaneous rate of change =

xy

x

0

lim

12

12 )()(lim12 xx

xfxfxx

MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag

H

R

MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag

H

R

RH(a,b)

MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag

H

H

H

H

R

R

MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag

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MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag

MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag