1. (25 Points) be given. x(y-1) Let f(x,y) = _ x2+4(9–132 ; (x,y) = (0,1) à...
Cal 4 , ) and use this to 6. Let f(x,y) = x2 + y2 + 2x + y. (a) Find all critical points of f in the disk {(x,y) : x2 + y2 < 4). Use the second derivative test to determine if these points correspond to a local maximum, local minimum, or saddle point. (b) Use Lagrange multipliers to find the absolute maximum/minimum values of f(x, y) on the circle a2 +y -4, as well as the points...
1. (25 points) Let f (x, y) = x4 - 4xy + y2 (6 points) Find fr.fy a. b. (9 points) Find fxx fry, fry c. (6 points) Find all critical points. 1. (25 points) Let f (x, y) = x4 - 4xy + y2 (6 points) Find fr.fy a. b. (9 points) Find fxx fry, fry c. (6 points) Find all critical points.
Let f(x,y) = ((9/64)(root 2))(x2)(root y ), for 0 ≤ x ≤ 2, and 0 ≤ y ≤ 2, and f(x,y) = 0 otherwise. a) Determine if X and Y are independent. Explain your answer. b) Find P( X ≤ 1). c) Find P( Y ≤ 1). d) Find P( X + Y ≤ 1).
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...
8. Let y = x2 cos x, Find y' 9. Let g(x) = -2 cos x, Find g'(x) 10. Find F(x) = (4x + 3)5, Find F'(x) BONUS QUESTION (15 POINTS Let y = (4x - 3)(x - 1)5; Find y"
Let fx=x2-x-2(x2-4) if x≠±2c if x=2 Find c that would make f continuous at 1. For such c, prove that f is continuous at 1 using an ε-δ proof. x2-x-2 с 1. (10 marks) Let f(x) = (x2-4) if x # +2 if x = 2 Find c that would make f continuous at 1. For such c, prove that f is continuous at I using an E-8 proof.
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an Suppose f is a continuous and differentiable function on...
5.1 (10 points): Let f(x,y) = 4 – 22 – y? Find all extrema (both relative and absolute) on the square D = {(x, y): 0 535 2,0 Sy <2}. 5.2 (10 points): Let f(x,y) = ry–2x+3y+100. Classify all critical points (rela- tive minimum, relative maximum, saddle point), and find the absolute maximum and absolute minimum on the triangle enclosed by the lines x = -4, y = 4, and y=++3.
Let f : [0, 1] x [0, 1] → R be defined by f(x,y) - 1 if y=%, 0 if y#x2 Show that f is integrable on [0,1] [0,1]. You may take the previous problem as given
Given f : R → R4, f(x, y, z) = (x2 + y2, 2, x + 2, y2 + x3) (Hint: f is not linear) (a) (1 point) Find the kernel of f. (b) (1 point) Find the range of f. (c) (1 point) Is f is 1-1? Is f onto? Justify your answers.