Question

30 6 9 Compute the slope of the line tangent to the 36 Consider the upper half of the ellipsoid f(x,y) = and the point P on the level curve f(x,y) - level curve at P, and verify that the tangent line is orthogonal to the gradient at that point. 245 A. The slope is 5 OB. The slope is undefined, so the tangent line is vertical Verify that the tangent line is orthogonal to the gradient at P Select the correct choice below and fill in the answer boxes to complete your choice. y O A. The slope of the tangent line, m, can be used to write a vector in the direction of the tangent line, (1,m). The dot product of this vector and the gradient CHis 0, so they are orthogonal. OB. The tangent line slope is undefined, so it is a vertical line. The gradient is horizontal, so it is orthogonal to the tangent line. ce

Answer 1

Given data be Function for upper half of ellipsoid flagg -yh 9 o 36 Tunction for level calne fenyl 30 6 From and We can obtain 1 equation of of sihtace 30 ㅋ 12 ye 36 6 Senare om both sides 2 2 30 - 22 9 36 ㅋ - 12 y2 30 9 36 36 36-42² - gz 30 36 36 7 36-42² - y2 30 36 X 36 36-4u - y2 20 o IT 422 + y2 – 36 +30 4x² + y² - 6 . 3

Dift cuntiating a gives d da (4x² + y2 - u) 3x + 2y leer 27 dy day da 2y ㅋ dy - 42 2 y be the slope At point (+ 55.), the slope of tangent be dy Ta yl -40 y 2 -4 ( 2 ) dy us 15 Hence, slope is ☺ (on) al 2 ſs 5

25 Ang vector propotional to ta (1 has slope 255 and points in the direction of the tangent line 5 - The gradient vector be - It (2,4) y 451-2/4-92/16 1651-27/4-92/16/ 55 If (2,5) ( ) (411-4) / 4.851/16 16/1-45/4-57% 4 Sr - 4 / 4-5 116 9651-14 14-5/16 7 04 (7,5) 7 of ( 2 ) ( 2 JF 4 1 2 1-1/16 16/17 21-1116) 706 (1,5) Cam - 55 2 x 4 x 0.214 16X0,217 -7 (95) (0.5, ---844