MATH 184 Quiz 2’ October 30 Grade: First Name: Last Name: Student-No: Section: 184-104 Short answer questions — you must show your work 1. 6 marks Each part is worth 2 marks. (a) Differentiate y = (x3 + 4)x+2 . Solution: 0 3 y = (x + 4) x+2 d (x + 2)(3x2 ) 3 3 3 x+2 ln(x + 4) + (x + 2) ln(x + 4) = (x + 4) . dx x3 + 4 √ (x + 1)3 x + 2 (b) Differentiate y = ln . (x2 + 1)2 Answer: y 0 = 3 1 4x + − 2 x + 1 2(x + 2) x + 1 Solution: d y = dx 0 1 3 ln(x + 1) + ln(x + 2) − 2 ln(x2 + 1) 2 = 3 1 4x + − 2 x + 1 2(x + 2) x + 1 (c) You invest $2,000 in a certificate of deposit with an annual percentage yield of 2%, compounded continuously. How many years will it take for your investment to become $3,000? (A calculator-ready form will suffice.) Answer: ln 1.5 ln 1.02 Solution: y(t) = y0 ekt where y0 = 2000. We have y(1)/y0 = 1 + 0.02 = 1.02. Thus ek = 1.02, k = ln 1.02. We want y(t) = 3000, thus 3000 = 2000ekt , t= ln 1.5 ln 1.5 = ≈ 20.5 (years). k ln 1.02 Long answer question — you must show your work 2. 4 marks The price p (in dollars) and the demand q for a product are related by 2p2 + q2 = 2200. 100 If the current price per unit is $30, use the price elasticity of demand = E = whether the revenue will increase or decrease if the price is raised slightly. Answer: decrease Solution: We have p q = 10 2200 − 2p2 , dq 5(−4p) . =p dp 2200 − 2p2 When p = 30, we have q = 200 and = p dq 30 −20 · 30 9 = =− . q dp 200 20 2 Since || > 1, the revenue will decrease if the price is raised. p dq to decide q dp