(0) 3 -1 (a) ynt X(t) Y(E) ( Express y(t) as a function x(+). (b) verify...
х 0<I< 3. The tent function is defined by T(x) = 1 - < x < 1 2 otherwise (a) Express T(2) in terms of the Heaviside function. (b) Find the Laplace transform of T(x). (c) Solve the differential equation y" – y=T(x), y(0) = y'(0) = 0
the function y=f(x)={ 0-4), 14x+16, x20 x<0 Consider 1. (a) Sketch the graph off. (3 pts.) (b) Verify that the function is continuous everywhere using the properties of the definition and possibly calculating the limit at a particular point. (2 pts.) (c) Show f'(x) is not continuous at x-0. (5 pts.)
the function y=f(x)={ 0-4), 14x+16, x20 x
5. Let y E C2([0, T]; R), T > 0 satisfy y"(t) = 피t, y(0) = y'(0) = 0 e R. Use Picard-Lindelöf 1+t' to prove that a unique solution to the IVP exists for short time, as follows: (a) Let b E R2, A E M2 (R) . Show that any function g : R2 -R2.9(x) = Ax+b is Lipschitz. 1 mark (b) Transform the DE for y into a(t) Az(t) +b(t) for a suitable z, A, b. 2...
1. Verify that fxr (x,y) -2e-x-y 0 < x < 00, x < y < is joint probability density function 2. Compute the probability that X < 1.and Y < 2.
25. Given the following parametric curve X(t) = -1 + 3 cos(t) y(t) = 1 + 2 sin(t) 0<t<21 a) Express the curve with an equation that relates x and y. 7C b) Find the slope of the tangent line to the curve at the point t c) State the pair(s) (x,y) where the curve has a horizontal/vertical tangent line. 27.A particle is traveling along the path such that its position at any time t is given by r(t) =...
1. (20 pts) RVs X and Y have joint density function 22 f(x, y) =(0 if O <z<1 and 0<y<2 īf 0 < x < 1 and 0 < y < 2 otherwise (a) Find E(X), V(X), E(Y), and V(Y). (b) Find the covariance cov(X,Y) and the associated correlation ρ (c) Find the marginal densities fx and fy. (Be sure to say where they're nonzero.) (d) Find E(X | Y = 1.5). (e) Are X and Y independent? Give two...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the OLS estimator for {00, Bo}, the minimizer of E. (Y: - a - 3x), is . (X.-X) (Y-Y) and a-Y-3X. - (Xi - x) When the equation (1) is the true data generating process, {X}- are non-stochastic, and {e} are random variables with B (ei) = 0, B(?) = 0, and Ele;e;) = 0 for any i, j = 1,2,...,n and i j, we...
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a) Verify that the function y = ?? + is a solution of the differential equation zy' +2y 4x? (x > 0). b) Find the value ofe for which the solution satisfies the initial condition (2) - 5. = Submit Question a) Verify that the function y=x? + с 2 is a solution of the differential equation ry' + 2y = 4x², (x > 0). b) Find the value of c for which the solution satisfies the initial...
in the sub-space -2) Find the closest point to f = | 2 3 C) (Verify your answer) e R: +y+ = 0} W = { y
in the sub-space -2) Find the closest point to f = | 2 3 C) (Verify your answer) e R: +y+ = 0} W = { y
Let's consider a function described in terms of its displacement y(x,t) at t 0 by: where a, b and e are positive constants a) Write an expression for this wave profile, having a speed in the negative x-direction, as a function of position and time (b) Sketch the profile of the wave at t-0 s and t 2 s if v1 m/s (c) Determine if the following functions describe a travelling wave: (i) vr,t) (ar+ bt c), where a, b...