Written Homework 4 2 5 2 6 2 7 Page 1 of 3 Math 140 PRINT YOUR NAME From the questions below please choose and solve 5 problems only The homework is worth 10 points Each question is worth 2 points Show all of your work and put a box around your nal answer Number each attempted question clearly Write legibly that is suitably large and suitably dark if the grader can t read your answer it s consider uncompleted Question 1 For each part below calculate the limit or determine that it does not exist If a limit is or say so Show all necessary steps Question 2 Assume that lim x 2 f x 2 For each part below calculate the limit or determine that it does not exist If a limit is or say so Show all necessary steps a lim x 6 b lim x 6 c lim x 6 x2 36 x 6 x 36 x 6 2 x2 36 x 6 a lim x 2 b lim x 2 c lim x 2 cid 0 f x cid 1 4 1 cid 0 f x 2 cid 0 3x x2f x cid 1 d lim x 2 e lim x 2 f lim x 2 f x x 2 f x x 2 f x x 2 Question 3 For each of the following draw the graph of a function on 0 5 with the given properties a f is not continuous at x 1 but lim x 1 b f has a jump discontinuity at x 2 and lim x 2 f x and lim x 1 f x f 2 f x exist and are equal c f has a removable discontinuity at x 1 a jump discontinuity at x 2 and Question 4 Consider the function lim x 3 f x lim x 3 f x 2 f x x2 3 10 x 11 6x x2 x 1 1 x 2 x 2 x 2 a For which value s of a does lim x a f x NOT exist Math 140 Written Homework 4 2 5 2 6 2 7 Page 2 of 3 b For which value s of b is f x NOT continuous at x b Question 5 For each of the given functions nd the value of the constant c which would make the function Question 6 Evaluate each limit Write all of your steps and show all necessary details c Draw the graph of f continuous for all values of x cid 40 a x2 c 4x 2c x 5 x 5 b 5 sin x x ex cid 112 x 0 x2 c x 0 a lim x 1 x 1 cos cid 18 1 cid 19 x 1 b lim x 2 x cos x cid 17 sin cid 16 x ln 5x2 7 c lim x d lim x 2 e lim x f lim x 2 g lim x ln 5x2 7 cid 2 ln 5x2 7 ln 3x2 2 cid 3 cid 2 ln 5x2 7 ln 3x2 2 cid 3 cid 17 cid 17 cid 16 1 sin2 cid 16 x h lim x 0 x Question 7 According to the Michaelis Menten equation when an enzyme is combined with a substrate of concen tration s in millimolars the reaction rate in micromolars min is R s As K s for constants A and K a Show by computing lim s increases without bound is K millimolars be R s that A is the limiting reaction rate as the substrate concentration b Show that the reaction rate attains one half of the limiting value when the substrate concentration Question 8 Ecologists predict the population of white tailed deer in the state of Kentucky t years after 2004 will P t 1 072 764e0 2311t 0 19196 e0 2311t According to the ecologists model what is the long term population of white tailed deer Question 9 After an oil spill in the Gulf of Mexico it is estimated that the concentration of oil in the water will total C t 20 85e t 10 mg per liter where t is the number of years after the spill a A safe concentration for the environment is considered to be 30 mg per liter How long will it take before the concentration levels reach this amount b What is the concentration of oil in the water expected to be in the long term Math 140 Written Homework 4 2 5 2 6 2 7 Page 3 of 3 Question 10 Because the form is indeterminate knowing that functions f and g satisfy lim x f x lim x g x tells us very little about lim x f x g x a Find formulas for f and g such that lim x is an form that has a limit of b Find formulas for f and g such that lim x is an form that has a limit of 0 c Find formulas for f and g such that lim x is an form that has a limit of 1 d Find formulas for f and g such that lim x is an form that has a limit of 13 72 Hint you can answer each part by choosing f and g to be appropriate polynomials f x g x f x g x f x g x f x g x Question 11 Because the form 0 is indeterminate knowing that functions f and g satisfy lim x f x 0 lim x g x tells us very little about lim x f x g x a Find formulas for f and g such that lim x b Find formulas for f and g such that lim x c Find formulas for f and g such that lim x d Find formulas for f and g such that lim x f x g x is a 0 form that has a limit of f x g x is a 0 form that has a limit of 0 f x g x is a 0 form that has a limit of 1 f x g x is a 0 form that has a limit of 13 72 Hint you can answer each part by choosing f x and g x to each be of the form Cxp for appro priate constants C and p