![3. let that fila, c]->IR and Cive Confinuous fuanetions with fces =gle) differentiable on la,c),giš ni(a,b)R by acceb and sup](http://img.homeworklib.com/questions/a5045490-b9d3-11eb-b26f-77649b3a9576.png?x-oss-process=image/resize,w_560)
Homework Answers
First of all g(x) is defined on [c,b] such that it is continuous on [b,c] and differentiable on [b,c].since f(x) is also continuous on [a,c] and differentiable on (a,c). Therefore ,for differentiability of h(x) at x=c is that it must be continuous at x=c and left hand derivative of h(x) at x=c must be equal to right hand derivative of h(x) at x=c.it mean that f'(c)= g'(c).
Hence the necessary and sufficient condition for differentiablity of h(x) is h'(c)=0 , i.e. f'(c)=h'(c)
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3. let that fila, c]->IR and Cive Confinuous fuanetions with fces =gle) differentiable on la,c),giš ni(a,b)R...
3. Let la, b) on [a, b]. Define | lo, by R and Cla, b be the space of continuously differentiable real-valued functions lsup () sup |f'() rEla,b rela,b Prove that (Cl a) is a Banach space...
3. Let la, b) on [a, b]. Define | lo, by R and Cla, b be the space of continuously differentiable real-valued functions lsup () sup |f'() rEla,b rela,b Prove that (Cl a) is a Banach space
3. Let la, b) on [a, b]. Define | lo, by R and Cla, b be the space of continuously differentiable real-valued functions lsup () sup |f'() rEla,b rela,b Prove that (Cl a) is a Banach space
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show...
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Exercise 1. Let f : R R be differentiable on la, b, where a, b R...
Exercise 1. Let f : R R be differentiable on la, b, where a, b R and a < b, and let f be continuous on [a, b]. Show that for every e> 0 there exists a 6 > 0 such that the inequality f(x)- f(c) T-C holds for all c, x E [a, 히 satisfying 0 < |c-x| < δ
3] Let f : (0, oo) + R be differentiable on (0,0o). Define the difference function...
3] Let f : (0, oo) + R be differentiable on (0,0o). Define the difference function (af)(z) := f(z + 1)-f(x), x > 0. If linnof,(z-0, find linn (Sf)(x).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f...
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
Let f: a, b R be a function, continuous on a, b and differentiable on (a,...
Let f: a, b R be a function, continuous on a, b and differentiable on (a, b). Show that 3c E (a, b) such that f (b) f(c) <0 f (e) f(a) 3s E (a, b) s.t f'(s) 0.
Let B C R" be any set. Define C = {x € R" | d(x,y) <...
Let B C R" be any set. Define C = {x € R" | d(x,y) < 1 for some y E B) Show that C is open.
Logic (a) Let f : [a, b] → R be a continuous function. Prover that there...
Logic
(a) Let f : [a, b] → R be a continuous function. Prover that there exists ce [a, b] such that con la silany - be a criminatoria per le sue in elan 5(e) = So gladde (b) Define F:R+R by F(x) = [** V1+e=i&t. Prove by citing the appropriate theorem(s) that F is differentiable on R, and calculate F'(c). Be sure to justify your reasoning at every stage.
4. Let S = {1,2,3). Define a relation R on SxS by (a, b)R(c,d) iff a...
4. Let S = {1,2,3). Define a relation R on SxS by (a, b)R(c,d) iff a <c and b <d, where is the usual less or equal to on the integers. a. Prove that R is a partial order. Is R a linear order? b. Draw the poset diagram of R.
3. (a) Let f be an infinitely differentiable function on R and define х F(x) =...
3. (a) Let f be an infinitely differentiable function on R and define х F(x) = e-y f(y) dy. Find and prove a formula for F(n), the nth derivative of F. (b) Show that if f is a polynomial then there exists a constant C such that F(n)(x) = Cem for sufficiently large n. Find the least n for which it is true.
3. Let la, b) on [a, b]. Define | lo, by R and Cla, b be the space of continuously differentiable real-valued functions lsup () sup |f'() rEla,b rela,b Prove that (Cl a) is a Banach space
3. Let la, b) on [a, b]. Define | lo, by R and Cla, b be the space of continuously differentiable real-valued functions lsup () sup |f'() rEla,b rela,b Prove that (Cl a) is a Banach space
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Exercise 31: (Chain rule) Let g : la,b] → R be differentiable and strictly increasing and f : R-IR be continuous. Show that gr) F(x) :=| f(t)dt Jg(a) is differentiable and compute its derivative
Exercise 1. Let f : R R be differentiable on la, b, where a, b R and a < b, and let f be continuous on [a, b]. Show that for every e> 0 there exists a 6 > 0 such that the inequality f(x)- f(c) T-C holds for all c, x E [a, 히 satisfying 0 < |c-x| < δ
3] Let f : (0, oo) + R be differentiable on (0,0o). Define the difference function (af)(z) := f(z + 1)-f(x), x > 0. If linnof,(z-0, find linn (Sf)(x).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f).
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
Let B C R" be any set. Define C = {x € R" | d(x,y) < 1 for some y E B) Show that C is open.
Logic
(a) Let f : [a, b] → R be a continuous function. Prover that there exists ce [a, b] such that con la silany - be a criminatoria per le sue in elan 5(e) = So gladde (b) Define F:R+R by F(x) = [** V1+e=i&t. Prove by citing the appropriate theorem(s) that F is differentiable on R, and calculate F'(c). Be sure to justify your reasoning at every stage.
4. Let S = {1,2,3). Define a relation R on SxS by (a, b)R(c,d) iff a <c and b <d, where is the usual less or equal to on the integers. a. Prove that R is a partial order. Is R a linear order? b. Draw the poset diagram of R.
3. (a) Let f be an infinitely differentiable function on R and define х F(x) = e-y f(y) dy. Find and prove a formula for F(n), the nth derivative of F. (b) Show that if f is a polynomial then there exists a constant C such that F(n)(x) = Cem for sufficiently large n. Find the least n for which it is true.
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