Problem 1.25 Calculate the Laplacian of the following functions: (a) Ta = x2 + 2xy +...
Problem for the following potentials, calculate Laplacis first a Tollowing potentials, calculate Laplacian first and then calculate volume density at respective points. cart (a) V= 4yz P (1,2,3) X²V=? x²+1 el (b) V= 5p² cos & P (3, Ţ; 2) - C) V= 20 (0.5, sao) om V= (A p'+ Bp") sin 4% P (05, 45,3) LOODUUUUU -oviE E Juin V = 4 al spherical condenaly) PROR 2 : Find
10. Use the limit definition of the derivative to calculate the derivatives of the following functions. a. f(x) = 2x2 – 3x + 4 b. g(x) = = x2 +1 1 x2 +1 c. h(x) = 3x - 2 a. 11. Find the derivative with respect to x. x² - 4x f(x)= b. y = sec v c. 5x2 – 2xy + 7y2 = 0 1+cos x 1-cosx cos(Inu) e. S(x) = du 1+1 + + f. y =sin(x+y) g....
Problem 1.11 Find the gradients of the following functions: (a) f(x, y, z) x2 + y3 + z4. (b) f(x, y,-x23-4 (c) f(x, y, z) e* sin(y) In(z).
Solve the following differential equations (a) dy – (1 – x2)(1+y?) (b) (2xy + 1) + (x2 + 3y2) dy = 0 (c) com + 4y= 22
solution for all 4 please
In Problems 1-3, solve the given DE or IVP (Initial-Value Problem). [First, you need to determine what type of DE it is. 1. (2xy + cos y) dx + (x2 – x sin y – 2y) dy = 0. 1 dy 2. + cos2 - 2.cy y(y + sin x), y(0) = 1. + y2 dc 3. [2xy cos (2²y) – sin x) dx + x2 cos (x²y) dy = 0. (1+y! x" y® is...
Question 4(25 marks) Find the critical points of following function, then determine whether they are relative maximum, relative minimum or saddle points i. f(x,y) 3x2-2xyy2- 8y [Smarks] [5marks] [5marks] iii. f(x,y)--2x + 4y-x2-4y2 + 9 b) Find the divergence and curl of the following vector fields i. F(x, y, z) = x2 yi + 2y3zj + 3zk [5marks] ii. F(x, y,z) x sin y i+4xyz j - cos 3z k [5marks]
Q1: 4pnts Evaluate the following integrals along the given curve C. (a) (32) ds. C : The section of the parabola y = x2 from the origin to the point (3,9) (b) yds,C:2 4 with y 20 0S152 (c) / C:x=cos t, y = sin t, z = t, ysin z ds, 0 t〈2π C : x e-t cos t, y = e-t sin t, z = e-t,
Q1: 4pnts Evaluate the following integrals along the given curve C. (a)...
Find the gradient and hessian of each of the following functions and use the hessians to check whether the functions are convex. (b) f:R2 + R given by f(x,y) = x3 – 2xy - y6. (c) f: R3 + R given by f(x1, x2, x3) = cos(x1) + 222} eigenvalues of the hessian). (you may use a computer to find the
(d) The line integral [(x+y?)dx + (x2 + 2xy)dy, where the positively oriented curve C is the boundary of the region in the first quadrant determined by the graphs of x=0, y=x2 and y=1, can be converted to A 2xdydx 0 0 BJ 2 xdxdy 0 0 С -2x)dyda 00 D none of the above (e) Consider finding the maximum and minimum values of the function f(x, y) = x + y2 - 4x + 4y subject to the constraint...
1. (a) Write the following two functions in terms of , and say where they are holomorphic: i. x2 + y2 - y - 2+ ix ii. (x + 7txt) +i(y-2) (b) Show that the function f(x) = eri-y (cos(2«ry)+isin(2xy)) is entire, and find its deriva- tive. Write it in terms of .