I have Five-questions for each question there's an exact example with a solution.

Question 1:

For the function ay) := 11:23:2 + 16:3y+ 9:2y+ 533 + 18y compute the following partial derivatives and evaluate them at the point (1,1): 6r _ amenEn? we III? 3+1 For the function f(x, y) := 14x y + 7x y+ 18x y+ 12x + 4y + 46 compute the following partial derivatives and evaluate them at the point (1, 1) : of (1, 1) = 121 of (1, 1) = dy 65 Feedback Computing the partial derivatives we find of - 28ry2 + 21x2y + 36xy + 36x2, of = 28x2y + 7x3+ 18x2+12y2. ay Evaluating at (1, 1) we obtain (1, 1) = 121, of (1, 1) = 65. ayFor the function f(x, y, z) = 4x y z + 5xy824+ 2xz2 + 6y624 + 10x9 + 17yall + 10ryz + 43 compute fryz and evaluate it at the point (1, 1, 1): fayz (1, 1, 1) =For the function f(x, y, z) = 7x y3z + 8xy824 + 18xz2 + 20y624 + 3x9 + 17yall + 10xyz + 47 compute fxyz and evaluate it at the point (1, 1, 1): fayz (1, 1, 1) = 371 Feedback Computing the indicated third partial derivative we have fxyz =105x y + 256y 23 + 10. Evaluating at the point (1, 1, 1) we obtain fayz (1, 1, 1) = 371.For the function f(x, y, z) := 10x3 + 10y + 52 + 12xy + 18yz + 6xz + 18 find the gradient at the point (1, 1, 1): Vf(1, 1, 1) = - Find the directional derivative of f at the point (1,1,1) in the direction of the vector (2,1,2): 86 enm. Notice that you might need to normalize the vector (2, 1, 2). To derivate in the direction of a given vector means to derivate along the vector of unit length pointing in the same direction. Feedback First we compute the partial derivatives of 3': f3 = 24:2 + 18y + 12z, fy = 30y2 + 182 + 12z, fz = 21;2 + 12y + 122. Then at the point (1,1,1) we have Vf(1,1, 1) = (54, so, 45). To nd the directional derivative along (2, 1, 2) we need to take the dot product Vf(1,1,1) - (2,1, 2) = (54,60,45) . (2,1,2) = 258, and divide by the length |(2, 1, 2)| = 3. The answer is 88. Given the composite function z = f(x, y), x = 9(u, v), y = h(u, v), where f(x, y) := 2x y+ 4x + 3xy + 3xy, 9 (u, v) := 4uv + 4u v + 4us, h(u, v) := 3u v2 + 4uv, compute the following partial derivatives evaluated at the given points 29 (1, 1) = du 2g (1, 1) = oh (1, 1) = du oh (1, 1) = of (9(1, 1), h(1, 1)) = of (9(1, 1), h(1, 1)) = Now use the chain rule to compute the partial derivatives of the composite function at the point (1, 1) : Oz (1, 1) = du oz (1, 1) =Given the composite function z = f(x, y), x = g(u, v), y = h(u, v), where f(x, y) := 3x y + 4x + 2xy + 2xy, g(u, v) := 5uv + 3u v + 375 h(u, v) := Au v + 4uv, compute the following partial derivatives evaluated at the given points 0g (1, 1) = du 29 0g (1, 1) = 13 oh (1, 1) = 16 oh (1, 1) = 12 of (9(1, 1), h(1, 1)) = 760 of (9(1, 1), h(1, 1)) = 737 Now use the chain rule to compute the partial derivatives of the composite function at the point (1, 1) : az (1, 1) = du 33832 Oz (1, 1) = 18724 Feedback Computing the partial derivatives we find of = 6xy + 8x+ 2y-+ 2y, Of = 3x2 + 4xy + 2x, ay ag = 502 + 9u2v + 15u*, 09 = 10uv + 3us, av ah =12u-v2 + 4v, oh = 8uv + 4u.Evaluating at (u, v) = (1, 1) we have of of g(1, 1) = 11, h(1, 1) = 8, (g(1, 1), h(1, 1)) = 760, ay (g(1, 1), h(1, 1)) = 737, ag ag oh ah au (1, 1) = 29, = 13, (1, 1) = 16, au (1, 1) = 12. av Then, using the chain rule, we find the partial derivatives of the composite function az of of oh (g(1, 1), h(1, 1)) . du (1, 1) = 760 . 29 + 737 . 16 = 33832, au (1, 1) = (1, 1) + ay (g(1, 1), h(1, 1)) . of of ah az ag an (1, 1) (g(1, 1), h(1, 1)) . an (1, 1) = 760 . 13 + 737 . 12 = 18724. av (1, 1) + dy (g(1, 1), h(1, 1)) .Using the equation 2x2 - 12x + 2y2 - 40y = 609, 318 to implicitly define y as a function of , compute the derivative of y with respect to x at the given point dy -(-549, 18) =Using the equation 322 542 + 33,2 60y = 9,531 to implicitly dene y as a function of z , compute the derivative of y with respect to x at the given point Feedback We use the following formula to compute the derivative: where F = 0 is the equation giving us the implicit dependence of 3; on 2. In our case F(z, y, z) = 332 542 + 3y2 60y 9, 561 Fz=6254, Fy=yG, and evaluating at ( 49, 12) we obtain 6'3; 4,12 =2. d2( 9 ) 9