Math 234 Spring 2015 Exam 1 Version 1Monday, February 16th, 2015Name: UIN:Circle the section you are registered for:ADA (Vanessa 9AM) ADB (Vanessa 10AM) ADC (Chris 11AM) ADD (Chris 12PM)ADE (Mingyu 1PM) ADF (Mingyu 2PM) ADG (Derek 3PM) ADH (Derek 4PM)BDA (Mary 9AM) BDB (Mary 10AM) BDC (Ian 11AM) BDD (Matthew 12PM)BDE (Matthew 1PM) BDF (Ian 2PM) BDG (Corrine 3PM) BDH (Corrine 4PM)CDA CDB (Shiyu 9AM) CDC (Shiyu 11AM) CDD (Panu 12PM)CDE (Panu 1PM) CDF (Neriman 2PM) CDG (Neriman 3PM) CDH (Joseph 11AM)(1) No baseball caps, hoodies, etc. or dark sunglasses. All hats are to be removed.(2) All bags, purses, jackets, and other personal items should be at the front of the room.(3) If you have a question, raise your hand and a proctor will come to you. If you haveto use the bathroom, do so NOW. You will not be permitted to leave the room andreturn during the exam.(4) No cells phones, i-Pods, MP3 players. Turn them off now. If you are seen these itemsin 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) Be sure to print your proper name clearly at the top of this page. Also circle thesection for which you are registered.(6) If you finish early, quietly and respectfully get up and hand in your exam.(7) When time is up, you will be instructed to put down your writing utensil, close yourexam and remain seated. Anyone seen continuing to write after time is called willhave their exam marked and lose all points on the page they are writing on.(8) To ensure that you receive full credit, show all of your work.(9) Good luck. You have 60 minutes to complete this exam.1(1) During the summer, a group of students knits sweaters for penguins in a convertedgarage. The rental for the garage is $100 for the summer, and the materials neededto knit 1 penguin sweater cost $2. The sweaters can be sold for $4 apiece.(a) How many sweaters must the students sell to break even?(b) How many sweaters must the students sell to make a profit of at least $500?(c) After working hard and selling 100 penguin sweaters, the students take theirprofit and invest it with the All Creatures Great and Small Credit Union. Thecredit union offers them a 5% interest rate compounded quarterly. How longmust they keep the money in the account in order for it to at least double?2(2) Find the following limits or state they do not exist. If the limit is infinite, specify ifit tends towards +∞ or −∞.(a) limx→1x3− 2x7+ 2x − 1(b) limx→1−1x − 1(c) limx→2x2− x − 6x2− 4(d) limx→2x2− 5x + 6x2− 4(e) limx→−∞x2− 3x + 14 − 2x23(3) (a) For a function f(x), write down the definition for the derivative of f (x).f0(x) =(b) Use the limit definition of derivative to find f0(x) when f (x) = 3x2− 2x + 1.Show all work.(c) Find f0(1) and interpret it graphically.(d) When x =1, is the function f(x) increasing, decreasing, or neither? Explain why.(e) A function has maximum/minimum at x=a when f0(a) = .....? Use this fact tofind the x-coordinate where f (x) has a maximum/minimum.4(4) As advances in technology result in the production of increasingly powerful cellphones, the price of cell phones currently on the market drops. Suppose that xmonths from now, the price of a certain cell phone model will be P dollars per unit,whereP (x) = 100 +30x(a) What will the price be 3 months from now?(b) By how much will the price drop during the third month?(c) What happens when x gets larger and larger? Interpret your result.(d) Find the instantaneous rate of change of price in 1 month.5(5) The boutique Hat’s It! can special order a certain fancy hat from the supplier at acost of $2 per hat. The store has been selling the hat at the price of $15 per hat,and at this price, has been selling 10 hats a month. (Since this hat is a special order,the number of hats they buy from the supplier is equal to the number of hats theysell). They are planning to lower the hat price to stimulate sales, and estimate thatfor each $1 reduction in the price, 5 more hats will be sold each month.(a) Let x denote the number of $1 price decreases to the hat. Find an expression,p(x), for the price of the hat after x-many price decreases.(b) Find an expression, n(x), for the number of hats sold after x-many price de-creases. (n(x) also equals the number of hats purchased after x-many pricedecreases).(c) Find an expression, R(x), for the total revenue after x price decreases.(d) Find an expression, P (x), for the total profit after x price decreases.(e) Suppose you find that P (x) = 100 + 20x − 2x2. When is profit maximized? (i.efor what price should you sell the hat to maximize