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) Verify your formula for 1"n directly from the definition of the convolution product by using in...
reasonable chance of getting good scores on the exams. References: Lectures 8.1 and 8.2. Sections 9.1, 9.2, and 9.3 in the textbook. You can make free use of the table of Laplace Transforms on Page 500 but you must justify any other formulas for Laplace transforms. When we speak of "the" inverse Laplace transform of a function F(s) we mean the unique continuous function / : [0,00) → R (if there is one!) whose Laplace transforn is F(s) (for sufficiently...
Solution steps plz 3. Derive the convolution product e" * cos bt by using the formula for the Laplace transform of the convolution.
Induction proofs. a. Prove by induction: n sum i^3 = [n^2][(n+1)^2]/4 i=1 Note: sum is intended to be the summation symbol, and ^ means what follows is an exponent b. Prove by induction: n^2 - n is even for any n >= 1 10 points 6) Given: T(1) = 2 T(N) = T(N-1) + 3, N>1 What would the value of T(10) be? 7) For the problem above, is there a formula I could use that could directly calculate T(N)?...
4) For each of the following utility functions derive directly from the definition not using the formula(s) from class or the text - the MRS (marginal rate of substitution) of y for x at the points (1, 1) and (2,4 (a) ua(x, y)-y (b) us(x, y) x + 2y (c) uc (z, y) = 3x + y
6. (20 pts) se the definition of convolution to compute tu(t) * u(t); Check your answer using Laplace transform. 7. (40 pts) Solve the following ODE. y" 4y 4y = e-4*[u(t) - u(t - 1)] (0) = 0; y'(0)= -1
MATH 211 1. Verify that sint using the definition of the Laplace Transform. 2. Find the Laplace Transforms using the table and simplify your answers as much as possible. (a) g(t) = tsin 2t - 2tº (b) g(t) = 3tuſt - 3) 1b. (c) h(t) = cost. ut - ) (d) m(t) = e-uſt - 1)
5) For each of the following utility functions derive directly from the definition not using the formula(s) from class or the tert - the MRS (marginal rate of substitution) of y for x at the point (2,4) (a) ua(x,y)=xy (c) ue(z,y)=z"y (d) ua(x, y)-
** From the definition (3.16) of its density function, verify that an N(11,02) random variable has mean 3.8. and variance ?2 3.9. Show that N(x) in (3.18) satisfies N(a) +N(-a)
[&r, a1]7, that V2u = V.Vu = 6.4. Verify directly from the gradient operator that V ux+u-see Definition 6.5 Definition 6.5 (Two-Dimensional Heat or Diffusion Equation). Consider the open do- main (x,y) W. Using the continuity equation (1.4) the flux rule (6.13) yields DV u+R. (6.14) where V2u V.Vu u +lyy is the linear Laplacian operator The boundary conditions come in the three types: conditions on u, conditions on flux, and mixed as we are familiar with from Chapter 4....
Compute the following binomial probabilities directly from the formula for b(x; n, p). (Round your answers to three decimal places.) (a) b(3; 8, 0.3) (b) b(5; 8, 0.6) (c) P(3 ≤ X ≤ 5) when n = 7 and p = 0.65 (d) P(1 ≤ X) when n = 9 and p = 0.15