Math 19 Practice Final B Winter 2011 SUID Name Complete the following problems In order to receive full credit please show all of your work and justify your answers You need to include the units in your answers when relevant You do not need to simplify your answers unless specifically instructed to do so You may use any result from class that you like but if you cite a theorem be sure to verify the hypotheses are satisfied You have 3 hours This is a closed book closed notes exam No calculators or other electronic aids will be permitted If you finish early you must hand your exam paper to a member of teaching staff Please check that your copy of this exam contains 14 numbered pages and is correctly stapled If you need extra room use the back sides of each page If you must use extra paper make sure to write your name on it and attach it to this exam Do not unstaple or detach pages from this exam Please sign the following On my honor I have neither given nor received any aid on this examination I have furthermore abided by all other aspects of the honor code with respect to this examination Signature The following boxes are strictly for grading purposes Please do not mark Question 1 2 3 4 5 6 7 8 9 10 11 Total Points 10 10 8 11 25 14 8 14 10 15 12 137 Score Math 19 Winter 2011 Practice Final B Winter 2011 Page 1 of 14 1 10 points Find each of the following limits with justification show steps If there is an infinite limit then explain whether it is or 2 x 3x 2 5 a 5 points lim x 1 3x2 21x 18 p b 5 points lim 4x2 x 2x x Math 19 Winter 2011 Practice Final B Winter 2011 Page 2 of 14 2 10 points a 3 points What are the critical numbers of a function f x b 8 points Find the global maximum and the global minimum of the function f x x 2 e x on the domain 0 10 Math 19 Winter 2011 Practice Final B Winter 2011 Page 3 of 14 3 8 points Mark each statement below as true or false by circling T or F No justification is necessary T F The function f x 2x ln x is one to one T F If lim f x and lim g x then lim f x g x 0 T F The function ln x is even T F The function x 7 is differentiable everywhere T F If f 00 1 0 then f 0 x is increasing at x 1 T F If f 0 2 0 then f x has a local maximum or a local minimum at x 2 T F If two functions have the same derivative then they must be equal T F The derivative of ln x3 is x 0 x 0 x 0 3 x Math 19 Winter 2011 Practice Final B Winter 2011 4 11 points For the function f x x 2 x 2 3 3x2 5x 3 if x 2 if x 2 if x 2 a 6 points Find lim f x if it exists x 2 b 3 points Is f x continuous at x 2 Justify your answer Page 4 of 14 Math 19 Winter 2011 Practice Final B Winter 2011 5 25 points Consider the function f x Page 5 of 14 x 1 2 x2 1 a 2 points Find the domain of f x b 5 points Where is f x increasing And decreasing Find the x coordinates of all the local maxima and minima of f x 4x3 12x Where is f x concave up And concave down x2 1 3 Find the x coordinates of all the inflection points of f x c 5 points It is known that f 00 x Math 19 Winter 2011 Practice Final B Winter 2011 d 5 points Find all the asymptotes of f x e 2 points Find the y intercept of f x and the x intercepts if there are any Page 6 of 14 Math 19 Winter 2011 Practice Final B Winter 2011 f 6 points Sketch the graph of f x in the axes provided below Page 7 of 14 Math 19 Winter 2011 Practice Final B Winter 2011 Page 8 of 14 6 14 points Find the derivatives of the following functions using any method you like that works You do not need to simplify your answers x a 4 points f x ln x4 5 e e b 6 points g x arcsin x x c 4 points Z x e1 x x2 Math 19 Winter 2011 Practice Final B Winter 2011 7 8 points We know that g 0 3 2 and g 3 4 Estimate the value of mation Please simplify your answer Page 9 of 14 1 using linear approxig 3 1 Math 19 Winter 2011 Practice Final B Winter 2011 Page 10 of 14 8 14 points A student can not use a calculator for the final exam so every time he has to do a calculation with he uses 3 instead Assume that the error is 0 15 in this quantity then a 6 points Using differentials estimate the maximum error in his computation of the area of a circle of radius 4 cm 2 b 8 points He needs to know which number is bigger 2 or 1 6 so he changes by 3 and gets to the conclusion that the second number must be bigger Using differentials find the possible error in each of these approximations and explain why his conclusion may not be true Hint The real number will be in the interval centered at the value computed using 3 and with length given by twice the propagated error Math 19 Winter 2011 Practice Final B Winter 2011 Page 11 of 14 9 10 points Find the coordinates x y of the two points in the curve x2 xy y 2 1 where the tangent line is horizontal Math 19 Winter 2011 Practice Final B Winter 2011 Page 12 of 14 10 15 points A company has three processing plants near a city One of them is located 1 km north of the city another 1 km south of the city and the third one is located 3 km east of the city The company plans to construct a distribution center at some point in between the city and the third factory or even at the city or at the third factory Where should the center be located in order to minimize the sum of the distances to the three processing plants Math 19 Winter 2011 Practice Final B Winter 2011 Page 13 of 14 11 12 points A certain population of rodents numbers P 100 30 3t 4 32t after t years a 1 point What was the original population b 6 points What is the rate of growth of the population after 1 year And after 2 years Interpret the sign of your …