Consider the function f(x) = Σ (a) Where is f defined? (b) Where is f continuous? (c) Where is f ...
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y) = if (x, y) (0, 0) (a) Prove that f is continuous at (0,0) (b) Calculate the partial derivatives (0,0) and (0,0) directly from the definition of partial derivatives. (c) Prove that f is not differentiable at (0,0).
Consider the function f : R → R defined by f(x) = !x if x is rational −x if x is irrational. Find all c ∈ R at which f is continuous. Consider the function f :R → R defined by .. х if x is rational f(x) = -2 if x is irrational. Find all c ER at which f is continuous.
x2 +7x+12 1. Consider the function: f(x)= x +3 a. Is this function continuous at x = -3? b. Does this function have a limit at x = -3? dito c. Is this function differentiable at x = -3? d. Sketch a graph of the function in the space below. Be sure to include all pertinent features.
7) [f function g is defined by g(x) = Ta'an, xa, thon g'(1) = ? (A) 1 (B) 2 (C) 3 (D)O (E) Does not exist The graph of function g is shown below. Which of the following statements about function g is true? A) Atx=b, function g is differentiable. BY Atxa, function g is continuous, but not differentiable. C) Atx=a, function g is both continuous and differentiable Function g is not differentiable at any point. E) Atx=c, function g...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
9. [7 points) Consider the function f(x) defined by f(x) = xeAs + B if x <3 C(x - 3)2 if 3 < x < 5 130 if > 5. C Suppose f(x) satisfies all of the following: f(x) is continuous at x = 3. • lim f(x) = 2 + lim f(x). 3+5+ 3-5- lim f(x) = -4. Find the values of A, B, and C. . 24-O
Let x ~ Nk(0, Σ) with pdf f(x) where Σ = {Σ defined as . The entropy h(x) is h(x) =-J f(x) In f(x) In(2me)"E! (a) Show that h(x) ( b) Hence, or otherwise, show that |E| s 11k! Σί, with equality holding if and only if Σ¡j 0, for i j [Hadamard's inequality]
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,...
Consider the function f:R + R defined by if x is rational f(x) = if x is irrational. Find all c € R at which f is continuous. C
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...