tutorial 6 solution - NATIONAL UNIVERSITY OF SINGAPORE...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 6 (Solution Notes) 1. Imagine you are visiting a country in the winter season. Let T ( x, y ) = 36 1 5 [ x 2 + ( y 5) 2 ] be the temperature at location ( x, y ) in a 10ft × 10ft hotel room with a heater on at night. One corner of the room is at (0 , 0) and the opposite corner is at (10 , 10). (i) What is the domain of the temperature function? (ii) Where is the likely location of the heater? (iii) Suppose you like to sleep within the temperature range of 20 C to 25 C. Where would you put your bed? (iv) Determine the locations in the room where the temperature is lowest. Solution (i) As the temperature function is only valid within the hotel room, its domain is { ( x, y ) : 0 x 10 , 0 y 10 } . (ii) The heater is at the location where the temperature is highest. It is clear that T ( x, y ) = 36 1 5 [ x 2 + ( y 5) 2 ] 36 . Note that the equation holds if and only if x = 0 and y = 5. That is, the largest value of T ( x, y ) is 36 at (0 , 5). (iii) We know the radius is given by r = x 2 + ( y 5) 2 , then T = 36 1 5 [ x 2 + ( y 5) 2 ] = 36 1 5 r 2 . We want to find the radius r , such that 20 T 25. T = 36 1 5 r 2 25 = 1 5 r 2 36 25 = r 55; T = 36 1 5 r 2 20 = r 80 = 4 5. Then, the bed should be put in the range 55 r 4 5. 130
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(iv) Suppose the temperature is c , then c = 36 1 5 [ x 2 + ( y 5) 2 ] , and so the (temperature) level curves of c is given by x 2 + ( y 5) 2 = 5(36 c ) . These are circles centered at (0 , 5), the values of c decreasing as the radius increases. The largest circle intersecting the domain intersects the domain at (10 , 0) and (10 , 10), so these points have the lowest temperature which is 11. 131
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2. In an electric circuit, the voltage of V volts (V), current of I amperes (A), and resistance of R ohms (Ω) are governed by Ohm’s Law V = I × R. (i) If the resistance is fixed at 15 Ω, how fast is the current increasing with respect to voltage? (ii) If the voltage is fixed at 120 V, how fast is the current increasing with respect to resistance at the instant when resistance is 20 Ω? (iii) If the resistance is slowly increasing as the resistor heats up, how is the current changing at the moment when R = 400Ω, I = 0 . 08A, dV/dt = 0 . 01 V / s and dR/dt = 0 . 03 Ω / s? Preliminaries Partial Derivative DEFINITION Partial Derivative with Respect to x The partial derivative of ƒ ( x , y ) with respect to x at the point is provided the limit exists. 0 ƒ 0 x ` s x 0 , y 0 d = lim h : 0 ƒ s x 0 + h , y 0 d - ƒ s x 0 , y 0 d h , s x 0 , y 0 d Partial derivatives are used for multi-variable functions. For example, to find ∂f ( x, y ) ∂x , we use the following scheme: (a) Regard y as a constant, then the function f ( x, y ) is a one-variable function; (b) Consider the derivative of f ( x, y ): df dx , and change d to . Remark: This scheme just tells you an intelligible way to get the partial derivatives.
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