derivative slides
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8/7/2019 Derivative Slides
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Differentiability and Derivative: If I is an interval of R, then a
function f : I → R is said to be differentiable at x 0 ∈ I if
limx →x 0
f (x )−f (x 0)x −x 0
(or, equivalently limh →0
f (x 0+h )−f (x 0)h ) exists in R.
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Differentiability and Derivative: If I is an interval of R, then a
function f : I → R is said to be differentiable at x 0 ∈ I if
limx →x 0
f (x )−f (x 0)x −x 0
(or, equivalently limh →0
f (x 0+h )−f (x 0)h ) exists in R.
If f is differentiable at x 0, then the derivative of f at x 0 is
f (x
0) = limx →x 0
f (x )
−f (x 0)
x −x 0 = limh →0
f (x 0+h )
−f (x 0)
h .
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Differentiability and Derivative: If I is an interval of R, then a
function f : I → R is said to be differentiable at x 0 ∈ I if
limx →x 0
f (x )−f (x 0)x −x 0
(or, equivalently limh →0
f (x 0+h )−f (x 0)h ) exists in R.
If f is differentiable at x 0, then the derivative of f at x 0 is
f (x
0) =limx →x 0
f (x )−f (x 0)
x −x 0 =limh →0
f (x 0+h )−f (x 0)
h .
f : I → R is said to be differentiable if f is differentiable at each
x 0 ∈ I .
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Differentiability and Derivative: If I is an interval of R, then a
function f : I → R is said to be differentiable at x 0 ∈ I if
limx →x 0
f (x )−f (x 0)x −x 0
(or, equivalently limh →0
f (x 0+h )−f (x 0)h ) exists in R.
If f is differentiable at x 0, then the derivative of f at x 0 is
f (x
0) =limx →x 0
f (x )−f (x 0)
x −x 0 =limh →0
f (x 0+h )−f (x 0)
h .
f : I → R is said to be differentiable if f is differentiable at each
x 0 ∈ I .
Result: If f : I
→R is differentiable at x 0
∈I , then f is
continuous at x 0.
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Examples:
1. For n = 1, 2, 3, let f n (x ) =
x n sin 1
x if x = 0,
0 if x = 0.
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Examples:
1. For n = 1, 2, 3, let f n (x ) =
x n sin 1
x if x = 0,
0 if x = 0.
2. f (x ) = x 2 if x ∈ Q,
0 if x ∈ R \Q.
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Examples:
1. For n = 1, 2, 3, let f n (x ) =
x n sin 1
x if x = 0,
0 if x = 0.
2. f (x ) = x 2 if x ∈ Q,
0 if x ∈ R \Q.
Ex. Let f (x ) = x 2|x | for all x ∈ R. Examine the existence of
f (x ), f (x ) and f (x ), where x ∈ R.
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Examples:
1. For n = 1, 2, 3, let f n (x ) =
x n sin 1
x if x = 0,
0 if x = 0.
2. f (x ) = x 2 if x ∈ Q,
0 if x ∈ R \Q.
Ex. Let f (x ) = x 2|x | for all x ∈ R. Examine the existence of
f (x ), f (x ) and f (x ), where x ∈ R.
Rules for finding derivatives:
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Definition: f : D → R has a local maximum (resp. minimum) at
x 0
∈D if there exists δ > 0 such that f (x )
≤f (x 0)
(resp. f (x 0) ≤ f (x )) for all x ∈ (x 0 − δ,x 0 + δ) ∩ D .
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Definition: f : D → R has a local maximum (resp. minimum) at
x 0 ∈ D if there exists δ > 0 such that f (x ) ≤ f (x 0)
(resp. f (x 0) ≤ f (x )) for all x ∈ (x 0 − δ,x 0 + δ) ∩ D .
Result: If f : D → R has a local maximum or local minimum at
an interior point x 0 of D and if f is differentiable at x 0, then
f (x 0) = 0.
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Definition: f : D → R has a local maximum (resp. minimum) at
x 0 ∈ D if there exists δ > 0 such that f (x ) ≤ f (x 0)
(resp. f (x 0) ≤ f (x )) for all x ∈ (x 0 − δ,x 0 + δ) ∩ D .
Result: If f : D → R has a local maximum or local minimum at
an interior point x 0 of D and if f is differentiable at x 0, then
f (x 0) = 0.
