Geometrical Applications

of Differentiation The First Derivative

Geometrical Applications

of Differentiation The First Derivative

dx

dyxf

dx

dxfy ,,,

Geometrical Applications

of Differentiation The First Derivative

dx

dyxf

dx

dxfy ,,,

curve a o tangent t theof slope themeasures dx

dy

Geometrical Applications

of Differentiation The First Derivative

dx

dyxf

dx

dxfy ,,,

curve a o tangent t theof slope themeasures dx

dy

increasing is curve the,0 If xf

Geometrical Applications

of Differentiation The First Derivative

dx

dyxf

dx

dxfy ,,,

curve a o tangent t theof slope themeasures dx

dy

increasing is curve the,0 If xf

decreasing is curve the,0 If xf

Geometrical Applications

of Differentiation The First Derivative

dx

dyxf

dx

dxfy ,,,

curve a o tangent t theof slope themeasures dx

dy

increasing is curve the,0 If xf

decreasing is curve the,0 If xf

stationary is curve the,0 If xf

Geometrical Applications

of Differentiation The First Derivative

dx

dyxf

dx

dxfy ,,,

curve a o tangent t theof slope themeasures dx

dy

increasing is curve the,0 If xf

curve sketch the Hence

nature. their determine and

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

decreasing is curve the,0 If xf

stationary is curve the,0 If xf

Geometrical Applications

of Differentiation The First Derivative

dx

dyxf

dx

dxfy ,,,

curve a o tangent t theof slope themeasures dx

dy

increasing is curve the,0 If xf

curve sketch the Hence

nature. their determine and

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

decreasing is curve the,0 If xf

stationary is curve the,0 If xf

236 xxdx

dy

0 occur when points Stationary dx

dy

0 occur when points Stationary dx

dy

036 .. 2 xxei

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1)

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1)

(-9)

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9)

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

2,4

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

2,4

x

dx

dy

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

2,4

x

dx

dy

2

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

2,4

x

dx

dy

2

0

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

2,4

x

dx

dy

2

0

2

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

2,4

x

dx

dy

2

0

2 2

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

2,4

x

dx

dy

2

0

2 2(1)

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

2,4

x

dx

dy

2

0

2 2(1)

(3)

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

2,4

x

dx

dy

2

0

2 2(1) (3)

(3)

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

2,4

x

dx

dy

2

0

2 2(1) (3)

(3) (-9)

0 occur when points Stationary dx

dy

036 .. 2 xxei

2or 0

023

xx

xx

2,4 and 0,0at occur points stationary

0,0

x

dx

dy

0

0

0 0(-1) (1)

(-9) (3)

point turningminimum a is 0,0

2,4

x

dx

dy

2

0

2 2(1) (3)

(3) (-9)

point turningmaximum a is 2,4

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