Question 1 20 points Save Answer Mark all true statements (there might be more than one...
5. 5 Let f(x) be continuous and differentiable on f(5) = 4. Mark TRUE or FALSE for the following statements and JUSTIFY. (No points will be given without the correct justification) [0, 10] with f(0) = f(10) 0 and (E) There is some c E (0,5) such that f'(c) = 5. 5 Let f(x) be continuous and differentiable on f(5) = 4. Mark TRUE or FALSE for the following statements and JUSTIFY. (No points will be given without the correct...
Let A= = {* : nen}u(2,3). PA Please mark all true statements (there might be more than one statement that is true LEO A for all nEN. The interior of Ais Int(A)={0}U(2,3). Points and 2 are accumulation points of A. The boundary of Ais A = {0,2,3}. The closure of A in R is A=Au{0,2,3).
Problem 1: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If a continuous function f:R +R is bounded, then f'(2) exists for all x. (b) Suppose f.g are two functions on an interval (a, b). If both f + g and f - g are differentiable on (a, b), then both f and g are differentiable on (a,b). Problem 2: Define functions f,g: RR by: x sin(-),...
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...
1 Let f: R R be a continuously differentiable map satisfying ilf(x)-FG) ll 리1x-vil, f Rn. Then fis onto 2. f(RT) is a closed subset of R'" 3, f(R") is an open subset of RT 4. f(0)0 or all x, y E 5) S= (xe(-1,4] Sin(x) > 0). Let of the following is true? I. inf (S).< 0 2. sup (S) does not exist Which . sup (S) π ,' inf (S) = π/2 1 Let f: R R be...
true or false The real valued function f : (1,7) + R defined by f(x) = 2is uniformly contin- uous on (0,7). Let an = 1 -1/n for all n € N. Then for all e > 0) and any N E N we have that Jan - am) < e for all n, m > N. Let f :(a,b) → R be a differentiable function, if f'() = 0 for some point Xo € (a, b) then X, is...
The multiple-choice question below might have more than one correct answer. Assume that F= (P, Q, R) where P,Q,R have continuous partial derivatives Which of the following statements are true? If Fis a conservative vector field and C is a closed curve then fer F: dr = 0. If Fis a conservative vector field then curlF = 0. div curlF = 0 If Fis a conservative vector field then 2 F is a conservative vector field. If Fis a conservative...
question starts at let. than one variable. Let f:R? → R3 be the function given by f(x, y) = (cos(x3 - y2), sin(y2 – x), e3x2-x-2y). (a) Let P be a point in the domain of f. As we saw in class, for (x, y) near P, we have f(x, y) f(P) + (Dpf)(h), where h = (x, y) - P. The expression on the right hand side is called the linear approximation of f around P. Compute the linear...
Question 16 Which of the following statements is True? iff has a local maximum or minimum at a point (a,b) and the partial derivatives fly exist then(a,b) - fy(a,b) - 0 A local maximum is always larger than a local minimum A local minimum is always smaller than a local maximum, off has a critical point at (a,b) and / (0,6) - 4,1yyla,b) - 2. /vy(a,b) – 3, then f has a local minimum at (a,b) none of the given...
please show all work, even trivial steps. Here are definitions if needed. do not write in script thank you! 4. Letf: R2 → R2, by f(x,y) = (x-ey,xy). a. Find Df (2,0). b. Find DF-1(f (2,0)) Inverse Function Theorem: Suppose that f:R" → R" is continuously differentiable in an open set containing a and det(Df(a)) = 0, then there is an open set, V, containing a and an open set, W, containing f(a) such that f:V W has a continuous...