Problem 3. (a) Let P(3:,y) be a point in 2D. Its coordinates are 3 and 4 with an error of at most 0.01. What is the error with which the polar coordinates 7" and 9 are known? (b) The sides of a rectangular box have been measured to be a = 1, b = 2, and c = 3. To which of these measurements is the volume most sensitive? So, if all a, b, and c are known with an error of 0.01, which one creates the largest error for the volume? Problem 4. (a) Suppose z = f (3:,y) has continuous partial derivatives. Let CE = Tcos(6) and y = Tsin(6). We want to compute the partial deriva- tives of f(a:(r, 6), y(7", 9)) with respect to 7' and 6. Explain why we have 8f _ 1 3f 3f 37" cos(6l)8$ + s1n(0) By and, further, 8_f 3f 3f 86 =Tsin(0 )833 +TCOS(6 )8y Finally, show that (2%)2+%(Z)2=(3) +3( :5) (b) Let f(:v y) = 1n(:c:+ | 2y2). Use (a) to compute( (93: 2 2 (c ) Let g( a: y) eV$ .(Use )to compute (99) + (gait . M E\") to + A 03E cab, v K) 8:1