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Homework 10 1. Suppose f is continuous on (a, b] and lim f (x) exists. Show that x→a+ R ∞ sin2 x result to show that 0 x2 dx exists. Rb a f (x) exists. Use this 2. Determine whether each integral is convergent or not. Evaluate those that are convergent. R∞ x (a) 0 e2xe +3 dx. R∞ (b) −∞ cos πtdt. R∞ 1 dv. (c) 2 v2 +2v−3 3. Find all the first partial derivatives of the following functions. (a) g(x, y, z) = exz sin(y/z). (b) s(u, v) = (c) f (x, y) = 4. Find ∂z ∂x u+2v u2 +v2 . Ry x and 2 et dt. ∂2z ∂x∂y at (0, 0) if z is defined implicitly as a function of x and y by xy 2 z 3 + x3 y 2 z = x + y + z. 5. Find the tangent plane of the given surfaces at the specified points. (a) z = ex cos y at (0, 0, 1). (b) z = 3x2 − y 2 + 2x at (1, −2, 1). (c) xy + yz + zx = 3 at (1, 1, 1). (d) sin(xyz) = x + 2y + 3z at (2, −1, 0). 6. If z = f (u, v), where u = xy, v = y/x, and f has continuous second partial derivatives, show that x2 2 ∂2z ∂2z ∂z 2∂ z − y = −4uv + 2v . 2 2 ∂x ∂y ∂u∂v ∂v 7. Find the gradient and the Hessian matrix for the following functions at the specified points. (a) f (x, y, z) = x3 y 2 z at (1, 1, 1). (b) v = r cos(s + 2t) at (1, 0, π2 ). (c) z = 4x3 − xy 2 at (−1, 1). 1