Q. 3 (a) An adult riding a roller-coaster moves with a velocity of i = 31-...

 

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Q. 3 (a) An adult riding a roller-coaster moves with a velocity of i = 31- 27. The rider also experiences a force F = 31-7-k when the roiler coaster goes through a loop. Express as a sum of two vectars, one parallel and one orthogonal to F. You may use a vector diagram to aid your explanation and numerical working. (b) A metal block is being beated with the heat source placed at the center of the block The temperature S (measured in "C) at the point ( x, y =) (each measured in inches) is expressed by the formula S (x, y, z) = 100 e-2r-ay-. O If you were asked to identify and sketch the level curves for this temperature function, explain clearly how you would do this (you do not need to sketch the actual curves). Give any one observation/comment on the level curves far this particular function. (1) Find Da S at the paint A (2,-3, -1) in the direction of the point B (1,0, 1). Explain the physical significance of Dg S in the context of the question. (i) In which direction would you expect S to increase fastest at the point A? Why? (iv) Hence, explain how you would find this macimum rate of increase of S at point A. You do not need to calculate the actual value. (C) () Let z = f (4, v), u = g (x.y), v = h (x. y). You're interested in finding and Copy and complete the following diagram to show the derivatives along each route of the az and az branches shown. Hence, write the expressions for . showing clearly all ay contributions along the individual paths. (1) Given w = (r+ y')xy, x=r sin t and y =r cost, use the result derived in part (1) to Hence evaluate at the point (r,t) = (2.) find the expression for ar Q. 3 (a) An adult riding a roller-coaster moves with a velocity of i = 31- 27. The rider also experiences a force F = 31-7-k when the roiler coaster goes through a loop. Express as a sum of two vectars, one parallel and one orthogonal to F. You may use a vector diagram to aid your explanation and numerical working. (b) A metal block is being beated with the heat source placed at the center of the block The temperature S (measured in "C) at the point ( x, y =) (each measured in inches) is expressed by the formula S (x, y, z) = 100 e-2r-ay-. O If you were asked to identify and sketch the level curves for this temperature function, explain clearly how you would do this (you do not need to sketch the actual curves). Give any one observation/comment on the level curves far this particular function. (1) Find Da S at the paint A (2,-3, -1) in the direction of the point B (1,0, 1). Explain the physical significance of Dg S in the context of the question. (i) In which direction would you expect S to increase fastest at the point A? Why? (iv) Hence, explain how you would find this macimum rate of increase of S at point A. You do not need to calculate the actual value. (C) () Let z = f (4, v), u = g (x.y), v = h (x. y). You're interested in finding and Copy and complete the following diagram to show the derivatives along each route of the az and az branches shown. Hence, write the expressions for . showing clearly all ay contributions along the individual paths. (1) Given w = (r+ y')xy, x=r sin t and y =r cost, use the result derived in part (1) to Hence evaluate at the point (r,t) = (2.) find the expression for ar