Homework Answers
Add Answer to:
Let f be defined on an open interval I containing a point a (1) Prove that if f is differentiable on I and f"(a) exists, then lim h-+0 (a 2 h2 (2) Prove that if f is continuous at a and there...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint:...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing funct...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
(2) Suppose that f and 9 are differentiable on an open interval I and that a...
(2) Suppose that f and 9 are differentiable on an open interval I and that a € R either belongs to I or is an endpoint of I. Suppose further that g and g' are never zero on I\{a} and that lim f(x) is of the form 0/0. (a) If there is an M ER such that f'(2)/'(x) < M for all x E I\{a}, prove that \$(r)/g(x) < M for all x € I\{a}. (b) Is this result true...
2. Let I be an interval and let f be a function which is differentiable on...
2. Let I be an interval and let f be a function which is differentiable on I. Prove that if f' is bounded on I then f is uniformly continuous on I. 3. Give an example to show that the converse of the result in the previous question is false, i.e., give an example of a function which is differentiable and uniformly continuous on an interval but whose derivative is not bounded. (Proofs for your assertions are necessary, unless they...
Analysis problem (b) Let f, q be defined on A to R and let c be...
Analysis problem
(b) Let f, q be defined on A to R and let c be a cluster point of A i. Show that if both lim f and lim (f + g) exist, then lim g exists. c I>c ii. If lim f and lim fg exist, does it follow that lim g exists? -c (c) Suppose that f and g have limits in R as x -> o and that f(x) < g(x) for all x € (a,...
Suppose that f' exists and is continuous on a nonempty open interval (a,b) with f'(c) +...
Suppose that f' exists and is continuous on a nonempty open interval (a,b) with f'(c) + 0 for all 2 € (a,b). | Prove that f is one-to-one on (a, b) and that f((a,b)) is an open interval II: if (c,d) is the open interval from (i), show that f-1EC'((c,d)), i.e. f-1 has a continuous first derivative on (a, b).
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the...
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the partial derivatives Dh,(x) exist, for all , SO , . . . , n, and and J-: に1 The subscripts hi, 9i, k denote the coordinates of the functions h, g, f relative to...
(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.
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.
(c) Give an example of an open interval I, a function h : 1 R, and a point c ε 1 such that lim h(...
(c) Give an example of an open interval I, a function h : 1 R, and a point c ε 1 such that lim h(ar) exists if and only if a c. Lightly justify your example. points
(c) Give an example of an open interval I, a function h : 1 R, and a point c ε 1 such that lim h(ar) exists if and only if a c. Lightly justify your example. points
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
(2) Suppose that f and 9 are differentiable on an open interval I and that a € R either belongs to I or is an endpoint of I. Suppose further that g and g' are never zero on I\{a} and that lim f(x) is of the form 0/0. (a) If there is an M ER such that f'(2)/'(x) < M for all x E I\{a}, prove that \$(r)/g(x) < M for all x € I\{a}. (b) Is this result true...
2. Let I be an interval and let f be a function which is differentiable on I. Prove that if f' is bounded on I then f is uniformly continuous on I. 3. Give an example to show that the converse of the result in the previous question is false, i.e., give an example of a function which is differentiable and uniformly continuous on an interval but whose derivative is not bounded. (Proofs for your assertions are necessary, unless they...
Analysis problem
(b) Let f, q be defined on A to R and let c be a cluster point of A i. Show that if both lim f and lim (f + g) exist, then lim g exists. c I>c ii. If lim f and lim fg exist, does it follow that lim g exists? -c (c) Suppose that f and g have limits in R as x -> o and that f(x) < g(x) for all x € (a,...
Suppose that f' exists and is continuous on a nonempty open interval (a,b) with f'(c) + 0 for all 2 € (a,b). | Prove that f is one-to-one on (a, b) and that f((a,b)) is an open interval II: if (c,d) is the open interval from (i), show that f-1EC'((c,d)), i.e. f-1 has a continuous first derivative on (a, b).
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the partial derivatives Dh,(x) exist, for all , SO , . . . , n, and and J-: に1 The subscripts hi, 9i, k denote the coordinates of the functions h, g, f relative to...
(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.
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.
(c) Give an example of an open interval I, a function h : 1 R, and a point c ε 1 such that lim h(ar) exists if and only if a c. Lightly justify your example. points
(c) Give an example of an open interval I, a function h : 1 R, and a point c ε 1 such that lim h(ar) exists if and only if a c. Lightly justify your example. points
Most questions answered within 3 hours.
-
Sparky, Co. purchased land as a factory site for $600,000.
Sparky paid $42,000 to tear down...
asked 29 seconds from now -
A Chi-square distribution with 14 degrees of freedom is a
correct model for
Question 8 options:...
asked 28 minutes ago -
In a group of 45 mice, there are 10 that have a certain genetic
character. suppose...
asked 10 minutes ago -
Topic: Hydrogenic Atoms
The wavefunction of one of the d orbitals is proportional to sin
θ...
asked 10 minutes ago -
6. Suppose that the Bank of Canada conducts an open market
purchase of $2000 from a...
asked 25 minutes ago -
A) Suppose U=X∙Y3. Find X* and Y*.
B) Suppose U=X3∙Y. Find X* and Y*.
C) Suppose...
asked 33 minutes ago -
The only quantities of good 1 that Barbara can buy are 1 unit or
zero units....
asked 23 minutes ago -
c. Node Admittance matrix and its use in different calculations
of power transmission system. Display the...
asked 31 minutes ago -
Try to get the code down to less than 40 lines. (PYTHON)
import random
fave_number =...
asked 31 minutes ago -
2.
Regression analysis was applied between sales (in $1,000) and
advertising (in $100), and the following...
asked 39 minutes ago -
I just took a final for chemistry 2. There were alot of
questions on cell potential....
asked 1 hour ago -
A spherical weather balloon is filled with hydrogen until its
radius is 2.30 m. Its total...
asked 36 minutes ago