Please prove by setting up the theorem below (Chain Rule)
Theorem 15.34 The Chain Rule Let O be an open subset of R and suppose that the mapping F:OR is continuously differentiable. Suppose also thatU is an open subset of Rm and that the functiong:u-R is continuously differentiable. Finally, suppose that F(O) is contained in U. Then the composition g o F:OR is also continuously differentiable. Moreover, for each point x in O and each index i such that 1 s i n OF (15.24) axi Экі j-1 that is, V(g o F)(x) -Vg(F(x))DF(x) (15.25)
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V:RR is continuously differentiable. Define the Suppose that the function function g : R2R by 8(s...
2) If F Uc R2R is a 1-1 continuously differentiable map of an open subset U of the plane and A is...
2) If F Uc R2R is a 1-1 continuously differentiable map of an open subset U of the plane and A is a measurable subset of U, then the area of F(A) is given by area(F(A)) - A det Jrx, )dm(x, y) If F is the induced map of a holomorphic f, what is the resulting formula?
2) If F Uc R2R is a 1-1 continuously differentiable map of an open subset U of the plane and A is a...
(a) Suppose f is continuously differentiable on the closed and bounded interval I = [0, 1]....
(a) Suppose f is continuously differentiable on the closed and bounded interval I = [0, 1]. Show that f is uniformly continuous on I. (b) Suppose g is continuously differentiable on the open interval J = (0,1). Give and example of such a function which is NOT uniformly continuous on J, and prove your answer.
linear optimization Assume that f : D → R is twice continuously differentiable for all x...
linear optimization
Assume that f : D → R is twice continuously differentiable for all x D, where the domain D off is an open, convex subset of Rn. Sh ▽2f(x), is symmetric positive-semi-definite for all x E D if and only if f is a convex function on D Moreover, if its Hessian matrix. ▽2 (x), is symmetric positive-definite for all x E D, then f is a strictly convex function on D Show that the converse of this...
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...
4. Let F be a continuously differentiable function, and let s be a fixed point of...
4. Let F be a continuously differentiable function, and let s be a fixed point of F (a) Prove if F,(s)| < 1, then there exists α > 0 such that fixed point iterations will o E [s - a, s+a]. converge tO s whenever x (b) Prove if IF'(s)| > 1, then given fixed point iterations xn satisfying rnメs for all n, xn will not converge to s.
differentiable function and there exists 0 <A < 1 (6) Suppose that f : R" -> R" is a such that |f'...
differentiable function and there exists 0 <A < 1 (6) Suppose that f : R" -> R" is a such that |f'(x)|< A, for all x E R". Prove that the function F(x)= x - f(x) maps R" one-to-one and onto R". (Suggestion: Use the Contraction Mapping Principle Why not use the Inverse Function Theorem?)
differentiable function and there exists 0
Suppose that g is differentiable at x for all x ∈ R. Let f(x) = |g(x)|....
Suppose that g is differentiable at x for all x ∈ R. Let f(x) = |g(x)|. Use the Chain Rule to find f′(x).
5. Let f a, b R be a 4 times continuously differentiable function. For n even, consider < tn = b, a to < t< an...
5. Let f a, b R be a 4 times continuously differentiable function. For n even, consider < tn = b, a to < t< an uniform partition of [a, b] with b- a , i = 0,1,.. , n - 1 h t Let T denote the composite Trapezoidal rule associated with the above partition which approx imates eliminate the term containing h2 in the asymptotic expansion. Interprete the result which you obtain as an appropriate numerical quadrature rule...
4. (a) Assume a function h is differentiable at some point to. Is it true that...
4. (a) Assume a function h is differentiable at some point to. Is it true that h is continuous on some open-neighbourhood of xo? Provide either a proof or a counterexample. (b) Let f be twice differentiable on R and assume that f" is continuous. Show that for all x ER S(x) = S(0) + s°C)x + [ (x - 1))"(dt. (C) Deduce that for any twice continuously differentiable function f on R and any positive x > 0, x...
Suppose that g is differentiable, invertible, with g'(x) 0. Suppose that f is any function so...
Suppose that g is differentiable, invertible, with g'(x) 0. Suppose that f is any function so that the function h()f(g(a))g'(a) is integrable with anti-derivative H. Prove that f is integrable, and that nog . Is an antiderivative for J.
2) If F Uc R2R is a 1-1 continuously differentiable map of an open subset U of the plane and A is a measurable subset of U, then the area of F(A) is given by area(F(A)) - A det Jrx, )dm(x, y) If F is the induced map of a holomorphic f, what is the resulting formula?
2) If F Uc R2R is a 1-1 continuously differentiable map of an open subset U of the plane and A is a...
(a) Suppose f is continuously differentiable on the closed and bounded interval I = [0, 1]. Show that f is uniformly continuous on I. (b) Suppose g is continuously differentiable on the open interval J = (0,1). Give and example of such a function which is NOT uniformly continuous on J, and prove your answer.
linear optimization
Assume that f : D → R is twice continuously differentiable for all x D, where the domain D off is an open, convex subset of Rn. Sh ▽2f(x), is symmetric positive-semi-definite for all x E D if and only if f is a convex function on D Moreover, if its Hessian matrix. ▽2 (x), is symmetric positive-definite for all x E D, then f is a strictly convex function on D Show that the converse of this...
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...
4. Let F be a continuously differentiable function, and let s be a fixed point of F (a) Prove if F,(s)| < 1, then there exists α > 0 such that fixed point iterations will o E [s - a, s+a]. converge tO s whenever x (b) Prove if IF'(s)| > 1, then given fixed point iterations xn satisfying rnメs for all n, xn will not converge to s.
differentiable function and there exists 0 <A < 1 (6) Suppose that f : R" -> R" is a such that |f'(x)|< A, for all x E R". Prove that the function F(x)= x - f(x) maps R" one-to-one and onto R". (Suggestion: Use the Contraction Mapping Principle Why not use the Inverse Function Theorem?)
differentiable function and there exists 0
5. Let f a, b R be a 4 times continuously differentiable function. For n even, consider < tn = b, a to < t< an uniform partition of [a, b] with b- a , i = 0,1,.. , n - 1 h t Let T denote the composite Trapezoidal rule associated with the above partition which approx imates eliminate the term containing h2 in the asymptotic expansion. Interprete the result which you obtain as an appropriate numerical quadrature rule...
4. (a) Assume a function h is differentiable at some point to. Is it true that h is continuous on some open-neighbourhood of xo? Provide either a proof or a counterexample. (b) Let f be twice differentiable on R and assume that f" is continuous. Show that for all x ER S(x) = S(0) + s°C)x + [ (x - 1))"(dt. (C) Deduce that for any twice continuously differentiable function f on R and any positive x > 0, x...
Suppose that g is differentiable, invertible, with g'(x) 0. Suppose that f is any function so that the function h()f(g(a))g'(a) is integrable with anti-derivative H. Prove that f is integrable, and that nog . Is an antiderivative for J.
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