Mathematics 21a Lecture : Problem Session Notes (Lagrange Multipliers, Extrema)

(1 point) Suppose that x(s, t) = 4s2-2t2, y a function of (s, t) with y(1, 1) = 1 and (1, 1) = 4 Suppose that u-xy, v a function of z, y with (2, 1)-1 Now suppose that f(s, t) -u(x(s, t), y(s, t)) and g(s, t)-v(r(s,t), y(s, t)). You are giver (1,1) 4, (1,1)=5. The value of (1,1) must be (a) -7 (b) -6 (c) -5 (d) -4 (e) 0

Consider F and C below. F(x, y)i 2y arctan(x) j 1 + X c: r(t) = t2 i + 3t j, 0 t 1 (a) Find a function f such that F = Vf f(x, y) = |y?arctan(x) 2 (b) Use part (a) to evaluate V. dr along the given curve C.

(3) Use Laplace transform to solve the following IVP for x = x(t): 1-5 1-3 | z,z(0)= (ii)z'= | 2 1-2 |x,x(0)=-1 ti(t) Note: In (i), r(t-1 -r2 (t) , l whereas in (ii), r(t) = | r2(t) | . 3(t)