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*5. A function f defined on an interval I = {x: a <x<b> is called increasing...
Definition 1. A function f(x) defined on (-L, L] is called piece-wise continuous if there are finitely many points xo =-L < x1 < x2 < < xn-L such that f is continuous on (xi, i+1) a...
Definition 1. A function f(x) defined on (-L, L] is called piece-wise continuous if there are finitely many points xo =-L < x1 < x2 < < xn-L such that f is continuous on (xi, i+1) and so that the limits lim f(z) and lim f(x) both crist for each a,. To save space we write lin. f(x) = f(zi-) ェ→z, lim, f(x) = f(zit), ェ→ Sub-problem 5. Let f(x)-x on (-2,-1), f(x) = 1 on (-1,0) and f(x)--z on...
(1 point) For the function f(x) = e2x + e- defined on the interval (-4, o),...
(1 point) For the function f(x) = e2x + e- defined on the interval (-4, o), find all intervals where the function is strictly increasing or strictly decreasing. Your intervals should be as large as possible. f is strictly increasing on f is strictly decreasing on (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10)) whenever r is near c on the left Find and classify all local max's and min's. (For the purposes...
Ja i3D1 3. A function is called increasing on an interval I if f(x) < f(y)...
Ja i3D1 3. A function is called increasing on an interval I if f(x) < f(y) for all in I. Suppose f is increasing on [a, b). Partition [a, b] into n equal pieces, each of width A = (b – a)/n. Find simple upper and lower bounds for Ja Ef(Ti)A. (Hint: In each [pi-1 – pi], f(pi-1) < f(x) < f(p;); also note that i=1 L(pi) – f(pi-1) = f(b) – f(a).) alsvmin ba i=1 taoo) ai eil diod...
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...
C-5.8 Let a set of intervals S (lao,bol), a,bla-1,b- of the interval [0, 1] be given, with 0-ai ...
C-5.8 Let a set of intervals S (lao,bol), a,bla-1,b- of the interval [0, 1] be given, with 0-ai 〈 bi I, for i = 0, î, , n-1. Suppose further that we assign a height hi to each interval [ai, biļin S. The upper envelope of S is defined to be a list of pairs |(xo,co), (Xi,on), (x2.c2), , (xnnCm), (xm+ 1,0)), with xo- 0 andx1, and ordered by xi values, such that, for each subinterval s- [Xi, Xi+1] the...
7 points Question 3. An Unusual Integrable Function (Show Working) Consider the function f : 10, ...
7 points Question 3. An Unusual Integrable Function (Show Working) Consider the function f : 10, 11 → R defined by 1 if r-for some nEN; f(x) = 0 for all other x E [0,1 (1 subpts) (a) Draw a rough diagram of the graph of f. When we study the formal definition of the continuity of a function later in the course, we will be able to prove that this function is discontinuous at those domain values r such...
Question 2 (6 Marks) (Chapter 2) Consider the function f : R3 -R defined as f(x1,2,3 +4eli++21), ...
this is an optimization subject.
that is example 2.33
Question 2 (6 Marks) (Chapter 2) Consider the function f : R3 -R defined as f(x1,2,3 +4eli++21), (G) Explain why f has a global minimum over the set Hint: Read Example 2.33 (i) Find the global minimum point and global minimum value of f over the set C. Example 2.33. Consider the function/(x1,x2)=xf+xỈ over the set The set C is not bounded, and thus the Weierstrass theorem does not guarantee the...
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d)...
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d) in the xy-plane that contains the point (xo, Yo), then there exists a solution y(x) to the initial-value problem dy = f(x, y), y(xo) = yo, dx that is defined in an open interval I = (a, b) containing xo. In addition, if the partial derivative Ofjay is continuous in R, then the solution y(x) of the given equation is unique. For the initial-value...
How do I approach this question? *69. Suppose that f'(x) > 0 and that f(a) <...
How do I approach this question?
*69. Suppose that f'(x) > 0 and that f(a) < 0 while f(6) > 0, so that f(x) = 0 has a root r in the interval (a,b). Newton's method for finding r, starting at c in (a, b) is as follows: let Xo = c, and for n> 1, define In = In-1 - f(xn-1)/f'(xn-1). a) Taking f(x) = x – 23, a=-1/13, b=1/13 and xo = 1/V5 find x0, 11,x2,.... b) If...
