5) The level curves of a function f(x,y) are given in the graph below. 2 X...
rty. I 5. [16 pointsj Consider the function f(x, y,z) Let S denote the level surface consisting of all points in space such that f(,y,z)-4, and let P- (2,-2,1), which is on S. a) Calculate Vf. b) Determine the maximum value of Daf(P), where u is any unit vector at P c) Find the angle between Vfp and PO, where O denotes the origin. d) Find an equation for the tangent plane to S at P rty. I 5. [16...
5. Consider the function f: R -> R given by f (x, y) := e°+v* _ 4. (a) Sketch the level curves of f. (5 marks) (b) Find Vf, the gradient of f, and determine at which points Vf is zero. Remark: These points are called the critical points of f (5 marks) (c) Determine whether the critical points of f are local minima, local maxima, or saddle points by considering the level curves of f. (5 marks) (d) Calculate...
5. These are the level curves of a smoothly varying function f(x, y) 4 (a) At P is fx positive or negative? Why? (b) At P is fxx positive or negative? Why? (c) Sketch Vf at P and Q on the level curve f(x, y) = 3. 5. These are the level curves of a smoothly varying function f(x, y) 4 (a) At P is fx positive or negative? Why? (b) At P is fxx positive or negative? Why? (c)...
e ana the gradient IV.1 The level curves of the function z fix, y) are sketched in the figure below: 20 50 100 10 150 10 20 30 Let u= (l,-) and v=죠(1,1) Estimate the derivatives at the indicated point: DJい IV.2 (The Directional Derivative) Compute the derivative of the function f(x,y,z)-sin(2x-y)+ cos(2y-2) Ft%r,-r) in the direction of the vector". (-2,-1, 2) at the point e ana the gradient IV.1 The level curves of the function z fix, y) are...
2. (20pt.) Let f(x, y) 2 (a) draw and describe the level curves of f of value c= 0,1,2,4 (b) sketch the graph of f (r.y) 2. (20pt.) Let f(x, y) 2 (a) draw and describe the level curves of f of value c= 0,1,2,4 (b) sketch the graph of f (r.y)
7) Let D be the domain described by -1 <y<1, x 2-1, r< y2 i) Sketch the domain D. neither? Is D horizontally simple, vertically simple, ii) Calculate the three integrals a = ffpxdxdy, b = SSpydxdy and iv) What physical meaning does the quantity What physical meaning does the point (a/c, b/c) have, related to the region D? or SSp drdy have, related to the region D? i) C= v) 5) The level curves of a function f(r,y) are...
[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...
7 Olet f(x,y)= -(--) Cy-x)+2 a) sketch flx,y) b) Draw the level curves f(x, y) = for t=2, 10, 1-2 08 c) Compute f (33) what point represents this computations, what are the signs (²) of fx (3,3), ty (3,3), +*x (3,3), toy (3,3)? point part b) ? a d) Without and
5. Given to the right is the graph of the function f(r, y) ye/2 on the domain 0r4 and 0 ys 4 (a) Use the graph to the right to rank the follow ing quantities in order from smallest to largest: f(3,2), f(1,2), f,(3,2), f,(1,2), 0 (b) Use algebra to calculate the numerical values of the four derivatives from part (a) and confirm that your answer to part (a) is correct. 5. Given to the right is the graph of...
2 Suppose that f(x, y) = - and the region D is given by {(x, y) |1<<3,3 <y < 6}. y D Q Then the double integral of f(x, y) over D is S1,612,)dady