#### Differential Equations

Question1.Does the functionhave a vertical or horizontal asymptote? If so find the equation of the asymptote(s). If not, explicitly state that there are none.

2.Find

3.Find the derivative of

4.Find the derivative ofusing the definition of the derivative (DO NOT USE the rules of differentiation).

1.Find the antiderivative of

2.Find the antiderivative of

3.Find the area under the curve for the functionon the interval [1,2].

4.Your DNA code is composed of a series of four nucleotides: adenine, guanine, thymidine and cytosine (A, G, T and C, respectively).

1.What is the probability an individual has the following nucleotide sequence: “TATATA” at any particular position? You may assume independence.

2.What is the probability that an individual has k T’s in their DNA code at any particular position? (Note: k can be any integer and you may assume independence).

1.A local bank requires that customers must create a password in order to use their online banking services. The password must have a length of 8 and be composed of lower case letters and/or numbers (letters and numbers can be repeated).

1.Suppose someone wants to access your online bank account. How many passwords would they have to try before being able to log into your online bank account?

2.Online security systems frequently tell customers not to use their birthdate as part of their passwords as it makes them more susceptible to hackers. Suppose a customer of the online bank uses part of their birthdate as the last 6 characters of their 8 character password (i.e. birthdate June 23, 1983 converted to 062383). If a hacker knew your birthdate and suspected this information was contained in the last 6 characters of your password, how many passwords would a hacker have to try before being able to log into your online bank account?

1.Professor Karl has n students in her classroom. At the end of the semester she decides to assign the students a group final project. If Professor Karl wants the students to pair into groups of 3, how many ways are there to choose the groups in which the order doesn’t matter (the group Mike, John and Matt is the same as the group Matt, Mike and John).