A function f:R HR is said to be strictly increasing if f(x1) < f(12) whenever I]...
A function f:R HR is said to be strictly increasing if f(x1) < f(12) whenever I] < 12. Prove: If a differentiable function f is strictly increasing, then f'(x) > 0. Then give counterexamples to show that the following statements are false, in general. (i) If a differentiable function f is strictly increasing, then f'(2) >0 for all 1. (ii) If f'(x) > 0 for all x, then f is strictly increasing -
Definition 1. A function f(x) defined on (-L, L] is called piece-wise continuous if there are finitely many points xo =-L < x1 < x2 < < xn-L such that f is continuous on (xi, i+1) and so that the limits lim f(z) and lim f(x) both crist for each a,. To save space we write lin. f(x) = f(zi-) ェ→z, lim, f(x) = f(zit), ェ→ Sub-problem 5. Let f(x)-x on (-2,-1), f(x) = 1 on (-1,0) and f(x)--z on...
(1 point) For the function f(x) = e2x + e- defined on the interval (-4, o), find all intervals where the function is strictly increasing or strictly decreasing. Your intervals should be as large as possible. f is strictly increasing on f is strictly decreasing on (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10)) whenever r is near c on the left Find and classify all local max's and min's. (For the purposes...
Ja i3D1 3. A function is called increasing on an interval I if f(x) < f(y) for all in I. Suppose f is increasing on [a, b). Partition [a, b] into n equal pieces, each of width A = (b – a)/n. Find simple upper and lower bounds for Ja Ef(Ti)A. (Hint: In each [pi-1 – pi], f(pi-1) < f(x) < f(p;); also note that i=1 L(pi) – f(pi-1) = f(b) – f(a).) alsvmin ba i=1 taoo) ai eil diod...
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...
C-5.8 Let a set of intervals S (lao,bol), a,bla-1,b- of the interval [0, 1] be given, with 0-ai 〈 bi I, for i = 0, î, , n-1. Suppose further that we assign a height hi to each interval [ai, biļin S. The upper envelope of S is defined to be a list of pairs |(xo,co), (Xi,on), (x2.c2), , (xnnCm), (xm+ 1,0)), with xo- 0 andx1, and ordered by xi values, such that, for each subinterval s- [Xi, Xi+1] the...
7 points Question 3. An Unusual Integrable Function (Show Working) Consider the function f : 10, 11 → R defined by 1 if r-for some nEN; f(x) = 0 for all other x E [0,1 (1 subpts) (a) Draw a rough diagram of the graph of f. When we study the formal definition of the continuity of a function later in the course, we will be able to prove that this function is discontinuous at those domain values r such...
this is an optimization subject.
that is example 2.33
Question 2 (6 Marks) (Chapter 2) Consider the function f : R3 -R defined as f(x1,2,3 +4eli++21), (G) Explain why f has a global minimum over the set Hint: Read Example 2.33 (i) Find the global minimum point and global minimum value of f over the set C. Example 2.33. Consider the function/(x1,x2)=xf+xỈ over the set The set C is not bounded, and thus the Weierstrass theorem does not guarantee the...
If f(x, y) is continuous in an open rectangle R = (a, b) x (c, d) in the xy-plane that contains the point (xo, Yo), then there exists a solution y(x) to the initial-value problem dy = f(x, y), y(xo) = yo, dx that is defined in an open interval I = (a, b) containing xo. In addition, if the partial derivative Ofjay is continuous in R, then the solution y(x) of the given equation is unique. For the initial-value...
How do I approach this question?
*69. Suppose that f'(x) > 0 and that f(a) < 0 while f(6) > 0, so that f(x) = 0 has a root r in the interval (a,b). Newton's method for finding r, starting at c in (a, b) is as follows: let Xo = c, and for n> 1, define In = In-1 - f(xn-1)/f'(xn-1). a) Taking f(x) = x – 23, a=-1/13, b=1/13 and xo = 1/V5 find x0, 11,x2,.... b) If...
A function f:R HR is said to be strictly increasing if f(x1) < f(12) whenever I] < 12. Prove: If a differentiable function f is strictly increasing, then f'(x) > 0. Then give counterexamples to show that the following statements are false, in general. (i) If a differentiable function f is strictly increasing, then f'(2) >0 for all 1. (ii) If f'(x) > 0 for all x, then f is strictly increasing -
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