3. Consider the sequences of functions f₁: [-π, π] → R, sin(n²x) An(2) n f pointwise as (i) Find a function ƒ : [-T,π] → R such that fn n∞. Further, show that fn →f uniformly on [-π,π] as n → ∞. [20 Marks] (ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7, 7]? Justify your answer. [10 Marks]
3. Consider the sequences of functions f₁: [-π, π] → R, sin(n²x) An(2) n f pointwise as (i) Find a function ƒ : [-T,π] → R such that fn n∞. Further, show that fn →f uniformly on [-π,π] as n → ∞. [20 Marks] (ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7, 7]? Justify your answer. [10 Marks]
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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Question
![3.
Consider the sequences of functions f₁: [-π, π] → R,
sin(n²x)
An(2)
n
f pointwise as
(i) Find a function ƒ : [-T,π] → R such that fn
n∞. Further, show that fn →f uniformly on [-π,π] as n → ∞.
[20 Marks]
(ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7, 7]?
Justify your answer.
[10 Marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c7573-6777-48ff-9bfb-9b3df1a769a6%2F633f8bcd-0f64-4b5f-83f1-2209b001f01b%2Frlt5so_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3.
Consider the sequences of functions f₁: [-π, π] → R,
sin(n²x)
An(2)
n
f pointwise as
(i) Find a function ƒ : [-T,π] → R such that fn
n∞. Further, show that fn →f uniformly on [-π,π] as n → ∞.
[20 Marks]
(ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7, 7]?
Justify your answer.
[10 Marks]
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