4. (31 marks) Let S be the surface given by the equation x +y-22 = 1....

    

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4. (31 marks) Let S be the surface given by the equation x² +y²-2²2 = 1. ƏI Ər a. (8 marks) Find a formula for and Əz CS Scanned with CamScanner b. (8 marks) Find the points on S (if any) where the tangent plane to S is parallel to the (y, z)-plane. 20 CS Scanned with CamScanner 5. (31 marks) A galaxy is modelled by R³. Scientists have determined that the temperature of any point (x, y, z) in this galaxy is given by T(x, y, z) = CS Scanned with CamScanner 1 1 + x² + 2y² + 3z² a. (8 marks) A small comet (which we may treat as a point) is travelling through this galaxy. Its position in this galaxy at time t is given by R(t) = (t + 1, 2t + 3, t²). Calculate the rate of change of the temperature of this comet at t = 0. 4 b. (8 marks) Find the points in this galaxy where the temperature is maximized and minimized. (If there are no maximums or minimums of the temperature in this galaxy, briefly explain why.) CS Scanned with CamScanner 4. (31 marks) Let S be the surface given by the equation x² +y²-2²2 = 1. ƏI Ər a. (8 marks) Find a formula for and Əz CS Scanned with CamScanner b. (8 marks) Find the points on S (if any) where the tangent plane to S is parallel to the (y, z)-plane. 20 CS Scanned with CamScanner 5. (31 marks) A galaxy is modelled by R³. Scientists have determined that the temperature of any point (x, y, z) in this galaxy is given by T(x, y, z) = CS Scanned with CamScanner 1 1 + x² + 2y² + 3z² a. (8 marks) A small comet (which we may treat as a point) is travelling through this galaxy. Its position in this galaxy at time t is given by R(t) = (t + 1, 2t + 3, t²). Calculate the rate of change of the temperature of this comet at t = 0. 4 b. (8 marks) Find the points in this galaxy where the temperature is maximized and minimized. (If there are no maximums or minimums of the temperature in this galaxy, briefly explain why.) CS Scanned with CamScanner