Math 140 Written Homework 5 2 8 3 1 3 2 Page 1 of 7 From the questions below please solve ALL 10 problems The homework is worth 10 points Each question is worth 1 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 f x x3e2x 1 write the limit de nition of f cid 48 4 Don t simplify or evaluate the limit For example if g x x3 you would write and that s it g cid 48 1 lim h 0 1 h 3 13 h Question 2 Suppose that f is a function given by f x x 5 x1 5 2x 62 if x 1 if 1 x 32 if x 32 a At which x values does f not have a limit b At which x values is f not continuous c At which x values is f not di erentiable Math 140 Written Homework 5 2 8 3 1 3 2 Page 2 of 7 Question 3 Some people who don t appreciate the ner things in life brew co ee in a percolator which results in subpar co ee since the balance of compounds is not ideal when extraction occurs at the boiling point of water As a connoisseur you prefer to brew your co ee at 93 C You pour over your co ee at t 0 where t is in minutes eventually transfer it to your insulated mug and nish drinking it after one hour The temperature of your co ee T f t in degrees Celsius at various times is given below t T f t 0 93 10 74 20 59 30 49 40 44 50 42 60 41 Assume that f t is a decreasing function on its domain 0 t 60 a Estimate f cid 48 20 and include units b Write a sentence explaining the practical interpretation of the statement f 35 47 c Write a sentence explaining the practical interpretation of the statement f 1 43 44 d Write a sentence explaining the practical interpretation of the statement f cid 48 10 1 7 Math 140 Written Homework 5 2 8 3 1 3 2 Page 3 of 7 Question 4 For the given graph of the function f on the left sketch the graph of of f cid 48 on the right 4 2 2 4 4 2 2 4 Question 5 For the given graph of the function f on the left sketch the graph of of f cid 48 on the right 4 2 2 4 4 2 2 4 Question 6 For the given graph of the function f on the left sketch the graph of of f cid 48 on the right 4 2 2 4 4 2 2 4 2 4 4 2 4 2 4 2 2 4 2 4 2 4 4 2 4 2 4 2 2 4 2 4 Math 140 Written Homework 5 2 8 3 1 3 2 Page 4 of 7 Question 7 The spread of a virus is modeled by V t t3 t2 12t where V t is the number of people in hundreds with the virus and t is the number of weeks since the rst case was observed a Sketch V t b What is a reasonable domain of t for this problem c Find the average rate of infection from t 0 to t 2 d Find the instantaneous rate of infection as a function of t using the limit de nition of the derivative e Find V 2 and V cid 48 2 Write a sentence interpreting V 2 and V cid 48 2 in terms of the number of infected people Make sure to include units f Sketch the tangent line to the graph you drew in a at the point 2 V 2 State the slope of the tangent line g Use V 2 and V cid 48 2 to estimate the value of V 2 1 h Find the maximum number of people infected at the same time and when the maximum occurs Determine the rate of infection at this time Math 140 Written Homework 5 2 8 3 1 3 2 Page 5 of 7 Question 8 Let P t be a deer population in thousands of an ecological region t years after 2000 A graph of the derivative P cid 48 t is give below Use it to answer the following questions P cid 48 6 4 2 2 2 4 6 8 10 12 14 t a Calculate P cid 48 8 and interpret that in terms of the deer population b Indicate the years during which the deer population declined c Order the following from least to greatest If there is a tie indicate that with an equals sign You do not need to nd the values themselves P cid 48 1 P cid 48 3 P cid 48 6 P cid 48 7 P cid 48 9 P cid 48 10 d Order the following from least to greatest If there is a tie indicate that with an equals sign You do not need to nd the values themselves P 0 P 2 P 3 P 5 P 7 P 9 e Order the following from least to greatest If there is a tie indicate that with an equals sign You do not need to nd the values themselves P 9 P 11 P 14 P 15 f Evaluate the limit lim h 0 P cid 48 2 h P cid 48 2 h Math 140 Written Homework 5 2 8 3 1 3 2 Page 6 of 7 Question 9 From the above graph of y f x nd all x value s at which f is not di erentiable Math 140 Written Homework 5 2 8 3 1 3 2 Page 7 of 7 Question 10 Use the Intermediate Value Theorem to show that there is a solution of the equation x3 4x 1 in 1 0