Copy of R - 25x<2, -2sys 2 Consider the functions f(x,y)= 3 - 3 xy -...
4. Let f(x, y) = 2 - 2x – y + xy. (a) Find the directional derivative of f at the point (2,1) in the direction (-1,1). [2] (b) Find all the critical points of the function f and classify them as local extrema, saddle points, etc. [2]
pleease solve them for me now pleeeaaase some of them have multiple answers x²-v If L= lim (x, y) + (0,0) X x+v then along y = mx, so limit exist L 1-mo 1+ m2 along path x = my, and limit does not exist L = m2 -1 0 m +1 along the path X = my and limit does not exist L m-1 0 m + 1 along y=mx? and limit does not exist L 1.-mo 1 +...
2. [8 pts] Consider the region R enclosed by the graphs of functions f(x) = 2 – 22 – 2x + 3 and g(x) = -2 +3 + 5 with points of intersection (-1,3) and (2,3), as shown in the figure. (a) Set up but do not evaluate the integral that repre- sents the volume of the solid resulting from revolving the region R about the vertical line r = 3. NA (b) Set up but do not evaluate the...
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x + y) (a) Prove that f and g are differentiable everywhere. You may use the theorem you stated in (b) Call F-fog. Properly use the Chain Rule to prove that F is differentiable at the point question (1c). (1,1), and write F'(1, 1) as a Jacobian matrix. 4. Consider the functions f : R2 R2 and...
Consider the system of coupled ODES: x' = x - y, y = x + xy - 6y (+) (a) Find the critical points (C+, Y*) € R2 of this system. [3 marks] Hint: One critical point is (0,0) and there are two more critical points. (b) For each critical point, find the approximate linear ODE system that is valid in a small neighbourhood of it. [6 marks] (c) Find the eigenvalues of each of the linear systems found in...
Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolating polynomial determined by m equidistant interpolation points, (2) an interpolating polynomial determined by interpolation at the m zeros of the Chebyshev polynomial T_m(x), and (3) by interpolating by cubic splines instead of by a polynomial. Estimate the approximation error by evaluation max_i |f(z_i)-p(z_i)| for many points z_i on [-1,1]. For instance, you could use 10m points z_i. The cubic spline interpolant can be determined in...
log(2 - 2) Consider the function f(x, y,z) (a) What is the maximal domain off? (Write your answer in set notation.) Find ▽f. (b) Find the tangent hyperplanes Ta2.1,f(r, y, 2) and To-ef(r, y, 2). Find the intersection (c) On (z, y, z)-axes, draw arrows representing the vector field F = Vf at the points (1,0,1), (d) Find the level set of f which has value ("height") wo 0, and describe it in words and of these two hyperplanes, and...
1.The region R is the region bounded by the functions y=x-3 and x=1+y^2. find the volume of the solid obtained by rotating the region R about the y axis. Please include a graph. 2.Find the volume of the solid obtained by rotating the region bounded by the graphs of y=x and y=sqrt(x) about the line x=2. Please include a graph
3. Problems 2.3 Suppose that f(x,y)=xy, with the constraint that x and y are constrained to sum to 1. That is, x +y = 1 Given this constraint, which of the following functions of x is equivalent to the original functionfx,y)=xy? f (x) = x2 f (x) = 1-r f (x) -x-x2 f(x) = x + x2 using the first order condition that f . (x) = 0, the value of x that maximizes f(x) (andfx,y)) is x- corresponding value...
C) Find the absolute maxima and minima of the function f (x, y) = xy-y^2 over the region -2 < x < 2, -5 < y < 5. on the square -2Srs 2,-5US5 1. (10 points) For the function fa.)y- (a) (4 points) Shotch the region described by the inequalities -ss2,-5 svs5 Label the boundaries of the region and write down their equations (b) (5 points) Find and classify all the interior critical points of fe, y) as local maxima...