11x1 t09 01 first derivative
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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
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