Rolle’s theorem: If f : [a ,b ] → R is continuous, if f isdifferentiable on (a ,b ) and if f (a ) = f (b ), then there exists
ξ ∈ (a ,b ) such that f (ξ) = 0.
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Definition: f : D → R has a local maximum (resp. minimum) at
x 0 ∈ D if there exists δ > 0 such that f (x ) ≤ f (x 0)
(resp. f (x 0) ≤ f (x )) for all x ∈ (x 0 − δ,x 0 + δ) ∩ D .
Result: If f : D → R has a local maximum or local minimum at
an interior point x 0 of D and if f is differentiable at x 0, then
f (x 0) = 0.
Rolle’s theorem: If f : [a ,b ] → R is continuous, if f isdifferentiable on (a ,b ) and if f (a ) = f (b ), then there exists
ξ ∈ (a ,b ) such that f (ξ) = 0.
Example: The equation x 2 = x sin x + cos x has exactly two
real roots.
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Definition: f : D → R has a local maximum (resp. minimum) at
x 0 ∈ D if there exists δ > 0 such that f (x ) ≤ f (x 0)
(resp. f (x 0) ≤ f (x )) for all x ∈ (x 0 − δ,x 0 + δ) ∩ D .
Result: If f : D → R has a local maximum or local minimum at
an interior point x 0 of D and if f is differentiable at x 0, then
f (x 0) = 0.
Rolle’s theorem: If f : [a ,b ] → R is continuous, if f isdifferentiable on (a ,b ) and if f (a ) = f (b ), then there exists
ξ ∈ (a ,b ) such that f (ξ) = 0.
Example: The equation x 2 = x sin x + cos x has exactly two
real roots.Ex. Find the number of real roots of the equation
x 4 + 2x 2 − 6x + 2 = 0.
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Mean value theorem: If f : [a ,b ] → R is continuous and if f is
differentiable on (a ,b ), then there exists ξ
∈(a ,b ) such that
f (b ) − f (a ) = f (ξ)(b − a ).
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Mean value theorem: If f : [a ,b ] → R is continuous and if f is
differentiable on (a ,b ), then there exists ξ
∈(a ,b ) such that
f (b ) − f (a ) = f (ξ)(b − a ).
Result: If f : I → R is differentiable and f (x ) = 0 for all x ∈ I ,
then f is constant on I .
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Mean value theorem: If f : [a ,b ] → R is continuous and if f is
differentiable on (a ,b ), then there exists ξ
∈(a ,b ) such that
f (b ) − f (a ) = f (ξ)(b − a ).
Result: If f : I → R is differentiable and f (x ) = 0 for all x ∈ I ,
then f is constant on I .
Result: Let f : I →R be differentiable.
(i) If f (x ) ≥ 0 for all x ∈ I , then f is increasing on I .
(ii) If f (x ) ≤ 0 for all x ∈ I , then f is decreasing on I .
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Mean value theorem: If f : [a ,b ] → R is continuous and if f is
differentiable on (a ,b ), then there exists ξ
∈(a ,b ) such that
f (b ) − f (a ) = f (ξ)(b − a ).
Result: If f : I → R is differentiable and f (x ) = 0 for all x ∈ I ,
then f is constant on I .
Result: Let f : I →R be differentiable.
(i) If f (x ) ≥ 0 for all x ∈ I , then f is increasing on I .
(ii) If f (x ) ≤ 0 for all x ∈ I , then f is decreasing on I .
Example: sin x ≥ x − x 3
6 for all x ∈ [0, π2 ].
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Mean value theorem: If f : [a ,b ] → R is continuous and if f is
differentiable on (a ,b ), then there exists ξ
∈(a ,b ) such that
f (b ) − f (a ) = f (ξ)(b − a ).
Result: If f : I → R is differentiable and f (x ) = 0 for all x ∈ I ,
then f is constant on I .
Result: Let f : I →R be differentiable.
(i) If f (x ) ≥ 0 for all x ∈ I , then f is increasing on I .
(ii) If f (x ) ≤ 0 for all x ∈ I , then f is decreasing on I .
Example: sin x ≥ x − x 3
6 for all x ∈ [0, π2 ].
Ex. If f (x ) = x 3 + x 2 − 5x + 3 for all x ∈ R, then show that f is
one-one on [1, 5] but not one-one on R.
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Intermediate value property of derivatives: Let f : I → R be
differentiable and let a ,b ∈I with a < b . If f
(a ) < k < f
(b ),
then there exists c ∈ (a ,b ) such that f (c ) = k .
