Homework Answers
a) The power P dissipated in a resistor is given by P(V, R) = y, where...
a) The power P dissipated in a resistor is given by P(V, R) = y, where V is the voltage and Ris the resistance. If V = 156 Vand R = 412, find the change AP in Pand its approximation dPresulting from a change SV = 2 Vin V and a change of SR= -0.1812 in R. 13 marks b) Calculate all first- and second-order partial derivatives of the function f(x, y, z) = xy: cos(yz2). 12 marks
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given ...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
8.) (10 Points) Given the contour diagram z = f(x,y). 2 1 2 3 4 -2...
8.) (10 Points) Given the contour diagram z = f(x,y). 2 1 2 3 4 -2 R a. Find i. f(-1,1) 11. a value of x for which f(x, 1) = 3 iii. a value of y for which f(0,y) = -2 b. The given graph has a local maximum value. At which point (x,y) does this occur? c. Determine the sign (positive or negative) of the following partial derivatives. i. (1,0) ii. fy(0,1)
a. Use the Chain Rule to find the indicated partial derivatives. z = x4 + x2y, x...
a. Use the Chain Rule to find the indicated partial derivatives. z = x4 + x2y, x = s + 2t − u, y = stu2; ∂z ∂s ∂z ∂t ∂z ∂u when s = 1, t = 2, u = 3 b. Use the Chain Rule to find the indicated partial derivatives. w = xy + yz + zx, x = r cos(θ), y = r sin(θ), z = rθ; ∂w ∂r ∂w ∂θ when r = 8, θ = pi/2 c. Use the...
emergency w R; 20 V 200 E + a) For the circuit shown, By using Millman's...
emergency
w R; 20 V 200 E + a) For the circuit shown, By using Millman's Theorem: i. Find the power dissipated by Ru. ii. What is the current in RL if R, is an open circuit? R 2000 E3 10 V E: 400 V R 1000 b) Find the equivalent resistance Rab between terminals a and b. w 1 ohm 3 ohm 3 ohm 3 ohm 3 ohm 3 ohm c) Find the value of R for maximum power...
*Let f : R2 -R be given by z, y)(0,0 r, y)- 2y and f(0,0) =...
*Let f : R2 -R be given by z, y)(0,0 r, y)- 2y and f(0,0) = 0. (a) Decide if both partial derivatives of f exist at (0, 0) (b) Decide if f has directional derivatives along all v R2 and if so compute these. (c) Decide if f is Fréchet differentiable at (0, 0)? (d) What can you infer about the continuity of the partial derivatives at (0, 0)? て
1. Given a vector V with x- and y- components Vx = -6.2 and Vy =...
1. Given a vector V with x- and y- components Vx = -6.2 and Vy = 2.9. What is the magnitude of vector V? 2. Given a vector V with x- and y- components Vx = -4.5 and Vy = 3. What is the direction of vector V with respect to the +x-axis counterclockwise?
5. Consider the function f: R -> R given by f (x, y) := e°+v* _...
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...
9-2. Five items are to be loaded in a vessel. The weight w, volume v, and value r for item i are ...
9-2. Five items are to be loaded in a vessel. The weight w, volume v, and value r for item i are tabulated below. Item i Unit weight, w, (tons)Unit volume, v, (yd) Unit worth, r, ($100) 4 The maximum allowable cargo weight and volume are 112 tons and 109 yd" respectively. Formulate the ILP model, and find the most valuable cargo. hatf fll and 7 empty You would like to
9-2. Five items are to be loaded in a...
4) A general intermolecular pair potential is given by: w(r) = A/r12 – B/r6 = 4ε[(σ/r)12...
4) A general intermolecular pair potential is given by: w(r) = A/r12 – B/r6 = 4ε[(σ/r)12 – (σ/r)6] a) What is the potential energy at r = σ b) In terms of σ, where does the minimum energy occur? c) What is the minimum energy (in terms of ε)? d) At the minimum energy, what is the van der Waals contribution?
a) The power P dissipated in a resistor is given by P(V, R) = y, where V is the voltage and Ris the resistance. If V = 156 Vand R = 412, find the change AP in Pand its approximation dPresulting from a change SV = 2 Vin V and a change of SR= -0.1812 in R. 13 marks b) Calculate all first- and second-order partial derivatives of the function f(x, y, z) = xy: cos(yz2). 12 marks
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
8.) (10 Points) Given the contour diagram z = f(x,y). 2 1 2 3 4 -2 R a. Find i. f(-1,1) 11. a value of x for which f(x, 1) = 3 iii. a value of y for which f(0,y) = -2 b. The given graph has a local maximum value. At which point (x,y) does this occur? c. Determine the sign (positive or negative) of the following partial derivatives. i. (1,0) ii. fy(0,1)
emergency
w R; 20 V 200 E + a) For the circuit shown, By using Millman's Theorem: i. Find the power dissipated by Ru. ii. What is the current in RL if R, is an open circuit? R 2000 E3 10 V E: 400 V R 1000 b) Find the equivalent resistance Rab between terminals a and b. w 1 ohm 3 ohm 3 ohm 3 ohm 3 ohm 3 ohm c) Find the value of R for maximum power...
*Let f : R2 -R be given by z, y)(0,0 r, y)- 2y and f(0,0) = 0. (a) Decide if both partial derivatives of f exist at (0, 0) (b) Decide if f has directional derivatives along all v R2 and if so compute these. (c) Decide if f is Fréchet differentiable at (0, 0)? (d) What can you infer about the continuity of the partial derivatives at (0, 0)? て
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...
9-2. Five items are to be loaded in a vessel. The weight w, volume v, and value r for item i are tabulated below. Item i Unit weight, w, (tons)Unit volume, v, (yd) Unit worth, r, ($100) 4 The maximum allowable cargo weight and volume are 112 tons and 109 yd" respectively. Formulate the ILP model, and find the most valuable cargo. hatf fll and 7 empty You would like to
9-2. Five items are to be loaded in a...
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