1. [4 marks] Evaluate the following limits using calculus and showing all working. 3. [7 marks] (a) lim 2 + 2x - 3 (a) Sketch the graph of f (x) = VI. 2- 1 4x - 1-+ 30 (b) lim 1-+00 20(1 - 12) (b) Using translations and reflections, sketch the graph of g(x) = 1 - -(4 -r), showing and labeling intermediate sketches. 2. [4 marks] Find the domain, giving reasons, of the function f(x) = (e* -1) 1/2 and express your answer using the interval notation.4. [9 marks] Find for the following, showing all working: 5. [12 marks] Using calculus and showing all your working, sketch the function f( I) = - I +1 (a y- 12 -9 COS (T) noting the domain of f(x), any intercepts or asymptotes, regions where the function is increasing or decreasing and any relative extrema. (b) y = 2 In | sin(2r)| (c) 3x y - Vity - ery(c) Find h (t) and evaluate h' (3.5) exactly. Is the horse moving up or down at t = 3.5 and why? (d) Find the smallest value of t 2 0 for which h' (t) is a maximum, showing the working. Notes: . 5 marks are reserved for presentation in this assignment. . There are penalties for late assignments. You must contact your tutor before the due date if you have difficulties making the deadline.6. [6 marks] Find, using calculus and showing all working, the absolute maximum and the absolute minimum of the function f(x) = x - 1812 + 5 for r E [-2, 4).atics Final Examination - SP3 202 7. [8 marks] Find the following indefinite integrals: 8. [8 marks] Find the following definite integrals, using calculus and showing all working: (a) (212 - 2)et/2 dx (b) zVr2 + 1de ( b) ( + 1)e 3 do .[4 marks] As an exercise ball is inflated, its radius increases from 40cm to 41cm. Using 11. [9 marks] Find y(2) for the following: linear approximations estimate the change in volume of the ball. (The volume of a sphere of radius r is V = = mrs.) 4 (a) dy -e (317 - 1) with y(0) = 0 10. [5 marks] An ice-cube with 3cm sides is placed in a warm drink and starts to melt with the side length decreasing at a rate of 1mm per minute. At what rate is the volume of the (b) my' - 2y - 3x2 with y(1) = 1 ice-cube melting when its side length is 2.5cm?12. o marks 13. [12 marks] A cook has finished baking a cake and placed it on the bench to cool. The temperature in the room is 20 C and the temperature of the cake when it was taken out of (a) Using the trapezoidal rule with n - 4, estimate the integral e 2t da. the oven is 160 C (a) Given that the temperature of the cake is governed by Newton's law of cooling, write down a differential equation governing T(t), the temperature of the cake after t hours. What is the appropriate initial condition? (Newton's law of cooling: dT dt -K(T - Ta), where K is a constant and To is the ambient temperature.) (b) From you answer in part (a), derive the solution T(t) = 20 + 140e , where K is a constant. (b) Calculate the exact result and compare this with the approximate answer in part (a) by calculating the absolute error.MAS120 - Applied Mathematics Final Examination - SP3 20 14. [5 marks] Using calculus and showing all your working, find the area between the curves (c) Given that the cake has cooled to 90'C after 1 hour, determine the constant K. f(x) - 4vr and g(x) =12. d) The cook decides that the cake is cool enough to be taken out of the cake pan when its temperature lowers to 40 degrees C. Find when this will happen, both in exact form and as a decimal approximation to at least 2 decimal places, showing all working. 15. [4 marks] Find the volume of the solid of revolution formed by rotating f(r) = Etz-1 about the r-axis for 1