Ex. Let f : R→ R be differentiable such that f (−1) = 5,
f (0) = 0 and f (1) = 10. Prove that there exist ξ1, ξ2 ∈ (−1, 1)such that f (ξ1) =
−3 and f (ξ2) = 3.
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Intermediate value property of derivatives: Let f : I → R be
differentiable and let a ,b ∈I with a < b . If f
(a ) < k < f
(b ),
then there exists c ∈ (a ,b ) such that f (c ) = k .
Ex. Let f : R→ R be differentiable such that f (−1) = 5,
f (0) = 0 and f (1) = 10. Prove that there exist ξ1, ξ2 ∈ (−1, 1)such that f (ξ1) =
−3 and f (ξ2) = 3.
Local maximum & Local minimum : Sufficient conditions:
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Intermediate value property of derivatives: Let f : I → R be
differentiable and let a ,b ∈I with a < b . If f (a ) < k < f (b ),
then there exists c ∈ (a ,b ) such that f (c ) = k .
Ex. Let f : R→ R be differentiable such that f (−1) = 5,
f (0) = 0 and f (1) = 10. Prove that there exist ξ1, ξ2 ∈ (−1, 1)such that f (ξ1) =
−3 and f (ξ2) = 3.
Local maximum & Local minimum : Sufficient conditions:
1. First derivative test
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Intermediate value property of derivatives: Let f : I → R be
differentiable and let a ,b ∈I with a < b . If f (a ) < k < f (b ),
then there exists c ∈ (a ,b ) such that f (c ) = k .
Ex. Let f : R→ R be differentiable such that f (−1) = 5,
f (0) = 0 and f (1) = 10. Prove that there exist ξ1, ξ2 ∈ (−1, 1)such that f (ξ1) =
−3 and f (ξ2) = 3.
Local maximum & Local minimum : Sufficient conditions:
1. First derivative test
2. Second derivative test
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Intermediate value property of derivatives: Let f : I → R be
differentiable and let a ,b ∈I with a < b . If f (a ) < k < f (b ),
then there exists c ∈ (a ,b ) such that f (c ) = k .
Ex. Let f : R→ R be differentiable such that f (−1) = 5,
f (0) = 0 and f (1) = 10. Prove that there exist ξ1, ξ2 ∈ (−1, 1)such that f (ξ1) =
−3 and f (ξ2) = 3.
Local maximum & Local minimum : Sufficient conditions:
1. First derivative test
2. Second derivative test
Example: Find the local maxima and local minima of f , wheref (x ) = 1 − x 2/3 for all x ∈ R.
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L’Hopital’s rules:
1. Let f : (a ,b ) → R and g : (a ,b ) → R be differentiable at
x 0 ∈ (a ,b ). Also, let f (x 0) = g (x 0) = 0 and g (x 0) = 0. Thenlim
x →x 0
f (x )g (x ) = f (x 0)
g (x 0).
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L’Hopital’s rules:
1. Let f : (a ,b ) → R and g : (a ,b ) → R be differentiable at
x 0 ∈ (
a ,b ). Also, let
f (x
0) =g
(x
0) = 0 andg (x
0) = 0. Thenlimx →x 0
f (x )g (x ) = f (x 0)
g (x 0).
2. Let f : (a ,b ) → R and g : (a ,b ) → R be differentiable such
that limx →
a +f (x ) = lim
x →
a +g (x ) = 0 and g (x )
= 0 for all
x ∈ (a ,b ). If limx →a +
f (x )g (x ) = , then lim
x →a +
f (x )g (x ) = .
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L’Hopital’s rules:
1. Let f : (a ,b ) → R and g : (a ,b ) → R be differentiable at
x 0 ∈ (
a ,b ). Also, let f
(x
0) =g
(x
0) =0 and g
(x
0) =0. Then
limx →x 0
f (x )g (x ) = f (x 0)
g (x 0).
2. Let f : (a ,b ) → R and g : (a ,b ) → R be differentiable such
that limx →
a +f (x ) = lim
x →
a +g (x ) = 0 and g (x )
= 0 for all
x ∈ (a ,b ). If limx →a +
f (x )g (x ) = , then lim
x →a +
f (x )g (x ) = .
