MATH 1A - MOCK MIDTERM 3PEYAM RYAN TABRIZIANName:Instructions: This is a mock midterm, designed to give you an idea ofwhat the actual midterm will look like. Make sure you do it, the actual examwill be very similar to this one (in length and in difficulty)! Remember thatthis midterm is 2 hours long!Warning: The questions on the actual midterm may look completely dif-ferent from the questions on this mock midterm, although the basic structurewill be the same, so make sure to study the problems from the related ratesand optimization handouts as well!1 252 103 154 105 106 107 20Bonus 1 5Bonus 2 5Total 100Date: Friday, July 22nd, 2011.12 PEYAM RYAN TABRIZIAN1. (25 points) Sketch a graph of the function f(x) = e−x22. Your workshould include:- Domain- Intercepts- Symmetry- Asymptotes (no Slant asymptotes, though)- Intervals of increase/decrease/local max/min- Concavity and inflection pointsNote: This function is used a lot in statistics, it is called the normaldistribution function.MATH 1A - MOCK MIDTERM 3 3(This page is left blank in case you need more space to do ques-tion 1.)4 PEYAM RYAN TABRIZIAN2. (10 points) Use linear approximations (or differentials) to find anapproximate value of (3.01)3MATH 1A - MOCK MIDTERM 3 53. (15 points) Two people start moving from the same point. One per-son travels north at a speed of 3 mph and the other person travelseast at a speed of 4 mph. At what rate is the distance between thetwo people changing after 2 hours?6 PEYAM RYAN TABRIZIAN4. (10 points) Evaluate the following limits:(a) limx→0x+sin(x)1+cos(x)(b) limx→−∞xex(c) limx→0xxMATH 1A - MOCK MIDTERM 3 75. (10 points) Find the absolute maximum and minimum of f (x) =x3− 3x on [0, 2]8 PEYAM RYAN TABRIZIAN6. (10 points) Suppose f is an odd function and is differentiable every-where. Let b be given. Show that there is a number c in (−b, b) suchthat f0(c) =f(b)bHint: Let a = −bMATH 1A - MOCK MIDTERM 3 97. (20 points) Find the point on the line x + y = 1 that is closest to thepoint (−3, 1).10 PEYAM RYAN TABRIZIANBonus 1 (5 points) Use l’Hopital’s rule to show:limh→0f(x + h) − 2f (x) + f (x − h)h2= f00(x)Note: Careful! You’re differentiating with respect to h here, notx !MATH 1A - MOCK MIDTERM 3 11Bonus 2 (5 points) (Courtesy Adi Adiredja)Omar didn’t get the girl. Sad and frustrated he decided to runaround the track at the gym. When he got to the track, one girl whowas about to start running captured his attention. Omar hurried sothey could start running together. They started at the same time, butthen she started running faster. Omar sped up, but then she passedhim again. Next thing he knew, the two of them were racing. Atsome point she noticed the finish line, and yelled out,“I dont go forlosers!” Omar ran as fast as he could and the race ended in a tie. Af-ter the race she confidently came up to him and said, “I dont meanto be rude, but I only value smart guys. If you can prove the nextthing I say, I’ll go on a date with you.”She then said, “We started the race at the same time, and the raceended in a tie, I claim that at some point during the race we wererunning at the same speed.” She smiled at Omar and said, “Are yousmart enough?” Use Calculus to help Omar!Hint: Let g(t) and h(t) be the position functions of the two run-ners, and consider f(t) = g(t) −
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