EXAMPLE 3 Find EXAMPLE 3 Find vom av 2 ox. SOLUTION Let u= 1 – 182....
EXAMPLE 3 Find dx. 13 - 2x² SOLUTION Let u = 3 - 2x. Then du dx, so x dx du and 1 3 = 2x2 dx = = 1.I tu du 1 wrz du (27ū)+c 11 Il + C (in terms of x).
By using u substitution Find fav? + 1dc by using substitution. 1. Let u= 2. Then du = 3. Solve for a from part "1". x2 = (Answer needs to be in terms of u and du. ) 5. integrate, leave in terms of u. 4. Make the substitution into the integral. po vo? + ide = f"x++ido I 6. Change your answer in 5 so that it is in terms of a 1dx =
EXAMPLE 2 Find sin$(7x) cos”7x) dx. SOLUTION We could convert cos?(7x) to 1 - sin?(7x), but we would be left with an expression in terms of sin(7x) with no extra cos(7x) factor. Instead, we separate a single sine factor and rewrite the remaining sin" (7x) factor in terms of cos(7x): sin'(7x) cos”(7x) = (sinº(7x))2 cos(7x) sin(7x) = (1 - Cos?(7x))2 cos?(7x) sin(7x). in (7x) cos?(7x) and ich is which? Substituting u = cos(7x), we have du = -sin (3x) X...
Consider the second Galerkin Example (videos: GalerkinDiscrete-Example_1 to 3). Solve this example if u(0) = 0, du(2)/dx =0, and 0 ≤ x ≤2. Every single step must be shown. EXAMPLE Solve ODE using Galerkin method for two equal-length elements du u(0) = 0 +1 = 0, 0 < x < 1 dx2 du Boundary conditions (1) dx We know for three nodes: X2 = 0, X2=0.5, X3=1.0; displacement at nodes = Uy, U2, U3; length of elements L1=0.5, L2=0.5 -...
3. Find the derivative using the quotient rule. 2e* f(x) = x-1 4. Let u and y be differentiable functions of x. Find the value of the indicated derivative using the given information. Pay careful attention to notation. du Find dx v at x =1 if u(1) = 3, u'(1)=-5, v(1)=7, v'(1)=-3
(1 - r)dr- (1/2)2 3.1.1 For a bar with constant c but with decreasing f-|-x, find w(x) and u(x) as in equations (8-10). 3.1.2 For a hanging bar with constant f but weakening elasticity c(x)-1-, find the displacement u(x), The first step w even at x - I where there is no force. (The condition is w - c du/dx-0 at the free end, and (1 -x)f is the same as in (9), but there will be stretching c=0 allows...
U 12 1 . puy you tapi DU MIDU TOU DO 3. Let X have the pdf fx(x) = 33.52 Fr?e=22/B2, 0<I< for any B > 0. (a) Verify fx(x) is a pdf. (b) Find E(X) and Var(X). (c) Does My(t) exist? If so, find it.
Suppose U(x, y) = 4x2+ 3y2 1. Calculate ∂U/∂x, ∂U/∂y 2. Evaluate these partial derivatives at x= 1, y= 2 3. Calculate dy/dx for dU= 0, that is, what is the implied trade-off between x and y holding U constant? 4. Show U= 16 when x= 1, y= 2. 5. In what ratio must x and y change to hold U constant at 16 for movements away from x= 1, y= 2?
5. Given the initial-boundary value problem as below: ди ди at +u=k 0<x<1, 1>0, Ox?? Ou -(0,1) Ox Ou (1,t)=0, @x t>0, u(x,0) = x(1 - x) 0<x<1. where k is a non-zero positive constant. (i) By separation of variables, let the solution be in the form u(x,t) = X(x)T(t), show that one can obtain two differential equations for X(x) and T(t) as below: X"-cX = 0 and I' + (1 - ck)T = 0) where c is a constant....
DU .U . U U . 1). . . . 20. B={(-3, 2), (8, 4); and B' ={(-1.2), (2,-2); are two bases for R (a) Find the transition matrix from B' to B. (b) Find the transition matrix from B to B'. (c) let [V]8. = [-] find [V]