MATH 1A MIDTERM 3 PEYAM RYAN TABRIZIAN Name Instructions This midterm counts for 20 of your grade You officially have 110 minutes to take this exam although I will try to give you more time This is a fairly long exam so don t spend too much time on each question 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 to P NP depends on the score you ll receive on this exam and you would like to have this exam graded by 5 pm please be honest 1 2 3 4 5 6 7 Bonus 1 Bonus 2 Total 25 10 15 10 10 10 20 5 5 100 Date Friday July 29th 2011 1 2 PEYAM RYAN TABRIZIAN 1 25 points Sketch a graph of the function f x x ln x x Your work should include Domain Intercepts Symmetry Asymptotes no Slant asymptotes though Intervals of increase decrease local max min Concavity and inflection points Note You may use the axes provided on page 4 to draw your graph Make sure to label all important points Hint For the Asymptotes part it might help to notice that f x x ln x 1 Hint There should be a nice simplification when you calculate f 0 x If there s no simplification then you made a differentiation mistake MATH 1A MIDTERM 3 3 This page is left blank in case you need more space to work on question 1 4 PEYAM RYAN TABRIZIAN 1A Math 1A Summer Exams Axes png MATH 1A MIDTERM 3 5 2 10 points Use a linear approximation or differentials to find an approximate value of 99 6 PEYAM RYAN TABRIZIAN 3 15 points Assume the radius of a cone is increasing at a rate of 3 cm s while its height is decreasing at a rate of 1 cm s At what rate is its volume increasing decreasing when its radius is 2 cm and its volume is 4 cm3 3 Note If you don t remember the formula for the volume of a cone do this problem with a cylinder instead 14 points max If you re still stuck do it with a triangle instead 11 points max MATH 1A MIDTERM 3 4 10 points Find the following limits a limx 0 x2 ln x 1 b limx x x 7 8 PEYAM RYAN TABRIZIAN 5 10 points Find the absolute maximum and minimum of f on 0 2 where f x x4 4x 1 MATH 1A MIDTERM 3 6 10 points Show that if f 0 x 0 for all x then f is increasing Hint Assume b a and show that f b f a 9 10 PEYAM RYAN TABRIZIAN 7 20 points Find the dimensions of the rectangle of largest area that can be inscribed in put inside of a circle of radius 1 Note If you re completely stuck then you can do problem 8 on page 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 on problem 7 12 PEYAM RYAN TABRIZIAN 8 ONLY do this one if you got completely stuck on problem 7 If 12 cm3 of material is available to make a box with a square base and an open top find the largest possible volume of the box MATH 1A MIDTERM 3 1A Practice Exams Snake jpg 13 14 PEYAM RYAN TABRIZIAN Bonus 1 5 points Show that f x ln x x does not have a slant asymptote 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 15 Bonus 2 5 points Assume 1 f x 1 and f 0 x 6 1 for all x Show that f has exactly one fixed point Definition a is a fixed point of f if f a a Hints At least one fixed point Show that g x f x x has at least one 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 TABRIZIAN Discussion of Bonus 2 Bonus 2 is part of a more general theorem the Brouwer fixed point 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 f 0 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 will always be one snowflake which lands on exactly same position it started 2 If you stir a cocktail glass then there is always one molecule which never changes position And in most cases there is only one such molecule unless you re rotating the glass 3 Suppose there is a hurricane in New York and everyone gets sweeped to a different place Then there is one lucky person who gets sweeped to the same place he she started 4 Take an ordinary map of a country and suppose that that map is laid out on a table inside that country There will always be a You are Here point on the map which represents that same point in the country 5 Have you ever looked at two mirrors that are across from each other There seems to be an infinite number of smaller and smaller mirrors However there is always one point on all those mirrors which has always the same height roughly at your belly button 6 Brouwer s fixed point theorem is used to prove the fundamental theorem of differential equations namely that differential equations have solutions as well as the implicit function theorem MATH 1A MIDTERM 3 Any comments about this exam too long too hard 17