Problem 2. Consider the domain S defined by 1<21 < 3. Notice – 2 and 2...
Find the Laplace transform F(s) - {0} of the function: f(t) = 1-21 0314 2-34 4 <t<6 14 6 by splitting the integral into three pieces. Enter your answers in order of increasing domain.
2. Consider the cubic spline for a function f on [0, 2] defined by S(x) = { ={ (z. 2x3 + ax2 + rx +1 if 0 < x <1 (x - 1)3 + c(x - 1)2 + d(x - 1) + ß if 1 < x < 2 where r, c and d are constants. Find f'(0) and f'(2), if it is a clamped cubic spline.
3. Consider the vector-valued function: r(t) = Vt +1 i + pi a. State the domain of this function (using interval notation). b. Find the open intervals on which the curve traced out by this vector-valued function is smooth. Show all work, including r 't), the domain of r', and the other required steps. c. Provide a careful sketch of the path traced out by this function below. Include at least 3 points on the graph of this function. Assume...
(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)
10:53 homework7 11 Homework7: Problem 11 Previous Problem List Next (1 point) Consider the function if0<t<2 a. Use the graph of this function to write it in terms of the Heaviside function. Use h(t - a for the Heaviside function shifted a units horizontally f(t) help (formulas) b. Find the Laplace transform 0. F(s) = L U(t)) for s help (formulas) Note: You can earn partial credit on this problemm. Pr
Question 1 Consider a function f (2) with domain 3 << 7 and range -3 < f (x) < 9 a) Find the domain and range of g (x) = f (5x). Domain: <3< Range: 59(2) b) Find the domain and range of -h (2) + 7. Domain: sos Range: <9() <
Problem 3: Consider a continuous function x(t), defined for t 0. The Laplace Transform (LT) for x(t) is defined as: X(s) - Ix(t)e-st dt. Derive the following properties: a) LT(6(t))-1, the ?(t) is the Dirac-delta function b) LT(u(t))-1/s, where u(t) is the unit-step function c) LT(sin(wt))-u/(s2 + ?2) d) LT(x(t-t)u(t-t)) = e-stx(s), ? > 0. e LT(tx)-4x(s).
A periodic function f(x) with period 21 is defined by: X + -1<x< 0 2 f(x) = 0<x< 2 Determine the Fourier expansion of the periodic function f(x). X - TT
1. Consider the following initial-value problem. s y' = e(1+B)t In(1 + y2), 0<t<1 y (0) = a +1 a) b) t=0.5. Determine the existence and uniqueness of the solution. Use Euler's method with h = 0.25 to approximate the solution at
2. For the circuit in problem 1 above: a) Transform the circuit into the s-domain b) Using Laplace transform techniques applied directly to the circuit (not applied to the differential equation found in problem 1), find iz(t), t > 0. No credit for time- domain techniques. IX V 40 + - 5+10u(t) 10 H 1/4 F 2. For the circuit in problem 1 above: a) Transform the circuit into the s-domain b) Using Laplace transform techniques applied directly to the...