
6) Consider a function g(x,y) satisfying g(2,3)=-1 and g(1.98, 3.03) = -0.97. Calculate an approximate value...
6) Consider a function g(x,y) satisfying g(2,4)=-4 and g(2.02, 3.96)=-4.04 Calculate an approximate value of the derivative of this function at the point P(2,4) in the direction of ü=-31 +6j. (15P)
5) Let P(1,2,2) be a point, and f(x,y,z) and g(x,y,z) be two differentiable functions satisfying the following conditions. 1) f(P)=1 and g(P)=4 og IT) = -2 Oz IP III) The direction in which f increases most rapidly at the point Pis ū=4i - +8k , and the derivative in this direction is 3. IV) Equation of the plane tangent to the surface f(x,y,z)+3g(x,y,z)=13 at the int P is x+4y + 5z =19 According to this, calculate og Ox . (20P)
2) Show that a Green's function G(x,y) satisfying the problem a2G = 8(x - y), G (0,y) = 6,(1, y) = 0 does not exist, but a modified Green's function Ĝ(x,y) satisfying a2G 22 = (x - y) -1, G.(0,y)=G.(1,y) = 0 does. How would you use G to solve problem (1) when f satisfies the condition that you found for a solution to exist? Hint: is f(x) = f(u) (8(x - y) - 1) dy?
Problem 7. [13 points; 4, 4, 5.] Consider the function f(r, y) 2y ln(r- ). (i) Find the unit direction of steepest increase for f at the point P (2, 1) (ii) Find the directional derivative of f at the point P(2,1) in the direction u = S (iii) Linearly approximate the value f((2,1)00)
Problem 7. [13 points; 4, 4, 5.] Consider the function f(r, y) 2y ln(r- ). (i) Find the unit direction of steepest increase for f at...
Consider the function g(x,y)=5.7sinx−7.9y2+8. Find the directional derivative of g in the direction v⃗ =(56) at the point P=(0,1). Answer:
(5 marks) Consider a smooth function u(x, y) satisfying: Show that u attains its maximum on the boundary àS2
(5 marks) Consider a smooth function u(x, y) satisfying: Show that u attains its maximum on the boundary àS2
Consider the following function 6 f(x, y,z)=z - x? cos(my) + xy? (i) Find the gradient of the function f(x, y, z) at the point P,(2,-1,-7). (ii) Find the directional derivative of f(x, y, z) at P,(2,-1,-7) along the direction of the vector ū = 2î+j+2k. (iii) Find the equation of the tangent plane to the surface given below at the point P,(2,-1, -7). 6 :- xcos(ty) + = 0 xy
6. For a given function f(x, y), is noted that at the point P(1,1) the directional derivative in the direction towards (0,0) is 1, while the directional derivative towards (1.2) is -1. Find andf at
6. For a given function f(x, y), is noted that at the point P(1,1) the directional derivative in the direction towards (0,0) is 1, while the directional derivative towards (1.2) is -1. Find andf at
1 1 Consider the function f(x.y,z) 2x y 2 the point P(3,0,1), and the unit vector u 0 Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P c. Find the rate of change of the function in the direction of maximum increase at P d. Find the directional derivative at P in the direction of the given vector. a.
1 1 Consider the function...
(a) Find the following derivative: d In (5.y dz dB (6 marks) (b) Consider the following bivariate function: g(x,y)exp 8B In (4x2y2)] Find the elasticity of g with respect to x (6 marks)
(a) Find the following derivative: d In (5.y dz dB (6 marks) (b) Consider the following bivariate function: g(x,y)exp 8B In (4x2y2)] Find the elasticity of g with respect to x (6 marks)