Now assume that f(0) = 0 and f'(0) = 0. Prove that if f is twice differentiable and If"(x) < 1 for all x E R then 22 Vx > 0, f(x) < 2
12. What value of c make the function f(x) = (x2 – 3x when x > 2 continuous when x = 2? 14x + 2c when x < 2 a)-5 b)-3 c) 0 d) 1
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...
Suppose that the piecewise function J is defined by f(2)= {**** -1<<3 - 3x2 + 2x + 23, 2> 3 Determine which of the following statements are true. Select the correct answer below: O f() is not continuous at I = 3 because it is not defined at I = 3. Of() is not continuous at 2 = 3 because lim f(x) does not exist. f() is not continuous at I = 3 because lim f() f(3). ->3 f(x) is...
Exercise 1. Let f : R R be differentiable on la, b, where a, b R and a < b, and let f be continuous on [a, b]. Show that for every e> 0 there exists a 6 > 0 such that the inequality f(x)- f(c) T-C holds for all c, x E [a, 히 satisfying 0 < |c-x| < δ
The Laplace transform of the piecewise continuous function $4, 0<t<3 f(t) is given by 2, t> 3 1 L{f} (1 – 2e-st), 8 >0. S None of them L{f} = (1 – 3e®), s>0. 2 L{f} (3 - e-), 8 >0. S 2 L{f} (2-est), s >0. S
Given the cumulative distribution, x <7 1 F(x) 1, 7 5xs14 0, 1, X> 14 Determine the first quartile, Q1 a. 12.25 b. 0.0 C. 8.75 d. 0.04 e. 10.5
A continuous random variable X has a beta distribution with p.d.f : 1 f(x) = 0<<<1, a > 2 B(4, 5)22-1(1 – 2)8-1 Determine E (3) HINT: E possible. (-) + E(X) Please show your work and simplify your final answer as much as
8. Consider the circuit: t=0 212 f(t)t 222 11 Pilt) 1F yt (a) Show that the transfer function of the circuit for t > 0 is † (s) = F(*) 452 +55+2 (b) What are the characteristic modes of the circuit (c) Determine the response y(t) for t > 0 if f(t) = 1, y(0-) = 1 V and i(0-) = 0.
5. 6 pt Determine whether the function f(x) is continuous and/or differentiable at x = 1. (x2+1 f(x) = { 12, >1 1 <1