Calculus

Code: MT129 KSA – TMA Spring 23/24 Cut-Off Date: Based on the Published Deadline. Total Marks: …. marks turned to 15 marks Contents Warnings and Declaration…………………………………….……………………………………1 Question 1 ……………….…………………………………. ……………………………………..3 Question 2 ………………………………………………………………………………….…..…..4 Question 3 ………………………………………………………………………………….…..…..5 Question 4……………………………………………………………………………………………6 Question 5…………………………………………………………………………………………..7 Marking Criteria ……………..………………………………………………………….………..…2 Plagiarism Warning: As per AOU rules and regulations, all students are required to submit their own TMA work and avoid plagiarism. The AOU has implemented sophisticated techniques for plagiarism detection. You must provide all references in case you use and quote another person’s work in your TMA. You will be penalized for any act of plagiarism as per the AOU’s rules and regulations. Declaration of No Plagiarism by Student (to be signed and submitted by student with TMA work): I hereby declare that this submitted TMA work is a result of my own efforts and I have not plagiarized any other person’s work. I have provided all references of information that I have used and quoted in my TMA work. Name of Student:…………………………….. Signature:…………………………………………… Date MT129 / TMA Page 1 of 7 2023/2024 Spring MT129-TMA [A] Student Component Student Name : ____________________ Student Number : ____________ Group Number : _______ [B] Tutor Component Comments Weight Q_1 12 Q_2 12 Q_3 12 Q_4 12 Q_5 12 60 Comments: MT129 / TMA Page 2 of 7 2023/2024 Spring Please solve each question in the space provided. You should give the details of your solutions and not just the final answer. Q−1: [4+4+4 marks] a) If 𝑓(𝑥) = 𝑥 − 2 and 𝑔(𝑥) = 5𝑥 + √𝑥 find (𝑔 ∘ 𝑓)(𝑥) and write the domain of (𝑔 ∘ 𝑓)(𝑥). b) Find the zeros for the function : MT129 / TMA 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 − 18𝑥 Page 3 of 7 2023/2024 Spring Q−2: [6+3+3marks] a) Use the definition of the derivative to compute the derivative for the function 𝑓(𝑥) = 𝑥 3 b) Differentiate: 5 i. 𝑌(𝑥) = √(3𝑥 2 − 7)2 ii. MT129 / TMA 𝐹(𝑥) = 3(−5𝑥 2 +2𝑥)−4 Page 4 of 7 2023/2024 Spring Q−3: [4+4+4 marks] 4 a) let 𝑓(𝑥) = 𝑥 4 − 𝑥 3 3 i) Find the intervals on which 𝑓 is increasing or decreasing, and find the local maximum and minimum of 𝑓, if any. ii) Find the intervals on which the graph of 𝑓 is concave up or concave down, and find the inflection points, if any. b) As a spherical balloon is being inflated its radius (in centimeters) after 𝑡 3 minutes is given by 𝑟(𝑡) = 3 √𝑡 + 8 , where 0 ≤ 𝑡 ≤ 10. What is the rate of change with respect to 𝑡 of 𝑟(𝑡) at 𝑡 = 8? MT129 / TMA Page 5 of 7 2023/2024 Spring Q−4: [4+4+4 marks] a) Find an equation of the tangent line, at the designated point, to each of the following: i) 𝑓(𝑥) = ln(4 + 5𝑥 − 2𝑥 3 ), 𝑥 = 0. ii) 𝑦 2 = 3𝑥𝑦 − 5, 𝑃(2,1). b) Find the x-coordinates of all points on the graph of 𝑦 = 𝑥 3 + 2𝑥 2 − 4𝑥 + 5 at which the tangent line is parallel to the line 2𝑦 + 8𝑥 − 5 = 0. MT129 / TMA Page 6 of 7 2023/2024 Spring Q−5: [6+6 marks] a) Using logarithmic differentiation find 𝑓 ′ (𝑥) if 𝑓(𝑥) = 𝑥 3 (3−2𝑥 2 ) 3 √𝑥 3 +7 b) Solve for 𝑥 the equation ln( x) − ln (𝑥 2 ) + ln(3) = 0 MT129 / TMA Page 7 of 7 2023/2024 Spring

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