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Q1:Westside Energy charges its electric customers a base rate of $6.00 per month, plus 15€ per kilowatt-hour (kWh) for the first 300 kWh used and 2€ per kWh for all usage over 300 kWh. Suppose a customer uses x kWh of electricity in one month. (a) Express the monthly cost E as a piecewise defined function of x. (Assume E is measured in dollars.) E(x) = Need Help? (b) Graph the function E for 0 sxs 600. E 60 50 L 40 30 20 10 E 60p 50 40 30 20 10 if 0 sxs 300 Read It if 300 < x 100 200 X 300 400 500 600 100 200 300 400 500 600 Watch It E 100 80 60 40 20 X 100 200 300 400 500 600 E 60 50 40 V 30 20 10 3 100 200 300 400 500 600See Answer
Q2:2. CAUCHY SCHWARTZ INEQUALITY For Cauchy Scwhartz inequality read p19 of 0 Inner Product Fin Dim Question 3.[4, 2 marks] Consider the normed spaces (RN, ||- |1) and (RN, ||- ||2). Prove that: (a) x1 < Mx||2 for all x € RN for some constant M > 0. (HINT: Use the Cauchy-Schwartz inequality in the inner-product space (RN, ||-||2).) (b) Give a geometrical interpretation of (a). 1See Answer
Q3:Question 5.[3, 5 marks] Let X₁ := {fe C[0, 1]f(0) = 0}. Prove that: (i) XA is a subspace of (C[0, 1], (ii) XA is incomplete.See Answer
Q4: 2. Assume that P = (((p1^-P2) AP3) ⇒ P2). Write down the truth table for P. How many different interpretations make P true. Give the list of interpretations I under which P false. 3. A formula P is called satisfiable if there exists an interpretation I such that I(P) = 1. Which of the following formulas are satisfiable? Check using truth tables. \text { (a) }\left(\neg p_{1} \wedge p_{2}\right) \text { (c) }\left(p_{1} \Leftrightarrow \neg p_{1}\right) \text { (d) }\left(p_{1} \wedge\left(\neg p_{2} \vee \neg p_{1}\right)\right)See Answer

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