Question

Need help on these homework problems images attached. Disregard the question title I don't know what happened. Thanks!

at P(1, 1,-1) in the direction of the vector v = ( 4, 5,-272). Homework 3 1. (14 pts) Find an equation of the tangent plane of the function at (x, y) = = (0,4). f(x,y) = e *sin (y) 2. (16 pts) Find the linear approximation of the function at (x,y) = (1, 1) and use it to approximate f(1.1,0.9). b.(4 pts) What is the maximum rate of change of potential and what is the direction in which it occurs? f(x, y) = 4xºy - 7xy? 5. (12 pts) Find an equation of the tangent plane to the ellipsoid at the point (-4,-1,5). 2x2 +8y2 + 3:2 = 115 3.(12 pts) Find the gradient vector of the function at (x, y, z) = (-1,2,2). F(x, y, z) = zln(x + y) + 4x3 + y2 + 2z

Answer 1

-0 12 Given, fexiy) = exsinly) of exsing i te teosty)] a aflora) = + 2 + 3 = < -* , 1 ) Equation of the tangent plane of the function frany) at (6, 7) is given by; -2 (1-0) + 3 (4- 7 ) = 0 - 2 + 3y - BA * x-33y + 3x - 0 -Ans. IF(x1y, Z) = z m (x+y) + 4x² + y² + 22 = ( 1 y 8 + 12x2) 1 + ( e ty (2+2y) } + (ln(x+y) + 2 * A DF (-11 21+2) = (2+2) 7+(2+4) 3+ ( Ance) +2) K = 142 +63 +2k Hence gradient vector of the function at (-1,2,2) is; of (-1,1 2,2) = 147 +67+ 2R 3 Griven TF - AAR or of(-1,212) = <141612)