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4. Is the infinite Continuous on R? series of functions defined on R? Differentiable on R?...
(a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g? (a) Can there be differentiable functio...
(a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g?
(a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g?
real analysis 1,2,3,4,8please 5.1.5a Thus iff: I→R is differentiable on n E N. is differentiable on / with g'...
real analysis
1,2,3,4,8please
5.1.5a
Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
- Let V be the vector space of continuous functions defined f : [0,1] → R...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
4. Let f be a differentiable function defined on (0, 1) whose derivative is f'(c) =...
4. Let f be a differentiable function defined on (0, 1) whose derivative is f'(c) = 1 - cos (+) [Note that we can confidently say such an f exists by the FTC.) Prove that f is strictly increasing on (0,1). 5. Let f be defined on [0, 1] by the following formula: 1 x = 1/n (n € N) 0, otherwise (a) Prove that f has an infinite number of discontinuities in [0,1]. (b) Prove that f is nonetheless...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and conti...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
Problem 5. Given vi,v2,... ,Vm R", let RRm be defined by f(x)-x, v1), x, v2), (x, Vm where (x' y)...
Problem 5. Given vi,v2,... ,Vm R", let RRm be defined by f(x)-x, v1), x, v2), (x, Vm where (x' y) is the standard inner product of Rn Which of the following statement is incorrect? 1. Taking the standard bases Un on R": codomain: MatUn→Un(f)-(v1 2. Taking the standard bases Un on R: codomain: v2 vm) Matf)- 3. f is a linear transformation. 4. Kerf- x E Rn : Vx = 0 , where: Problem 8. Which of the following statements...
2. The vector space CO, 1) consisting of continuous functions defined on the interval 0く〈 1 is ...
2. The vector space CO, 1) consisting of continuous functions defined on the interval 0く〈 1 is infinite-dimensional since it is not spanned by a finite set of functions. Consider the subspace H {aet + bt2 fet _ 3t2) | a, b, c є R} of C10, 1]. Is H finite-dimensional or infinite- dimensional? If it is infinite dimensional, explain why. If it is finite dimensional, find a basis for H and explain your reasonsing. cle
2. The vector space...
Find the results of next functions 2.-Find the values of a and b such that fis differentiable at x 1 ax+b si 1s. Sol, a...
Find the results of next functions
2.-Find the values of a and b such that fis differentiable at x 1 ax+b si 1s. Sol, a- 2, b-1. f(x)=1si x<1 x-7 si 0<x Sb| f(x) =16/x si x< 3 If ... a) decide a value of b far which f is continuous b)fis differentiable in the value of b that ycu find in part a)? 4.- In the following functions determine what is requested sen(x) si x < mx+b si x...
Definition. Let fi, f2.83.... be a sequence of functions defined on an interval I. The series...
Definition. Let fi, f2.83.... be a sequence of functions defined on an interval I. The series fn(x) is said to have property 6 on I if there erists a convergent series of positive constants, Mn, satisfying \fu(x) S M for all values of n and for every or in the interval I. n=1 Theorem. If the series (1) has property C on the interval (a, b), and if the terms f(x) are continuous functions on (a, b), then nel 1...
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that 2. Let f R R and g R-R...
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
(a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g?
(a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g?
real analysis
1,2,3,4,8please
5.1.5a
Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
4. Let f be a differentiable function defined on (0, 1) whose derivative is f'(c) = 1 - cos (+) [Note that we can confidently say such an f exists by the FTC.) Prove that f is strictly increasing on (0,1). 5. Let f be defined on [0, 1] by the following formula: 1 x = 1/n (n € N) 0, otherwise (a) Prove that f has an infinite number of discontinuities in [0,1]. (b) Prove that f is nonetheless...
1) Suppose f (a, b) R is continuous. The Carathéodory Theorem says that f(x) is differentiable at -cE (a, b) if 3 (a, b)-R which is continuous, and so that, (a) Show, for any constant a and continuous function (x), that af(x) is continuous at z-c by finding a Carathéodory function Paf(x). (b) Show, for any constants a, B, that if g : (a, b) -R is differentiable at c, with Carathéodory function pg(z), then the linear combination of functions,...
Problem 5. Given vi,v2,... ,Vm R", let RRm be defined by f(x)-x, v1), x, v2), (x, Vm where (x' y) is the standard inner product of Rn Which of the following statement is incorrect? 1. Taking the standard bases Un on R": codomain: MatUn→Un(f)-(v1 2. Taking the standard bases Un on R: codomain: v2 vm) Matf)- 3. f is a linear transformation. 4. Kerf- x E Rn : Vx = 0 , where: Problem 8. Which of the following statements...
2. The vector space CO, 1) consisting of continuous functions defined on the interval 0く〈 1 is infinite-dimensional since it is not spanned by a finite set of functions. Consider the subspace H {aet + bt2 fet _ 3t2) | a, b, c є R} of C10, 1]. Is H finite-dimensional or infinite- dimensional? If it is infinite dimensional, explain why. If it is finite dimensional, find a basis for H and explain your reasonsing. cle
2. The vector space...
Find the results of next functions
2.-Find the values of a and b such that fis differentiable at x 1 ax+b si 1s. Sol, a- 2, b-1. f(x)=1si x<1 x-7 si 0<x Sb| f(x) =16/x si x< 3 If ... a) decide a value of b far which f is continuous b)fis differentiable in the value of b that ycu find in part a)? 4.- In the following functions determine what is requested sen(x) si x < mx+b si x...
Definition. Let fi, f2.83.... be a sequence of functions defined on an interval I. The series fn(x) is said to have property 6 on I if there erists a convergent series of positive constants, Mn, satisfying \fu(x) S M for all values of n and for every or in the interval I. n=1 Theorem. If the series (1) has property C on the interval (a, b), and if the terms f(x) are continuous functions on (a, b), then nel 1...
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
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