Applied Calculus Questions

MT129: Applied Calculus Tutor Marked Assignment Cut-Off Date: Total Marks: 60 Contents Feedback form ……….……………..……………………………..…………………….……….. Question 1 ……………………..……………………………………………………………..……… Question 2 ……………………………..……………………………………..……………………… Question 3 ………………………………..……………………………………..…………………… Question 4 ………………..…………………………………………………………..……………… Question 5 ……………………..……………………………………………………………..……… 2 3 4 5 6 7 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 students 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. Student Name Signature Date : _____________________ : _________________ : ___________ MT29 – Applied Calculus 2022-2023 / Summer 1 MT129 TMA Feedback Form [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 Mark 60 General Comments: Tutor name: MT29 – Applied Calculus 2022-2023 / Summer 2 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) Let f(x)= √𝑥 2 + 1and g(x)= √𝑥 − 1 Find the domain of f(x) and g(x). b) i. Find the zeros of the equation: 𝑥 5 − 𝑥 4 − 2𝑥 3 =0 𝑥 ii. Let f(x) = 2 𝑥 −1 1−𝑥 and g(x) = 1+𝑥 , find f(x).g(x). MT29 – Applied Calculus 2022-2023 / Summer 3 Q−2: [6+3+3marks] a) Use the definition of the derivative to compute the derivative for the 1 function f(x)= 𝑥−1 b) Differentiate: 7 i. Y=3√1 + 𝑥 + 𝑥 3 ii. F(x)= 2(𝑥 2 −3)−5 MT29 – Applied Calculus 2022-2023 / Summer 4 Q−3: [4+4+4 marks] a) let f(x) =3𝑥 4 − 8𝑥 3 + 6𝑥 2 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) There are $320 available to fence in a rectangular garden. The fencing for the side of the garden facing the road costs $6 per foot and the fencing for the other 3three sides costs $2 per foot. Find the dimensions of the largest garden. MT29 – Applied Calculus 2022-2023 / Summer 5 Q−4: [6+6 marks] Find an equation of the tangent line, at the designated point, to each of the following: a) 𝑓(𝑥) = ln(𝑥 2 + 𝑒), 𝑥 = 0. b) 𝑥. 𝑦 + 𝑦 3 = 14, 𝑃(3,2). MT29 – Applied Calculus 2022-2023 / Summer 6 Q−5: [6+6 marks]a) Using logarithmic differentiation find 𝑓 ′ (𝑥) if (2𝑥 + 1)(3𝑥 + 1) 𝑓(𝑥) = 𝑒 𝑥 . √4𝑥 2 + 1 b) Solve for 𝑥 the equation ln( x + 1) −ln (x-2)=1 MT29 – Applied Calculus 2022-2023 / Summer 7