Consider the function f(x, y) = x^3 − 2xy + y^2 + 5.
(a) Find the equation for the tangent plane to the graph of z = f(x, y) at the point (2, 3, f(2, 3)).
(b) Calculate an estimate for the value f(2.1, 2.9) using the standard linear approximation of f at (2, 3).
(c) Find the normal line to the zero level surface of F(x, y, z) = f(x, y) − z at the point (2, 3, f(2, 3))
Consider the function f(x, y) = x^3 − 2xy + y^2 + 5. (a) Find the...
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x, y, z) = x2 + y2 + x2 = 2a?, (1) where a is a fixed real positive constant, and the point u = (0,a,a) on the surface f(x, y, z) = 2a. a) Find the gradient of f(x, y, z) at the point u. b) Calculate the normal derivative of f(x, y, 2) at u. c) Find the equation of the tangent plane...
Consider the surface given as a graph of the function g(x, y) = x∗y 2 ∗cos(y). The gradient of g represents the direction in which g increases the fastest. Notice that this is the direction in the xy plane corresponding to the steepest slope up the surface, with magnitude equal to the slope in that direction. 1. At the point (2, π), find the gradient, and explain what it means. 2. Use it to construct a vector in the tangent...
Please explain b! 2. Let z = f(x, y) = ln(4x2 + y2) (a) Use a linear approximation of the function z = f(x,y) at (0,1) to estimate f(0.1, 1.2) (b) Find a point P(a,b,c) on the graph of z = f(x, y) such that the tangent plane to the graph of z = f(x,y) at the point P is parallel to the plane 2x + 2y – 2=3
2. [3 marks] Consider the function f(x,y) = log (– 2y). (a) Find the partial derivatives (b) Find an equation of the tangent (plane) to the surface of f at the point (3,1, f (3,1)).
3 4. (4 pts) Consider the surface z = z = x²y + y3. (a) Find the normal direction of the tangent plane to the surface through (1,1,2). (b) Find the equation of the tangent plane in (a). (e) Determine the value a so that the vector 7= -7+27 +ak is parallel to the tangent plane in (a). (d) Find the equation of the tangent line to the level curve of the surface through (1,1).
Question 1. 30% Given the function f(x, y) = e 1. Specify the domain and range of f. 2. Describe the level curves off and graph the one that passes through the point (2,4). 3. Find the limit, if possible, when (x,y) approaches (0,0) of the function f(x,y). 4. Find the equation of the tangent plane and the normal line to surface defined by at the point (1,1,e). 5. We now let x = 12 and y = In 3t...
[Question 1] Find and graph the domain of the function f(,y)-In-) Question 2] Graph a contour map of the function f(z, y)2s y 1 that contains four level curves. Make sure to find an equation for each level curve and label each one on the graph. IQuestion 3] The equation of the tangeat plane to the function z the equation: Using the form of the equatioa above, fiud the tangent plane to f(a,y)yat the point (2. ). Question 4] Find...
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
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...
Check that the point (1,-1,2) lies on the given surface. Then, viewing the surface as a level surface for a function f(x,y,z), find a vector normal to the surface and an equation for the tangent plane to the surface at (1,-1,2). 2x^2-3y^2+z^2=3. Vector normal? and tangent plane?