Geometrical Applications of Differentiation

The First Derivative

Geometrical Applications of Differentiation

The First Derivative dxdyxf

dxdxfy ,,,

Geometrical Applications of Differentiation

The First Derivative dxdyxf

dxdxfy ,,,

curve a o tangent t theof slope themeasures dxdy

Geometrical Applications of Differentiation

The First Derivative dxdyxf

dxdxfy ,,,

curve a o tangent t theof slope themeasures dxdy

increasingiscurvethe,0If xf

Geometrical Applications of Differentiation

The First Derivative dxdyxf

dxdxfy ,,,

curve a o tangent t theof slope themeasures dxdy

increasingiscurvethe,0If xf decreasingiscurvethe,0If xf

Geometrical Applications of Differentiation

The First Derivative dxdyxf

dxdxfy ,,,

curve a o tangent t theof slope themeasures dxdy

increasingiscurvethe,0If xf decreasingiscurvethe,0If xf stationaryiscurvethe,0If xf

Geometrical Applications of Differentiation

The First Derivative dxdyxf

dxdxfy ,,,

curve a o tangent t theof slope themeasures dxdy

increasingiscurvethe,0If xf

curvesketch the Hence nature. their determine and

points stationary theof all find ,3 curve For the e.g. 32 xxy

decreasingiscurvethe,0If xf stationaryiscurvethe,0If xf

Geometrical Applications of Differentiation

The First Derivative dxdyxf

dxdxfy ,,,

curve a o tangent t theof slope themeasures dxdy

increasingiscurvethe,0If xf

curvesketch the Hence nature. their determine and

points stationary theof all find ,3 curve For the e.g. 32 xxy

decreasingiscurvethe,0If xf stationaryiscurvethe,0If xf

236 xxdxdy

0 occur when points Stationary dxdy

0 occur when points Stationary dxdy

036 .. 2 xxei

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1)

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1)

(-9)

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9)

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

2,4

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

2,4x

dxdy

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

2,4x

dxdy

2

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

2,4x

dxdy

2

0

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

2,4x

dxdy

2

0

2

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

2,4x

dxdy

2

0

2 2

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

2,4x

dxdy

2

0

2 2(1)

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

2,4x

dxdy

2

0

2 2(1)

(3)

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

2,4x

dxdy

2

0

2 2(1) (3)

(3)

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

2,4x

dxdy

2

0

2 2(1) (3)

(3) (-9)

0 occur when points Stationary dxdy

036 .. 2 xxei

2or 0023

xx

xx

2,4and0,0at occur points stationary

0,0x

dxdy

0

0

0 0(-1) (1)

(-9) (3)

pointturningminimumais0,0

2,4x

dxdy

2

0

2 2(1) (3)

(3) (-9)

pointturningmaximumais2,4

x

y

x

y

x

y

2,4

x

y

2,4

323 xxy

Exercise 10A; 1, 2ace, 4, 5, 6ac, 7, 8, 9ace etc, 11, 12, 13, 15ace, 16, 17

Exercise 10B; 1ad, 2ac, 3ac, 5, 6, 7ac, 8, 10, 12