DSC1520 Assignment and Exam Pack 2022

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Quantitative Modelling (DSC1520)
Assignment 2, Semester 1, 2012

Unique Number: 886493
Question 1

Differentiate the function

𝑃(𝑥) = 3𝑥3𝑒2𝑥

Apply the Product Rule:

If 𝑃(𝑥) = 𝑢. 𝑣 then 𝑃′(𝑥) = 𝑢. 𝑑𝑣 + 𝑣. 𝑑𝑢

𝑢 = 3𝑥3 𝑑𝑢 = 9𝑥2

𝑣 = 𝑒2𝑥 𝑑𝑣 = 2𝑒2𝑥

𝑃′(𝑥) = 𝑢. 𝑑𝑣 + 𝑣. 𝑑𝑢

𝑃′(𝑥) = 3𝑥3. 2𝑒2𝑥 + 𝑒2𝑥. 9𝑥2

𝑃′(𝑥) = 6𝑥3𝑒2𝑥 + 9𝑥2𝑒2𝑥

𝑃′(𝑥) = 3𝑥2𝑒2𝑥(2𝑥 + 3)

Option [5]




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Question 2

Find the derivative of the function
𝑥
𝑃(𝑥) = 15 − 𝑥√6𝑥 + 1 +
1+𝑥

Apply the Product Rule on the derivative of 𝑥√6𝑥 + 1

Apply the Quotient Rule on the derivative of 𝑥
𝑥+1

𝑑 𝑑
(𝑥√6𝑥 + 1) = (𝑢. 𝑣) = 𝑢. 𝑑𝑣 + 𝑣. 𝑑𝑣
𝑑𝑥 𝑑𝑥
𝑢=𝑥 𝑑𝑢 = 1
1 1 −1 3
𝑣 = √6𝑥 + 1 = (6𝑥 + 1)2 𝑑𝑣 = (6)(6𝑥 + 1) 2 =
2 √6𝑥 + 1
𝑑 3 9𝑥 + 1
(𝑥√6𝑥 + 1) = 𝑢. 𝑑𝑣 + 𝑣. 𝑑𝑢 = 𝑥 ( ) + (√6𝑥 + 1 )(1) =
𝑑𝑥 √6𝑥 + 1 √6𝑥 + 1
𝑑 𝑥 𝑑 𝑢 𝑣. 𝑑𝑢 + 𝑢. 𝑑𝑣
( )= ( )= 2
𝑑𝑥 1 + 𝑥 𝑑𝑥 𝑣 𝑣

𝑢=𝑥 𝑑𝑢 = 1

𝑣 =1+𝑥 𝑑𝑣 = 1
𝑑 𝑥 𝑣. 𝑑𝑢 − 𝑢. 𝑑𝑣 (1 + 𝑥)(1) − (𝑥)(1) 1+𝑥−𝑥 = 1
( )= 2 = 2 = 2 2
𝑑𝑥 1 + 𝑥 𝑣 (1 + 𝑥) (1 + 𝑥) (1 + 𝑥)
𝑥
𝑃(𝑥) = 15 − 𝑥√6𝑥 + 1 +
1+𝑥
9𝑥 + 1 1
𝑃′(𝑥) = 0 − +
√6𝑥 + 1 (1 + 𝑥)2
9𝑥 + 1 1
𝑃′(𝑥) = − +
√6𝑥 + 1 (1 + 𝑥)2

Option [1]




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