1. Consider the function f defined by f(x, y) = x²y + y²x x² + y² 0 if (x, y) (0,0); " if (x, y)=(0,0). (a) Show that f is continuous at (0,0). (Hint: Evaluate the appro- priate limit by changing to polar coordinates.) (b) Using the principal definition of partial derivatives, show that (0,0) exists and equals 0. af მე (c) You may assume that af მყ (0, 0) exists and is equal to 0. Using the definition of differentiablity of a multivariable function, explain why f is not differentiable at (0,0). (Hint: Investigate the exis- tence or otherwise of the appropriate limit by changing to polar coordinates.) 2. Consider the region R in the xy-plane given by R={(x,y) R² x ≥ 0, y ≥ 0 and 1 ≤ x² + y² ≤ 2}. (a) Sketch the region R. (b) By changing to polar coordinates evaluate the double integral y sin (π (x² + y²)) √x² + y² dx dy. Recall that Cartesian and polar coordinates are related by (x, y) = (r cos 0, r sin 0), ✓ (1) where r = √√√x² + y², and 0 ≤ 0 < 27. You may assume that the Jacobian for the change of variables (1) is given by (x, y) ǝ (r, 0) = 1.

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Consider the following. Function: h(x, y) 9у cos (x - у) Point: (0, ) 3. (a) Find the gradient of the function. -9y sin(x - y) i + 9 cos(x - y) + 9y sin(x – y) j (b) Find the maximum value of the directional derivative at the given point. (Give your answer correct to 2 decimal places.)

Day traders typically buy and sell stocks (or other investment instruments) during the trading day and sell all investments by the end of the day. The following table shows the closing prices on September 22, 2015, of 12 stocks selected by your broker, Prudence Swift, as well as the change that day. Tech Stocks Close Change AAPL (Apple) $113.40 -1.81 ADBE (Adobe Systems) $84.66 1.34 EBAY (eBay) $25.61 -0.31 MSFT (Microsoft) $3.90 -0.21 S (Sprint) $4.40 0.02 WIFI (Boingo Wireless) $8.51 0.56 Non-Tech Stocks ANF (Abercrombie & Fitch) $21.81 -0.02 в (Воeing) $133.99 -2.03 F (Ford Motor Co.) $13.91 -0.40 GE (General Electric) $25.10 0.01 GIS (General Mills) $57.12 0.33 JNJ (Johnson & Johnson) $93.26 0.13 On the morning of September 22, 2015, Swift advised you to purchase a collection of three tech stocks and two non-tech stocks, all chosen at random from those listed in the table. You were to sell all the stocks at the end of the trading day. (a) How many possible collections are possible?…