4. Let F RNR be a mapping. (i) x ЄRN ? (ii) : What does it mean to say that F is differentiable at a point [1 Mark] In Theorem 5.4 in the Lecture Notes we proved that if F is differentiable at a point x E RN then F is continuous at x. Proof. Let (n) CRN be a sequence such that xn → x ЄERN as n → ∞. We want to show that F(xn) F(x), which means F is continuous at x. Denote hnxn - x, so that ||hn|| 0. Thus we find ||F(xn) − F(x)|| = ||F(x + hn) − F(x)|| * ||DF (x)hn + R(hn) || (**) ||DF(x)hn||+||R(hn)||| → 0, because the linear mapping DF(x) is continuous and for all large nЄ N, (***) ||R(hn) || ||R(hn) || ≤ → 0. ||hn|| (a) Explain in details why ||hn|| → 0. [3 Marks] (b) Explain the steps labelled (*), (**), (***). [6 Marks]
4. Let F RNR be a mapping. (i) x ЄRN ? (ii) : What does it mean to say that F is differentiable at a point [1 Mark] In Theorem 5.4 in the Lecture Notes we proved that if F is differentiable at a point x E RN then F is continuous at x. Proof. Let (n) CRN be a sequence such that xn → x ЄERN as n → ∞. We want to show that F(xn) F(x), which means F is continuous at x. Denote hnxn - x, so that ||hn|| 0. Thus we find ||F(xn) − F(x)|| = ||F(x + hn) − F(x)|| * ||DF (x)hn + R(hn) || (**) ||DF(x)hn||+||R(hn)||| → 0, because the linear mapping DF(x) is continuous and for all large nЄ N, (***) ||R(hn) || ||R(hn) || ≤ → 0. ||hn|| (a) Explain in details why ||hn|| → 0. [3 Marks] (b) Explain the steps labelled (*), (**), (***). [6 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...
Related questions
Question
![4.
Let F RNR be a mapping.
(i)
x ЄRN ?
(ii)
:
What does it mean to say that F is differentiable at a point
[1 Mark]
In Theorem 5.4 in the Lecture Notes we proved that if F
is differentiable at a point x E RN then F is continuous at x.
Proof. Let (n) CRN be a sequence such that xn → x ЄERN as n → ∞. We
want to show that F(xn) F(x), which means F is continuous at x.
Denote hnxn - x, so that ||hn|| 0. Thus we find
||F(xn) − F(x)|| = ||F(x + hn) − F(x)|| * ||DF (x)hn + R(hn) ||
(**)
||DF(x)hn||+||R(hn)||| → 0,
because the linear mapping DF(x) is continuous and for all large nЄ N,
(***) ||R(hn) ||
||R(hn) || ≤
→ 0.
||hn||
(a)
Explain in details why ||hn|| → 0.
[3 Marks]
(b)
Explain the steps labelled (*), (**), (***).
[6 Marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c7573-6777-48ff-9bfb-9b3df1a769a6%2F76b27e2f-24ff-4937-a34d-5aded4beab83%2Fzhhv7aa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.
Let F RNR be a mapping.
(i)
x ЄRN ?
(ii)
:
What does it mean to say that F is differentiable at a point
[1 Mark]
In Theorem 5.4 in the Lecture Notes we proved that if F
is differentiable at a point x E RN then F is continuous at x.
Proof. Let (n) CRN be a sequence such that xn → x ЄERN as n → ∞. We
want to show that F(xn) F(x), which means F is continuous at x.
Denote hnxn - x, so that ||hn|| 0. Thus we find
||F(xn) − F(x)|| = ||F(x + hn) − F(x)|| * ||DF (x)hn + R(hn) ||
(**)
||DF(x)hn||+||R(hn)||| → 0,
because the linear mapping DF(x) is continuous and for all large nЄ N,
(***) ||R(hn) ||
||R(hn) || ≤
→ 0.
||hn||
(a)
Explain in details why ||hn|| → 0.
[3 Marks]
(b)
Explain the steps labelled (*), (**), (***).
[6 Marks]
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