Differentiability

Differentiability

Project Type: Presentation (ppt)

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Differentiability - Overview

------------ PAGE 1 ------------ Differentiability ------------ PAGE 2 ------------ Differentiability A function is differentiable at a point if the curve is smooth continuous i. e. lim ƒv ƒw lim ƒv ƒw x a x a f x f x ƒ{ ƒy „_ „_ „S ƒ­ „S ------------ PAGE 3 ------------ Differentiability A function is differentiable at a point if the curve is smooth continuous i. e. lim ƒv ƒw lim ƒv ƒw x a x a f x f x ƒ{ ƒy „_ „_ „S ƒ­ „S y x 1 1 y ƒ­ x ƒ{ 1 ------------ PAGE 4 ------------ Differentiability A function is differentiable at a point if the curve is smooth continuous i. e. lim ƒv ƒw lim ƒv ƒw x a x a f x f x ƒ{ ƒy „_ „_ „S ƒ­ „S y x 1 1 y ƒ­ x ƒ{ 1 not differentiable at x = 1 ------------ PAGE 5 ------------ Differentiability A function is differentiable at a point if the curve is smooth continuous i. e. lim ƒv ƒw lim ƒv ƒw x a x a f x f x ƒ{ ƒy „_ „_ „S ƒ­ „S y x 1 1 y ƒ­ x ƒ{ 1 ƒv ƒw 1 lim 1 x f x ƒ{ „_ „S ƒ­ ƒ{ not differentiable at x = 1 ------------ PAGE 6 ------------ Differentiability A function is differentiable at a point if the curve is smooth continuous i. e. lim ƒv ƒw lim ƒv ƒw x a x a f x f x ƒ{ ƒy „_ „_ „S ƒ­ „S y x 1 1 y ƒ­ x ƒ{ 1 ƒv ƒw 1 lim 1 x f x ƒ{ „_ „S ƒ­ ƒ{ ƒv ƒw 1 lim 1 x f x ƒy „_ „S ƒ­ not differentiable at x = 1 ------------ PAGE 7 ------------ Differentiability A function is differentiable at a point if the curve is smooth continuous i. e. lim ƒv ƒw lim ƒv ƒw x a x a f x f x ƒ{ ƒy „_ „_ „S ƒ­ „S y x 1 1 y ƒ­ x ƒ{ 1 ƒv ƒw 1 lim 1 x f x ƒ{ „_ „S ƒ­ ƒ{ ƒv ƒw 1 lim 1 x f x ƒy „_ „S ƒ­ not differentiable at x = 1 y x 2 y ƒ­ x ------------ PAGE 8 ------------ Differentiability A function is differentiable at a point if the curve is smooth continuous i. e. lim ƒv ƒw lim ƒv ƒw x a x a f x f x ƒ{ ƒy „_ „_ „S ƒ­ „S y x 1 1 y ƒ­ x ƒ{ 1 ƒv ƒw 1 lim 1 x f x ƒ{ „_ „S ƒ­ ƒ{ ƒv ƒw 1 lim 1 x f x ƒy „_ „S ƒ­ not differentiable at x = 1 y x 2 y ƒ­ x 1 ------------ PAGE 9 ------------ Differentiability A function is differentiable at a point if the curve is smooth continuous i. e. lim ƒv ƒw lim ƒv ƒw x a x a f x f x ƒ{ ƒy „_ „_ „S ƒ­ „S y x 1 1 y ƒ­ x ƒ{ 1 ƒv ƒw 1 lim 1 x f x ƒ{ „_ „S ƒ­ ƒ{ ƒv ƒw 1 lim 1 x f x ƒy „_ „S ƒ­ not differentiable at x = 1 y x 2 y ƒ­ x 1 differentiable at x = 1 ------------ PAGE 10 ------------ Implicit Differentiation ------------ PAGE 11 ------------ Implicit Differentiation df df dy dx dy dx ƒ­ „e ------------ PAGE 12 ------------ Implicit Differentiation df df dy dx dy dx ƒ­ „e e. g. ( i) x ƒ­ y 2 ------------ PAGE 13 ------------ Implicit Differentiation df df dy dx dy dx ƒ­ „e e. g. ( i) x ƒ­ y 2 ƒv ƒw ƒv ƒw 2 d d x y dx dx ƒ­ ------------ PAGE 14 ------------ Implicit Differentiation df df dy dx dy dx ƒ­ „e e. g. ( i) x ƒ­ y 2 ƒv ƒw ƒv ƒw 2 d d x y dx dx ƒ­ ƒv ƒw ƒv ƒw 2 2 d d dy y y dx dy dx ƒ­ „e ------------ PAGE 15 ------------ Implicit Differentiation df df dy dx dy dx ƒ­ „e e. g. ( i) x ƒ­ y 2 1 2 dy y dx ƒ­ ƒv ƒw ƒv ƒw 2 d d x y dx dx ƒ­ ƒv ƒw ƒv ƒw 2 2 d d dy y y dx dy dx ƒ­ „e ------------ PAGE 16 ------------ Implicit Differentiation df df dy dx dy dx ƒ­ „e e. g. ( i) x ƒ­ y 2 1 2 dy y dx ƒ­ ƒv ƒw ƒv ƒw 2 d d x y dx dx ƒ­ 1 2 dy dx y ƒ­ ƒv ƒw ƒv ƒw 2 2 d d dy y y dx dy dx ƒ­ „e ------------ PAGE 17 ------------ Implicit Differentiation df df dy dx dy dx ƒ­ „e e. g. ( i) x ƒ­ y 2 1 2 dy y dx ƒ­ ƒv ƒw ƒv ƒw 2 d d x y dx dx ƒ­ 1 2 dy dx y ƒ­ ƒv ƒw ƒv ƒw 2 2 d d dy y y dx dy dx ƒ­ „e ƒv ƒw 2 3 ( ii) d x y dx ------------ PAGE 18 ------------ Implicit Differentiation df df dy dx dy dx ƒ­ „e e. g. ( i) x ƒ­ y 2 1 2 dy y dx ƒ­ ƒv ƒw ƒv ƒw 2 d d x y dx dx ƒ­ 1 2 dy dx y ƒ­ ƒv ƒw ƒv ƒw 2 2 d d dy y y dx dy dx ƒ­ „e ƒv ƒw 2 3 ( ii) d x y dx ƒv ƒw ƒv ƒw ƒv ƒw 2 2 3 3 2 dy x y y x dx „¹ „É ƒ­ „º „Ê ƒy „» „Ë ------------ PAGE 19 ------------ Implicit Differentiation df df dy dx dy dx ƒ­ „e e. g. ( i) x ƒ­ y 2 1 2 dy y dx ƒ­ ƒv ƒw ƒv ƒw 2 d d x y dx dx ƒ­ 1 2 dy dx y ƒ­ ƒv ƒw ƒv ƒw 2 2 d d dy y y dx dy dx ƒ­ „e ƒv ƒw 2 3 ( ii) d x y dx ƒv ƒw ƒv ƒw ƒv ƒw 2 2 3 3 2 dy x y y x dx „¹ „É ƒ­ „º „Ê ƒy „» „Ë 2 2 3 3 2 dy x y xy dx ƒ­ ƒy ------------ PAGE 20 ------------ ƒv ƒw ƒv ƒw 22 Find the equation of the tangent to 9 at the point 1,22 iiixy ƒy ƒ­ ------------ PAGE 21 ------------ ƒv ƒw ƒv ƒw 2 2 Find the equation of the tangent to iii x ƒy y ƒ­ 9 at the point 1,2 2 2 2 x ƒy y ƒ­ 9 ------------ PAGE 22 ------------ ƒv ƒw ƒv ƒw 2 2 iii Find the equation of the tangent to x ƒy y ƒ­ 9 at the point 1,2 2 