Background This is an incredible story. MindBoolt went from a start up company in January 2005 to an IF'CI in January Elli-3. Its stock is performing amazineg well. Answer the questions below to help this rm predict its stock price. maximize its prots and analyze its investments. Limits 6 marks Derivatives: ti marks Optimization: En marks Curve Sketching: 3 marks 1isectors and Equations of Lines: 3 marks Tots] Marks: 34 Pm; A: Limi To predict the price of the stock. a model is developed that includes the past behaviour of the stool: and also the expectations for the future. The graphical model below shows the past behaviour of the stock price {red graph} and the future expectation {blue graph} over time. Answer the questions below: [El] a} Determine the price of the stock at t=4 based on passed behaviour? Use proper limit notations. b] Determine the price of the stock at t=4 based on future expectations? Use proper limit notations. o} Is the function discontinuous at t=4'? How do you know? Use proper limit notations. Pg B: Derivative; gf fgngtigns The stout: price is performing well however it is quite volatile. The stock price over the next year is modelled by the function r} = at\": 5: [s] a] How fast is the stack prise she nging att = 5? b] Is the function above differentiable everywhere? Explain using derivatives. P C: imiz in The rm is vent:r interested in maximizing prots. Its profit model is given by the function: f(x) = Tsimc where it represents the number of new subscribers per year in thousands. Determine how many new su bsu'ibers are needed to maximize prot given that the number of subscribers supported has to be between sum and moon? [5] P D:Crv It hin To visualize the stock price behaviour you need to graph this rational function that models the behaviour of the function in the next few years. [3] 4 - i s1 s 1 2? 3: +9 air {x2 +9): \"if (1'2 +9]: a} Graph the function by rst determining the intervals of increase and decrease. the intervals of concavityr and anyr local exuema or points of inection. b] What asymptotss does this function have? What happens to the stock price over time? Fag E: Veggrg MindBook decides to diversify their portfolio and invest in the \"Robolvlap\" technology which is a robot that maps different types of ten'itories. The new CEO is trying to make some drastic changes to some of the software and hardware systems in the new version of the robot. You have been hired by the opposing faction in the hoard to lead an investigation on how the performance of the robot is affected by the new proposed changes. [3] a] The rst system that you will test is the navigation system. The robot maps the territory by calculating distances and angles whilst moving between various locations. it then stores on its system the shortest direct path from initial location to the intended target. During the test the robot navigates around a muddy surface in the following path: 55cm [NW] and then moves 45cm [sso'iw]. The robot sends back to the server the following overall shortest path: i'cm ['W1El\"S]. Lise triangle or parallelogram method to verify the precision of the mbot's navigation system. Diagram not to scale b] The next test involves the performance of the new battery. The robot encounters an obstacle on his journey. Using its arm it applies a force of 3511 at an angle of 25" below horizontal and is able to displace the object for 20m in a time of E5s. At the beginning of the test the battery's is displaying a total stored energy {abilityr to perform mechanical work} of 2500 him. What would be the remaining energy displayed at the end of the test? Useful formula: as = w [change in available energy is equal to the work done} c} The robot is also able to lift itself by pushing its arm against the ground in a rotational motion around an axis at the back wheel. This is useful to move on top of obstacles encountered during exploration. The robot applies a force [10.5.15] at a point where the arm vector is [-53.3]. Determine the torque vector. Diagram for ilusbslion erposas only d] One exploratoryr mapping session for a 1m by 1m can last 5 minutes. The new mapping system should be able to handle absolute coordinates in space as well. That is the compass directions might not be available. This mapping system uses cartesian system using as origin {c.co) the landing site. The robot starts the exploration at the point (1, 2. -3,'I and ends the exploration at the point [2. c, 1}. Describe the trsjectorzlr of the move using the equation of line between the starting and the nal location of the robot. Diagram for Ilustratlm purposes onlyI I iIiIIJN-Jln 3 Emit. '1' .u