3. This question concerns the L metric d and the L¹ metric e on a) Calculate d(f.g) and e(f.g), where g(x) = x² b Explain why we have e(f.g) ≤d(f.g) for all functions f and g. Under what conditions on f and g is e(f. g) = d(f.g)? C) d f(x) = 1 + x In the remainder of the question, we treat an explicit example of the function with a very narrow tall bump" of Figure 1.14. For each integer n 2, define a function fn € C[0, 1] by e) f(x) = for all f.ge CÍO. and nx+1-4 -1-2 C[0, 1] if-sest indsasi-f otherwise Sketch the graph of fa, and describe the graph off for any a. Let g(x) = 0 be the zero function. Calculate d) and ( 0). Deduce that there is no number 4 such that c(5.g) ≤d(/.s) shelf al
3. This question concerns the L metric d and the L¹ metric e on a) Calculate d(f.g) and e(f.g), where g(x) = x² b Explain why we have e(f.g) ≤d(f.g) for all functions f and g. Under what conditions on f and g is e(f. g) = d(f.g)? C) d f(x) = 1 + x In the remainder of the question, we treat an explicit example of the function with a very narrow tall bump" of Figure 1.14. For each integer n 2, define a function fn € C[0, 1] by e) f(x) = for all f.ge CÍO. and nx+1-4 -1-2 C[0, 1] if-sest indsasi-f otherwise Sketch the graph of fa, and describe the graph off for any a. Let g(x) = 0 be the zero function. Calculate d) and ( 0). Deduce that there is no number 4 such that c(5.g) ≤d(/.s) shelf al
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![3. This question concerns the L metric d and the L metric e on C0, 1)
a) Calculate d(f. 9) and e(f.9), where
f(z) = 1+z
and
9(1) 2
b,
Explain why we have e(f.g) S d(f.g) for all functions f and g Uder what
conditions on f and g is e(f,g) d(f.g)?
In the remainder of the question, we treat an explicit example of the function with
a very narrow tall bump" of Figure 1.14. For each integer n 2 2, define a function
fnE C10, 1] by
nz +1-号
-Asest
fn(z) =
otherwise.
Sketch the graph of f, and describe the graph of for any n.
d)
Let g(z) 0 be the zero function. Calculate d ) and e( o).
Deduce that there is no number 4 such that
for all f,g e Cto, 1].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2548eb1e-2424-4d48-82ae-76b755e59ca5%2F1a6e7868-2fd7-4423-b84b-daadec6fcfa3%2Fa9757u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. This question concerns the L metric d and the L metric e on C0, 1)
a) Calculate d(f. 9) and e(f.9), where
f(z) = 1+z
and
9(1) 2
b,
Explain why we have e(f.g) S d(f.g) for all functions f and g Uder what
conditions on f and g is e(f,g) d(f.g)?
In the remainder of the question, we treat an explicit example of the function with
a very narrow tall bump" of Figure 1.14. For each integer n 2 2, define a function
fnE C10, 1] by
nz +1-号
-Asest
fn(z) =
otherwise.
Sketch the graph of f, and describe the graph of for any n.
d)
Let g(z) 0 be the zero function. Calculate d ) and e( o).
Deduce that there is no number 4 such that
for all f,g e Cto, 1].
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