using
the given information the problem is solved, do comment for any
difficulty.
Find the centroid of the infinite lamina with > 1 and 0 <y<r-3 with density 8(x, y) = 1.
Let y'(x)y(x)g'(x) = g(x)g'(x), y(0) = 0, x e í, where f'(x) denotes ar(X) and g(x) is a given non- 4. dx constant differentiable function on R with g(0) = g(2) = 0. Then find the value of y(2)
Let y'(x)y(x)g'(x) = g(x)g'(x), y(0) = 0, x e í, where f'(x) denotes ar(X) and g(x) is a given non- 4. dx constant differentiable function on R with g(0) = g(2) = 0. Then find the value of y(2)
1. Find the area and also the centroid of (a) ((x, y): x' sysx+2)
1. Find the area and also the centroid of (a) ((x, y): x' sysx+2)
1. Let f : [0, 1]2 → R be given by: 1 f(x,y) -»-< if x = 0 if x + y Show that f is integrable on [0, 1]2 and compute the value of the integral.
S-1. Consider the functions f: RR defined by f(x, y, 2) 2-y2- and g(r,y,) -2 Describe the sets of regular values of f and g, respectively. Which of the sets f-1(0), g(0), and g (1) are regular surfaces? Describe the images of f10), 0)\(0,0,0)), and g(1) under the Gauss-map.
S-1. Consider the functions f: RR defined by f(x, y, 2) 2-y2- and g(r,y,) -2 Describe the sets of regular values of f and g, respectively. Which of the sets f-1(0),...
Question 12
11. Show that if F is continuous on Rn and F(X + Y) = F(X) + F(Y) for all X : in R", then A is linear. HINT: The rational numbers are dense in the reals. 12. Find F and JF. Then find an affine transformation G such that F(X)-G(Y) lim =0. T x2+y+2z (a) F(x, y,z)coxy. Xo- (1,-1,0) e*yz ex cos y (b) Fe*sin y 1, xo=(0, π/2) 13. Find F. g1 (x)
11. Show that if...
Find the centroid of the region bounded by y = {x + Ź, y = x”, and x = 1 Find the centroid of the region bounded by (x - 2)2 + (y + 3)2 = 25.
Suppose that f(x, y) = cx, for 0 y x 2. (a) Find c. (b) Find P(x > 1 and Y < (c) Find the marginal pdf of X. (d) Find the conditional pdf of Y given that X = x. (e) Find E[Y IX x (f) Find E[E[YX]]. (g) Find Cov(X, Y) (h) Are X and Y independent?
Suppose that f(x, y) = cx, for 0 y x 2. (a) Find c. (b) Find P(x > 1 and Y
1. Let f(x,y) = (2-7-% and g(x,y) = v f(x,y). J(1)(4 points) Find the maximum value of g(y). |(272 points) At which point(s) (x,y) and in the direction of which unit vector(s) ů does the maximum value for the directional derivative Dif(x,y) occur?
1. Sketch the vector field F x, y+(y-x)j F(x, y x, y f 2. Find the gradient vector field of f(r, y)-xe"
1. Sketch the vector field F x, y+(y-x)j F(x, y x, y f 2. Find the gradient vector field of f(r, y)-xe"