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P a g e 1 | 4 Department of Mathematics and Philosophy of Engineering MHZ3551 Engineering Mathematics I Assignment 02 Academic Year: 2020/2021 Due Date: Will notify later Instructions Answer all the questions. Attach the cover page with your answer scripts. Use both sides of papers when you are answering the assignment. Please send the answer scripts of your assignment on or before the due date to the following address. Course Coordinator – MHZ3551, Dept. of Mathematics & Philosophy of Engineering, Faculty of Engineering Technology, The Open University of Sri Lanka, P.O. Box 21, Nawala, Nugegoda. P a g e 2 | 4 Q1 (a) The functions ? and ? are defined as follows. ?(?) = 3?2, where ? ∈ (−1, 1) and ?(?) = ? + 2,where ? ∈ (−1, 1). i) Find ?(?) and ?(?). ii) Draw the graphs of ? and ?. iii) Find ? ∘ ? and ? ∘ ?. iv) Show that ?(?) is not one to one function. (b) The function ℎ is defined as ℎ(?) = 3?2, where ? ∈ (−∞, 0]. i) Show that ℎ is a one to one function. ii) Find the inverse function ℎ−1of ℎ. iii) Show that, (α) ℎ−1 ∘ ℎ(?) = ?, where ? ∈ (−∞, 0]. (β) ℎ ∘ ℎ−1(?) = ?, where ? ∈ [0,∞). (γ) Are the functions ℎ−1 ∘ ℎ and ℎ ∘ ℎ−1 the same? Explain your answer. Q2 (a) Evaluate the following limits. i) lim ?→∞ √?2 − 2? + 5 − ?. ii) lim ?→∞ ? 2? . iii) lim ?→∞ 5?3 − 3? + 7 1 + 2? − ?3 . (b) By using the ratio test, discuss the convergence or divergence of the following series. i) ∑ 3?2 (2? + 1)! ∞ ?=1 . ii) ∑ 7? ?2 + ? . ∞ ?=1 (c) Using a suitable test, discuss the convergence or divergence of the following series. i) ∑ (−0.5)? √? ∞ ?=1 . ii) ∑ −3?3 + 7 2?3 + ? − 1 ∞ ?=1 . P a g e 3 | 4 Q3 (a) Prove the following by using the definition of limits. i) lim ?→3 ?2 = 9 ii) lim ?→5 2? − 3 7 = 1 (b) Evaluate the following limits. i) lim ?→0 ( ?2???(2?) − 1 √3? + 3 − √2? + 3 ) ii) lim ?→∞ (1 + 1 ? ) 2020? (c) Using the differentiation, prove that ? − ?3 3 < −1="">< −="" 3="" 3="" +="" 5="" 5="" ,="" where=""> 0. i) Using the sandwich rule prove that ??? ?→0+ ( ???−1(?) − ? ?3 ) = − 1 3 . ii) Deduce that ??? ?→0 ( ???−1(?) − ? ?3 ) = − 1 3 . (d) The function ? is defined on ℝ\{0} as follows: ?: ? → { ?2 3 − 3 if ? < 3="" 0="" if="" =="" 3="" 3="" −="" 27="" 2="" if=""> 3 . Show that ? is continuous at ? = 3. Q4 (a) Find the maxima, minima, and inflection points of ? = 3?5 − 5?3. Indicating those points, draw the graph of ? = 3?5 − 5?3. (b) An open lid tank to be made by concrete has width 50??, inside capacity of 4000 ?3 and square base. Find the inner dimension of the tank with the minimum volume of concrete. (c) Using definite integrals, design a mathematical model to find the inner volume of a circular tube of a vehicle wheel. (d) Draw the curves of ?2 = 4?? and ?2 = ?(8 − 4?) in the same diagram, where ? > 0. i) Find the area ? enclosed by the curves. ii) If the area ? is rotated of an angle 2? about: (α) ? − axis; (β) ? − axis; then find the volumes of the generated bodies. P a g e 4 | 4 Q5 (a) Show that lim (?,?)→(0,0) ?2? ?4 + ?2 does not exists. (b) Using the definition, show that lim (?,?)→ (0,0) 2?4 ?2 + ?2 = 0. (c) Let ?(?, ?) = { ? √?2 + ?2 if ? ≠ 0, ? ≠ 0 2 if ? = 0, ? = 0 Show that ?(?, ?) is not continuous at (0, 0). (d) If ? = sin−1 ( ? ? ) + tan−1 ( ? ? ) , then find ? ?? ?? + ? ?? ?? . End -Copyrights Reserved-