Final examination semester 2 academic year 2019-2020 course name Calculus 1 - ĐH Sư phạm Kỹ thuật

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Final examination semester 2 academic year 2019-2020 course name Calculus 1 - ĐH Sư phạm Kỹ thuật HCMC UNIVERSITY OF TECHNOLOGY AND FINAL EXAM, SEMESTER 2, 2019-2020 EDUCATION Subject: Calculus 1 HIGH QUALITY TRAINING FACULTY Course code: MATH141601E GROUP OF MATHEMATICS Number of pages: 02 pages. ------------------------- Duration: 90 minutes. Date of exam: 31/07/2020 Materials are allowed during the exam.Question 1 (1 point) Given f ( x ) = x 3 + 3, g ( x) = cos (2 x ) . a) f ( x) and g ( x) are even, odd or neither? b) Find ( f g )( x) and ( g f )( x) .Question 2 (1.5 points) a) Find the value of the constant m such that the following piecewise - defined functionis continuous everywhere.  e3 x − 1  , x≠0 f ( x) =  x . m , x=0  b) With m found in question a), find f ( x ) , ∀x.Question 3 (1 point) Let y be an implicit function of x satisfying the equation: x 2 + xy − y 3 = 1 Find the tangent line to the graph of the equation at the point M (1; −1) .Question 4 (1 point) Find the relative extrema of g ( x ) = ( x − 2)e − 0.5 x .Question 5 (1 point) x+2 Let f ( x ) = . Find the average value of f on the interval [ 0, 1] . 4 − x2Question 6 (1 point) Find the particular solution of the separable differential equation dy y ln y = 2x dx esatisfying the initial condition y ( 0 ) = e.Question 7 (1.5 points) Assume that the position at time t of an object moving along a line is given by s ( t ) = 2t 3 − 15t 2 + 36tfor t on [1, 4]. a) Find the initial velocity and acceleration for the object. b) When is the object advancing and retreating? c) When is the object accelerating and decelerating?No.: BM1/QT-PĐBCL-RĐTV Page: 1Question 8 (1 point) The volume of a spherical balloon is increasing at a constant rate of10 cm3 / s . At what rate is the radius of the balloon increasing when theradius is 3 cm?Question 9 (1 point) A cylinder box (Figure 1) is constructed with the volumeV = 24π cm3 . The cost of the material used for the bottom is $2/cm2, thecost of the material used for the lateral side is $2.5/cm2 and the cost of thematerial used for the top is $5/cm2. Find the dimension of the box r and hthat minimizes the total cost.Notice: Invigilators should not explain the questions on the exam papers.Expected Learning Outcomes Questions[G 2.1]: Present mathematical information using words, statements, 1numbers, formulas, graphs and diagrams[G 1.1, 1.3, 5.2]: Students are able to find basic limits and test the 2, 3, 4continuity of a function. Students are able to find derivative anddifferential.[G 3.1, 5.4] : Apply important rules and theorems effectively, such 5as the mean value. Students are able to apply theory to evaluateindefinite and definite integrals.[ELO 1.4, 5.4]: Students are able to solve basic differential 6equations.[G 4.1] Identify and analyze the information given in formulas, 7graphs and tables relating to (a) rectilinear motion; (b) linear; (c)optimization and applications in physics; (d) Riemann sum andintegration[G 5.3]: Students are able to use derivative to solve problems 8,9relating to rates of change and optimization July 24th, 2020 Head of group of mathematics Đáp ánQ1: (1 point)a) Because of f ( − x ) ≠ ± f ( x ) , f(x) is neither odd nor even. (0.25pt) Because of g ( − x) = g ( x ) , g(x) is even. (0.25pt)b) ( f g )( x) = (cos(2 x))3 + 3;( g f )( x) = cos(2( x 3 + 3)). (0.25pt+0,25pt)Q2:(1,5 points) D f = ℝ .a) x ≠ 0 : f ( x ) is continuous with all x ≠ 0 . f ( x) is continuous with all x ∈ ℝ ⇔ f ( x ) is continuous with at x = 0 .(0.25pt) ...