7. Consider the function f on U50 defined by f(x) = [x2] where [2²] repre- sents...
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...
Consider the function f(x) = Σ (a) Where is f defined? (b) Where is f continuous? (c) Where is f differentiable? Consider the function f(x) = Σ (a) Where is f defined? (b) Where is f continuous? (c) Where is f differentiable?
a) Consider the function f(x) = x2 defined over the interval [0,a]. What is the value of “a” for this to be a valid probability distribution function? Express your answer to four decimal places. b) develop the cumulative distribution function, F(x), and use it determine the probability that the random variable X is less than one.
(6) Consider the function f(x) = 1 2 x − 1 with its domain defined on the interval 2 ≤ x ≤ 4. (a) Draw the graph of f. (b) Verify that f is a probability density function for a continuous random variable X. (c) Compute P(X ≤ 3). (d) Compute P(X ≥ 3)
1. Consider the function defined by 1- x2, 0< |x| < 1, f(x) 0, and f(r) f(x+4) (a) Sketch the graph of f(x) on the interval -6, 6] (b) Find the Fourier series representation of f(x). You must show how to evaluate any integrals that are needed 2. Consider the function 0 T/2, T/2, T/2 < T. f(x)= (a) Sketch the odd and even periodic extension of f(x) for -3r < x < 3m. (b) Find the Fourier cosine series...
1. Consider the function defined by (1 -2, 0 r< 1, f(x) 1 < |x2 (0. and f(r) f(x+ 4) (a) Sketch the graph of f(x) on the interval -6,61 (b) Find the Fourier seriess representation of f(x). You must show how to evaluate any integrals that are needed. 1. Consider the function defined by (1 -2, 0 r
10. [12 Points) Properties of relations Consider the relation R defined on R by «Ry x2 - y2 = x - y (a) Show that R is reflexive. (b) Show that R is symmetric. (c) Show that R is transitive. (d) You have thus verified that R is an equivalence relation. What is the equivalence class of 3? (e) More generally, what is the equivalence class of an element x? Use the listing method. (f) Instead of proving the three...
48 The function f is defined by f(x) = for 3 <x< 7. The function g is defined by g(x) = 2x - 4 for .X-1 a<x<b, where a and b are constants. (i) Find the greatest value of a and the least value of b which will permit the formation of the composite function gf. [2] It is now given that the conditions for the formation of gf are satisfied. (ii) Find an expression for gf(x). [1] (iii) Find...
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x, y, z) = x2 + y2 + x2 = 2a?, (1) where a is a fixed real positive constant, and the point u = (0,a,a) on the surface f(x, y, z) = 2a. a) Find the gradient of f(x, y, z) at the point u. b) Calculate the normal derivative of f(x, y, 2) at u. c) Find the equation of the tangent plane...
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