2 2 x ƒy y ƒ­ 9 2 2 0 dy x y dx ƒy ƒ­ ------------ PAGE 23 ------------ ƒv ƒw ƒv ƒw 2 2 iii Find the equation of the tangent to x ƒy y ƒ­ 9 at the point 1,2 2 2 2 x ƒy y ƒ­ 9 2 2 0 dy x y dx ƒy ƒ­ 2 2 dy y x dx ƒ­ ƒ{ ------------ PAGE 24 ------------ ƒv ƒw ƒv ƒw 2 2 iii Find the equation of the tangent to x ƒy y ƒ­ 9 at the point 1,2 2 2 2 x ƒy y ƒ­ 9 2 2 0 dy x y dx ƒy ƒ­ 2 2 dy y x dx ƒ­ ƒ{ dy x dx y ƒ­ ƒ{ ------------ PAGE 25 ------------ ƒv ƒw ƒv ƒw 2 2 iii Find the equation of the tangent to x ƒy y ƒ­ 9 at the point 1,2 2 2 2 x ƒy y ƒ­ 9 2 2 0 dy x y dx ƒy ƒ­ 2 2 dy y x dx ƒ­ ƒ{ dy x dx y ƒ­ ƒ{ ƒv ƒw 1 at 1,2 2 , 2 2 dy dx ƒ­ ƒ{ ------------ PAGE 26 ------------ ƒv ƒw ƒv ƒw 2 2 iii Find the equation of the tangent to x ƒy y ƒ­ 9 at the point 1,2 2 2 2 x ƒy y ƒ­ 9 2 2 0 dy x y dx ƒy ƒ­ 2 2 dy y x dx ƒ­ ƒ{ dy x dx y ƒ­ ƒ{ ƒv ƒw 1 at 1,2 2 , 2 2 dy dx ƒ­ ƒ{ 1 required slope 2 2 ƒÌ ƒ­ ƒ{ ------------ PAGE 27 ------------ ƒv ƒw ƒv ƒw 2 2 iii Find the equation of the tangent to x ƒy y ƒ­ 9 at the point 1,2 2 2 2 x ƒy y ƒ­ 9 2 2 0 dy x y dx ƒy ƒ­ 2 2 dy y x dx ƒ­ ƒ{ dy x dx y ƒ­ ƒ{ ƒv ƒw 1 at 1,2 2 , 2 2 dy dx ƒ­ ƒ{ 1 required slope 2 2 ƒÌ ƒ­ ƒ{ ƒv ƒw 1 2 2 1 2 2 y ƒ{ ƒ­ ƒ{ x ƒ{ ------------ PAGE 28 ------------ ƒv ƒw ƒv ƒw 2 2 iii Find the equation of the tangent to x ƒy y ƒ­ 9 at the point 1,2 2 2 2 x ƒy y ƒ­ 9 2 2 0 dy x y dx ƒy ƒ­ 2 2 dy y x dx ƒ­ ƒ{ dy x dx y ƒ­ ƒ{ ƒv ƒw 1 at 1,2 2 , 2 2 dy dx ƒ­ ƒ{ 1 required slope 2 2 ƒÌ ƒ­ ƒ{ ƒv ƒw 1 2 2 1 2 2 y ƒ{ ƒ­ ƒ{ x ƒ{ 2 2 y ƒ{ 8 ƒ­ ƒ{ x ƒy 1 ------------ PAGE 29 ------------ ƒv ƒw ƒv ƒw 2 2 iii Find the equation of the tangent to x ƒy y ƒ­ 9 at the point 1,2 2 2 2 x ƒy y ƒ­ 9 2 2 0 dy x y dx ƒy ƒ­ 2 2 dy y x dx ƒ­ ƒ{ dy x dx y ƒ­ ƒ{ ƒv ƒw 1 at 1,2 2 , 2 2 dy dx ƒ­ ƒ{ 1 required slope 2 2 ƒÌ ƒ­ ƒ{ ƒv ƒw 1 2 2 1 2 2 y ƒ{ ƒ­ ƒ{ x ƒ{ 2 2 y ƒ{ 8 ƒ­ ƒ{ x ƒy 1 x ƒy 2 2 y ƒ{ 9 ƒ­ 0 ------------ PAGE 30 ------------ ƒv ƒw ƒv ƒw 2 2 iii Find the equation of the tangent to x ƒy y ƒ­ 9 at the point 1,2 2 2 2 x ƒy y ƒ­ 9 2 2 0 dy x y dx ƒy ƒ­ 2 2 dy y x dx ƒ­ ƒ{ dy x dx y ƒ­ ƒ{ ƒv ƒw 1 at 1,2 2 , 2 2 dy dx ƒ­ ƒ{ 1 required slope 2 2 ƒÌ ƒ­ ƒ{ ƒv ƒw 1 2 2 1 2 2 y ƒ{ ƒ­ ƒ{ x ƒ{ 2 2 y ƒ{ 8 ƒ­ ƒ{ x ƒy 1 x ƒy 2 2 y ƒ{ 9 ƒ­ 0 Exercise 7K; 1acegi, 2bdfh, 3a, 4a, 7, 8
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