MATH 1A - MIDTERM 3PEYAM RYAN TABRIZIANName:Instructions: This midterm counts for 20% of your grade. You officiallyhave 110 minutes to take this exam (although I will try to give you moretime). This is a fairly long exam, so don’t spend too much time on eachquestion! May your luck be maximized :)Note: Please check the following box if it applies to you:Today is the last day to change your grade to P/NP and vice-versa.Please check this box if your decision of changing your grade toP/NP depends on the score you’ll receive on this exam, and youwould like to have this exam graded by 5 pm (please be honest)1 252 103 154 105 106 107 20Bonus 1 5Bonus 2 5Total 100Date: Friday, July 29th, 2011.12 PEYAM RYAN TABRIZIAN1. (25 points) Sketch a graph of the function f(x) = x ln(x) −x. Yourwork should include:- Domain- Intercepts- Symmetry- Asymptotes (no Slant asymptotes, though)- Intervals of increase/decrease/local max/min- Concavity and inflection pointsNote: You may use the axes provided on page 4 to draw yourgraph! Make sure to label all important points!Hint: For the Asymptotes-part, it might help to notice thatf(x) = x(ln(x) − 1)Hint: There should be a nice simplification when you calculatef0(x). If there’s no simplification, then you made a differentiationmistake!MATH 1A - MIDTERM 3 3(This page is left blank in case you need more space to work onquestion 1.)4 PEYAM RYAN TABRIZIAN1A/Math 1A Summer/Exams/Axes.pngMATH 1A - MIDTERM 3 52. (10 points) Use a linear approximation (or differentials) to find anapproximate value of√996 PEYAM RYAN TABRIZIAN3. (15 points) Assume the radius of a cone is increasing at a rate of 3cm/s while its height is decreasing at a rate of 1 cm/s. At what rateis its volume increasing/decreasing when its radius is 2 cm and itsvolume is4π3cm3?Note: If you don’t remember the formula for the volume of acone, do this problem with a cylinder instead (14 points max). Ifyou’re still stuck, do it with a triangle instead (11 points max).MATH 1A - MIDTERM 3 74. (10 points) Find the following limits:(a) limx→0x2ln(x)(b) limx→∞x1x8 PEYAM RYAN TABRIZIAN5. (10 points) Find the absolute maximum and minimum of f on [0, 2],where:f(x) = x4− 4x + 1MATH 1A - MIDTERM 3 96. (10 points) Show that if f0(x) > 0 for all x, then f is increasing.Hint: Assume b > a and show that f(b) > f(a)10 PEYAM RYAN TABRIZIAN7. (20 points) Find the dimensions of the rectangle of largest area thatcan be inscribed in (put inside of) a circle of radius 1.Note: If you’re completely stuck, then you can do problem 8 onpage 12 instead, for a maximum of 12 points.MATH 1A - MIDTERM 3 11(This page is left blank in case you need more space to work onproblem 7.)12 PEYAM RYAN TABRIZIAN8. (ONLY do this one if you got completely stuck on problem 7.)If 12 cm3of material is available to make a box with a squarebase and an open top, find the largest possible volume of the box.MATH 1A - MIDTERM 3 131A/Practice Exams/Snake.jpg14 PEYAM RYAN TABRIZIANBonus 1 (5 points) Show that f (x) = ln(x) −x does not have a slant asymp-tote at ∞.Hint: Assume f(x) has a slant asymptote y = mx + b at ∞.Calculate m, then calculate b, and find a contradiction!MATH 1A - MIDTERM 3 15Bonus 2 (5 points) Assume −1 < f (x) < 1 and f0(x) 6= 1 for all x. Showthat f has exactly one fixed point.Definition: a is a fixed point of f if f(a) = aHints:At least one fixed point: Show that g(x) = f(x) − x has at leastone zero on [−1, 1].At most one fixed point: Assume f has 2 fixed points a and b,then f(a) = a and f (b) = b, and find a contradiction!Note: See the comments on the next page!16 PEYAM RYAN TABRIZIANDiscussion of Bonus 2:Bonus 2 is part of a more general theorem, the Brouwer fixedpoint theorem. It states that any function f with domain B(0, 1)(the open ball of center 0 and radius 1 in Rn) and range B(0, 1)has at least one fixed point. (in the previous problem, B(0, 1) =(−1, 1)). Moreover, if f0(x) 6= 1, then f has exactly one fixed point.Here are some cool applications of this theorem:(1) No matter how well you shake a snowglobe, then there willalways be one snowflake which lands on exactly same positionit started!(2) If you stir a cocktail glass, then there is always one moleculewhich never changes position. And in most cases there is onlyone such molecule (unless you’re rotating the glass).(3) Suppose there is a hurricane in New York, and everyone getssweeped to a different place. Then there is one lucky personwho gets sweeped to the same place he/she started!(4) Take an ordinary map of a country, and suppose that that mapis laid out on a table inside that country. There will always bea ‘You are Here” point on the map which represents that samepoint in the country.(5) Have you ever looked at two mirrors that are across from eachother? There seems to be an infinite number of smaller andsmaller mirrors! However, there is always one point on allthose mirrors which has always the same height (roughly atyour belly button).(6) Brouwer’s fixed point theorem is used to prove the fundamen-tal theorem of differential equations (namely that differentialequations have solutions), as well as the implicit function the-orem.MATH 1A - MIDTERM 3 17Any comments about this exam? (too long? too