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1. Evaluate the following indefinite integral: � 𝑙𝑙𝑙𝑙 2𝑥𝑥 𝑥𝑥6 2. Evaluate the following definite integral: � 12𝑥𝑥5 + 8𝑥𝑥3 − 16𝑥𝑥 + 4 𝑥𝑥6 + 𝑥𝑥4 − 4𝑥𝑥2 + 2𝑥𝑥 − 6 𝑑𝑑𝑑𝑑 10 5 3. For the following system of equations find solutions to the values of x, y, and z using the matrix inversion technique shown in this course. 1 23 2 3 2 xyz x yz xyz ++= + −= −−= 4. Find the partial derivative of the following equation with respect to x and y: 𝑧𝑧 = 𝑒𝑒𝑥𝑥𝑙𝑙𝑙𝑙�𝑥𝑥𝑥𝑥2 + 𝑆𝑆𝑆𝑆𝑆𝑆 𝑥𝑥𝑥𝑥2 5. Without substituting u or v into Z, please find the total derivative of Z with respect to x: 𝑧𝑧 = 𝑢𝑢𝑢𝑢𝑢𝑢(𝑢𝑢2 + 𝑣𝑣2) − ln � 𝑥𝑥2 𝑥𝑥2 − 6𝑥𝑥 + 3 � 𝑢𝑢 = 𝑥𝑥 𝑦𝑦 𝑣𝑣 = 2𝑥𝑥 − 3𝑦𝑦 6. Determine the critical points of functions for the following function: 𝑓𝑓(𝑥𝑥, 𝑦𝑦) = 4(𝑥𝑥 − 𝑦𝑦2) − 𝑥𝑥2 − 𝑦𝑦3 Determine whether or not they correspond to local maxima, minima, or saddle-points. Clearly distinguish the first-order conditions and the second-order conditions. 2 7. Given the constraint, find the stationary points for the following function and evaluate the second order conditions. Use the Lagrange technique. f xy x y ( , ) ln 2ln = + Subject to 2 2 x y + = 6