PROBLEM 1 The following three questions are unrelated a...

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FINAL EXAM: CALC II (BIO AND SOC. SCI.) PROBLEM 1. The following three questions are unrelated. a) Find the solution to the equation dy dt = 3 t - 1 which satisfies y (2) = 5. b) Solve the equation dy dt = 2(1 - y ), with y (0) = 2. c) The autonomous equation dy dt = y 5 has exactly one equilibrium solution. Deter- mine wether it is a stable or unstable equilibrium. PROBLEM 2. a) Determine which of the following matrices is invertible and find its inverse. A = - 1 1 1 - 3 and B = 1 2 2 4 b) With A as above, find X such that AX = - 2 3 . PROBLEM 3. In the xy -plane, consider the triangle that has vertices at coordi- nates P (1 , 2), Q (4 , 3) and R (3 , - 4). a) Determine the length of the segment QR . b) Determine the angle at P (i.e. the angle RPQ ). c) Determine the equation of the line that passes through P and is perpendicular on the line QR . PROBLEM 4. a) Compute f x and f y for f ( x, y ) = x sin( πxy ). b) Consider the function g ( x, y ) = x 2 y . Determine the directional derivative of g at ( - 1 , 2) in the direction of 1 1 . c) With g as above, find a unit vector v