ECN312 - PSO Summer 2023 Problem Set 0: Math Review Instructions . This set of problems covers various math-skills covered by the prerequisite courses, which we will make use of throughout the course. . This problem set is due on May 22nd at the beginning of class. . Be sure to show all of your work. Correct solutions without work shown will not receive credit. Incorrect solutions with work shown may be eligible for partial credit. . Please scan your solutions and submit them on Canvas as a single PDF file. Make sure your scanned solutions are readable! . Note: Throughout this problem set - and the rest of the semester - log() denotes the natural logarithm of r (not the base-10 logarithm) and log,(r) denotes the base-b logarithm of r. Algebra & Inverses 1) Solve for a in the following expressions, where b and c are arbitrary constants. (a) -273 + 16 2 0 (b) 232 + 7 = c (c) log(4.r?) + log(3x) - b=0 (d) e2x - 9et + 18 = 0 (Hint: Make use of the fact that ezz = (er)?) 2) Graph each of the following functions and then either obtain its inverse or briefly explain why the function is not invertible. (a) f(x) = 3x +4 (b) g(x) = 5(x -2)2 +7 (c) h(x) = 3log(x + 2) + 5 for = > -2 Derivatives 3) Take the derivative of the following functions, where a and b are arbitrary constants. For part (d), take partial derivatives both with respect to r and with respect to y. (a) f(x) - 273-1 (b) g(x) = 4x2er (c) h(x) = log (ar2 + b) (d) f(x,y) = 5raybECN312 - PSO Summer 2023 Optimization 4) Solve the following (unconstrained) optimization problem in one variable. Report both the maximum function value and the value(s) of a at which the function value is attained. max 75 + (3x + 5)' - (6x -2)2 5) Solve the following (unconstrained) optimization problem in one variable (x) where a and b are arbitrary constants. Report both the maximum function value and the value(s) of r at which the function value is attained. max log(2r) - 2 +3-62 a 6) Solve the following (unconstrained) optimization problem in two variables. Report both the maximum function value and the value(s) of (r, y) at which the function value is attained. max 20 - (x + 8)' - (y - 4)' + xy I,y