PROBLEM (7):Find the outward drawn unit normal vector n to the surface(x-1)+y?+(z+2) = 9 at the point ( 3, 1,-4)xy+2 x z = 4 at the point (2,- 2, 3)(i)%3D(ii)

To find- Find the outward drawn unit normal vector n^ to the surface x - 12 + y2 + z + 22 = 9 at…

Answered: PROBLEM (7): Find the outward drawn…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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8. The temperature in °C on the surface of a metal plate is given by T(x, y)= 20-4x² - y² , where x and y %3D are measured in inches. a) In what direction from (2,-3) does the temperature increase most rapidly? What is this rate of increase? b) Provide a contour map and draw the vector that indicates the direction of most rapid increase at (2,-3). c) What is the hottest point on the plate? Note that the vector does not point toward the hottest point on the plate. That is because the gradient provides only a local solution to finding a maximum increase.
O (a) O (b) O () Question 2 1 pts The unit vector tangent to the curve r(t) =(1,41²,3t² ) is (a) (0,4/5,3/5). (b) (0,8t,61) . O (a) O () O (b) Question 3 1 pts The vectors T and N, when defined,
orthogonal to the vector y = (2)? You may want to go through the proof of When is the vector x = projecting y on x and the implications of (intuition behind) "orthogonality".
(b) Let r(t)ti-j+(4-e) k, then the vector equation of the line tangent to the graph of r(t) at the point (4, 1,0) on the curve is A.r-(4i+ j) + t(-4i+j+4k) B. r-(4i- 3) + t(-41+j+ 4k) (e) The
4. (a) (b) (c) Consider f(x, y, z) = 2²ev - 2xy. Find the equation of the level surface through the point (0, 1, -1). Find a vector normal to this level surface at the point (0, 1,-1). Is the function increasing or decreasing in the direction of the vector you found in part (b)?
(a) For a given vector a + 0, describe the loci (the set of points 7 determined by the equations: (x, y, z)) on R³ (i) d (2—2, у+1, 2— 3) 3D 0. (ii) ||7– a|| 21. Z-. (b) A person throws a paper ball from a height of 4m from the ground. The wind is blowing and the acceleration is given by 2/3 a(t) = ( 12v3t? – 24t + (12/5P - ) -6 for t> 0. 3 7 (0), is thrown with an initial velocity of magnitude 4m/s at an angle 0 = rad with respect to the horizontal. All measurements are in SI units. me 3 rad 4 m basket
(2) Write parametrizations 7(u, v) for the following surfaces (a) a sphere with radius 5 (b) a cone (c) a cylinder (3) Write the tangent vectors r, and r, for the parameterizations in the last problem.
Question 4 Find r'(t) given the following vector function. r(t) = 2t³i+3t+j+37²k ○ r'(t)= 6t²i+12t³j+6tk ○ r'(t)= 6t²i+3t³j+3tk ○ r'(t) = 2t²i+3t³j+3tk ○ r'(t) = 2t²i+3tj+3t³k ○ r'(t) = 6t³i+12tªj + 6t²k Question 5 Find r'(t) r" (t) given the following vector function.
(5) Please explain in words how a student would determine the normal vector to the direction vector D[2, -1]

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PROBLEM (7): Find the outward drawn unit normal vector n to the surface (x-1) +y?+(z+2) = 9 at the point ( 3 ,1,-4) x'y+2 x z 4 at the point (2,-2,3) (i) %3D (ii)