Math 234 Spring 2014 Exam 1 Version 1 Monday February 17 2014 Name Circle the section you are registered for ADA 9 00 Neriman ADB 10 00 Neriman ADC 11 00 Itziar ADD 12 00 Itziar ADE 1 00 Malik ADF 2 00 Malik ADG 3 00 Alessandro ADH 4 00 Alessandro UIN BDA 9 00 Dileep BDB 10 00 Dileep BDC 11 00 Jiashun BDD 12 00 Sean BDE 1 00 Argen BDF 2 00 Argen BDG 3 00 Albert BDH 4 00 Albert CDA 8 00 Nima CDB 9 00 Nima CDC 11 00 Michael CDD 12 00 Michael CDE 1 00 Jaehoon CDF 2 00 Claire CDG 3 00 Claire CDH 11 00 Sean 1 You should have already checked into the exam and had your calculator approved No exam will be accepted from a student who does not check in before they start the exam 2 No baseball caps hoodies etc or dark sunglasses All hats are to be removed 3 All book bags should be closed and placed at the front of the classroom Do not reach into your bag for anything during the exam If you need extra pencils pull them out NOW 4 No cells phones Turn them off now If you are seen with a cell phone in hand during the exam it will be considered cheating and you will be asked to leave This includes using it as a time piece 5 No iPods or MP3s players etc Same rules as with cell phones 6 If you have a question raise your hand and a proctor will come to you If you have to use the bathroom do so NOW You will not be permitted to leave the room and return during the exam 7 Every exam is worth a total of 100 points Check to see that you have all of the pages Including this cover sheet each exam has 7 sides Each problem is worth 16 points You will receive 4 points for your name and section 8 Be sure to print your proper name clearly at the top of this page Also circle the section for which you are registered 9 If you finish early quietly and respectfully get up and hand in your exam 10 When time is up you will be instructed to put down your writing utensil close your exam and remain seated Anyone seen continuing to write after time is called will have their exam marked and lose all points on the page they are writing on 11 To ensure that you receive full credit show all of your work 12 Good luck You have 60 minutes to complete this exam Problem Problem Problem Problem Problem Problem Problem Problem Total 1 2 3 4 5 6 7 8 Points 12 12 12 12 12 12 12 12 12 1 1 Compute the following limits or state that they do not exist x2 x 2 a lim 2 x 2 x x 6 3x3 4x2 2 b lim x 2x 1 3 a x2 x 2 x 1 x 2 lim x 2 x2 x 6 x 2 x 3 x 2 x 1 lim x 2 x 3 2 1 2 3 3 5 lim b 3x3 4x2 2 3x3 4x2 2 lim x x 8x3 12x2 6x 1 2x 1 3 lim 3x3 4x2 2 lim x 8x3 12x2 6x 1 3 x4 x23 lim x 8 12 62 x x 3 0 0 8 0 0 0 3 8 2 1 x3 1 x3 1 x3 2 Charlotte s Sweet Shoppe specializes in producing boxes of chocolate covered toffee Each box costs 4 to produce and can be sold for 10 Assume that Charlotte s has fixed daily overhead costs of 150 a Write a function for Charlotte s total daily cost if she produces x boxes of chocolate covered toffee b Assuming that Charlotte sells every box of chocolate covered toffees that she produces what is the daily revenue function c How many boxes of toffee will she need to sell each day in order to break even a Cost 150 4x b Revenue 10x c The break even point is the point at which revenue is equal to cost We have Cost Revenue 150 4x 10x 150 6x 25 x So she needs to produce 25 boxes of chocolate covered toffee each day in order to break even 3 3 a Which of the following is the correct definition of the derivative of f x f x h f x h 0 h i f 0 x lim f x h f x h 0 h ii f 0 x lim iii f 0 x f x h f x h f x h f x x 0 x iv f 0 x lim b Use the definition of the derivative to compute f 0 x where f x x2 3x f x h f x h 0 h x h 2 3 x h x2 3x lim h 0 h 2 2 x 2xh h 3x 3h x2 3x lim h 0 h 2 2xh h 3h lim h 0 h lim 2x h 3 f 0 x lim h 0 0 f x 2x 3 4 4 A concert promoter is trying to decide what price to charge per ticket She finds that she can sell 1000 tickets at 50 each For every 1 increase in ticket price she will sell 10 fewer tickets Let x be the number of 1 price increases a Express p the price of each ticket as a function of x p 50 x b Find an expression for n the number of tickets sold after x price increases n 1000 10x c Find an expression for the total revenue after x price increases Revenue tickets sold price per ticket R 1000 10x 50 x 50 000 500x 10x2 d What value of x maximizes revenue The revenue will be maximized at the vertex of the parabola 500 500 b 25 2a 2 10 20 So the revenue is maximized at x 25 e What price should she charge for each ticket to maximize revenue How many tickets will she sell at that price When x 25 p 50 25 75 and n 1000 25 10 750 f What will her maximum revenue be When x 25 we have R 25 1000 10 25 50 25 750 75 56250 5 5 Find all values k so that f x is continuous for all real values of x x2 2x if x 2 f x 3 2 x kx 4 if x 2 Since both pieces of our piecewise defined function are polynomials the only possible point of discontinuity occurs when we transfer from the first function definition to the second So long as our one sided limits at this point align we will have continuity Thus we compute lim f x lim f x x 2 2 x 2 lim x 2x lim x3 kx2 4 x 2 2 x 2 3 2 2 2 2 k 22 4 8 12 4k 4 4k 1 k So our function is continuous everywhere when …