2 + (a) Determine and sketch the domain of the function f(x, y) = (x2 +...
QUESTION 5 a) Find and sketch the domain of the function f(x,y) = \n(x2 - y +1) + VÝ +1. (5 marks) b) Evaluate eży sin(3x +2y) lim (x,y) (-2,3) 3x +2y (6 marks)
2. Find and sketch the domain of the function f(x,y) = V x2 – y + 2.
1. Sketch a few of the level curves of the function f(x, y) = surface z = y2 and then use these to graph the f (x, y) 2. Evaluate the following limits if they exist. If they don't, explain why not. (a lim (x,y)(0,0) + 4y2 x4-y4 (b lim (x,y)(0,0) x2 + y2 cos 2 y2) - 1 lim (c (z,y)(0,0 2ry (x, y)(0,0) Is the function f(x, y) continuous at (0,0)? 3 = (х, у) — (0,0) 2x2y...
Find the absolute maximum and minimum of the function f(x, y) -ry1 on the domain D (r, y),y 20, x2 +y2< 1) rty+1
Find the absolute maximum and minimum of the function f(x, y) -ry1 on the domain D (r, y),y 20, x2 +y2
(x + 1)2 Consider the function f(x) -. The first and second derivatives of f(x) are 1 + x2 2(1 – x2) 4x(x2 - 3) f'(x) = and f" (2) Using this information, (1 + x2) (1 + x2)3 (a) Find all relative extrema. (4 points) Minimum: Maximum: (b) Find the intervals of concavity for f(x) and identify any inflection points for yourself. (5 points) Concave up: Concave down: (c) Using the fact that lim f(x) = 1, and our...
Answer All Questions.
(9) Let f(x,y) = 1+ 4- y2. Evaluate f(3,1), find and sketch the domain of f. (10) A thin metal plate, located in the xy-plane, has temperature T(x,y) at the point (x,y). The level curves of T are called isothermals because at all points on such a curve the temperature is the same. Sketch some isothermals if the temperature function is given by 100 T(x, y) = 1 + x2 + 2y2 (11) Show that lim (z2...
1. Find the first and second partial derivatives: A. z=f(x,y) = x2y3 - 4x2 + x2y-20 B. z=f(x,y) = x+ y - 4x2 + x2y-20 2. Find w w w x2 - 4x-z-5xw + 6xyz2 + wx - wz+4 = 0 Given the surface F(x,y) = 3x2 - y2 + z2 = 0 3. Find an equation of the plane tangent to the surface at the point (-1,2,1) a. Find the gradient VF(x,y) b. Find an equation of the plane...
7. The function z = f(x,y)= x2 +2 12 is restricted to the domain x2 + y2 =1, a circle of radius 1. Determine the global extreme points and global extreme values using the Lagrange multipliers method.
Question 2 (20 points): Consider the functions f(x, y)-xe y sin y and g(x, y)-ys 1. Show f is differentiable in its domain 2. Compute the partial derivatives of g at (0,0) 3. Show that g is not differentiable at (0,0) 4. You are told that there is a function F : R2 → R with partial derivatives F(x,y) = x2 +4y and Fy(x, y 3x - y. Should you believe it? Explain why. (Hint: use Clairaut's theorem)
Question 2...
Evaluate the line integral of the function f(x,y)= (x+y2)/(sqrt(1+x2)) over the curve C: y=x^2/2 ; from (1,1/2) to (0,0)