Examples: (i) limx →0
√1+x −1x (ii) lim
x →π2
1−sin x 1+cos2x
(iii) limx →0
x 2
sin1
x sin x (iv) lim
x →0( sin x
x ) 1x (v) limx →0
e −
1/x 2
x
(vi) limx →0
( 1sin x − 1
x ) (vii) limx →∞
x −sin x 2x +sin x
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Taylor’s theorem: Let f : [a ,b ]→R be such that f , f , f , ..., f (n )
are continuous on [a ,b ] and f (n +1) exists on (a ,b ).
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Taylor’s theorem: Let f : [a ,b ]→R be such that f , f , f , ..., f (n )
are continuous on [a ,b ] and f (n +1) exists on (a ,b ). Then there
exists ξ ∈ (a ,b ) such that
f (b ) = f (a ) + f (a )(b − a ) + f (a )2! (b − a )2 + · · · + f (n )(a )
n ! (b − a )n
+
f (n +1)(ξ)
(n +1)! (b −
a )
n +1
.
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Taylor’s theorem: Let f : [a ,b ]→R be such that f , f , f , ..., f (n )
are continuous on [a ,b ] and f (n +1) exists on (a ,b ). Then there
exists ξ ∈ (a ,b ) such that
f (b ) = f (a ) + f (a )(b − a ) + f (a )2! (b − a )2 + · · · + f (n )(a )
n ! (b − a )n
+
f (n +1)(ξ)
(n +1)! (b −
a )
n +1.
Example: 1 + x 2 − x 2
8 ≤ √1 + x ≤ 1 + x
2 for all x > 0.
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Taylor’s theorem: Let f : [a ,b ]
→R be such that f , f , f , ..., f (n )
are continuous on [a ,b ] and f (n +1) exists on (a ,b ). Then there
exists ξ ∈ (a ,b ) such that
f (b ) = f (a ) + f (a )(b − a ) + f (a )2! (b − a )2 + · · · + f (n )(a )
n ! (b − a )n
+
f (n +1)(ξ)
(n +1)! (b −
a )
n +1.
Example: 1 + x 2 − x 2
8 ≤ √1 + x ≤ 1 + x
2 for all x > 0.
Taylor series & Maclaurin series: Convergence
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Taylor’s theorem: Let f : [a ,b ]
→R be such that f , f , f , ..., f (n )
are continuous on [a ,b ] and f (n +1) exists on (a ,b ). Then there
exists ξ ∈ (a ,b ) such that
f (b ) = f (a ) + f (a )(b − a ) + f (a )2! (b − a )2 + · · · + f (n )(a )
n ! (b − a )n
+ f (n +1)(ξ)
(n +1)!(b −
a )n +1.
Example: 1 + x 2 − x 2
8 ≤ √1 + x ≤ 1 + x
2 for all x > 0.
Taylor series & Maclaurin series: Convergence
Example: Taylor series expansion of e x , sin x and cos x .
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Result on local maxima and local minima:
Let x 0∈
(a ,b ) and let n
≥2. Also, let f , f , ..., f (n ) be continuous
on (a ,b ) and f (x 0) = f (x 0) = · · · = f (n −1)(x 0) = 0 butf (n )(x 0) = 0.
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Result on local maxima and local minima:
Let x 0∈
(a ,b ) and let n
≥2. Also, let f , f , ..., f (n ) be continuous
on (a ,b ) and f (x 0) = f (x 0) = · · · = f (n −1)(x 0) = 0 butf (n )(x 0) = 0.
(i) If n is even and f (n )(x 0) < 0, then f has a local maximum
at x 0.
(ii) If n is even and f (n )(x 0) > 0, then f has a local minimum atx 0.
(iii) If n is odd, then f has neither a local maximum nor a local
minimum at x 0.
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Result on local maxima and local minima:
Let x 0∈
(a ,b ) and let n
≥2. Also, let f , f , ..., f (n ) be continuous
on (a ,b ) and f (x 0) = f (x 0) = · · · = f (n −1)(x 0) = 0 butf (n )(x 0) = 0.
(i) If n is even and f (n )(x 0) < 0, then f has a local maximum
at x 0.
(ii) If n is even and f (n )(x 0) > 0, then f has a local minimum atx 0.
(iii) If n is odd, then f has neither a local maximum nor a local
minimum at x 0.
Ex. Find all the local maximum and local minimum values of f ,where f (x ) = x 5 − 5x 4 + 5x 3 + 12 for all x ∈ R.