Farmer Dan Field loves his job! He has been a farmer for his entire life! The farm has been passed down from one generation to the next. The hilly farms have been in the family for 7 generations. Every time the farm is passed down, the next farmer has the farm for a longer amount of time because of the increased life expectancy. The amount of time each farmer owns the farm can be represented by y=√x. Where x represents the years since 1800 and y represents how long each farmer owns the farm. Every day he enjoys mowing the fields and planting new crops. He farms on many hills which can be defined as functions. It is always up and down for Farmer Dan.

Farmer Dan has been mysteriously losing his goats. He used to have 200 goats and now he only has 50. This is represented by the integral f(t) = 200e^(-kt). Farmer Dan thinks that the goats could be going missing for a multitude of reasons. His first hypothesis is that the goats have been killed by the local foxes. He is also suspicious that his rival neighbor, Farmer Billy Joe, has been stealing the goats for his own gain. Farmer Billy Joe and Farmer Dan have had a rivalry for the past seventeen years, since they were little boys. 

One day, Farmer Dan walked into town to go to the town meeting. He saw Farmer Billy Joe on his way and decided to confront him. Farmer Dan said, "Billy Joe! I know you have been stealing my goats. You are the worst neighbor ever. You are worse than an undefined function!". After getting yelled at, Farmer Billy Joe responded in defense, "I ain't no thief. I did not steal your goats, you crazy lunatic. In fact, you have been stealing my goats. I know that you are jealous of me because your goat places second to mine every year at the county fair."

"I am definitely not stealing your goats!" said Farmer Dan. 

As the accusations continued, Farmer Dan and Farmer Billy Joe resulted to the only thing they could think of. They realized the only way to solve their problems was to compete in a good ole country math competition. They needed someone to conduct the competition, who would have no bias. So they plotted the age of every person in their small town vs. the amount of land they owned to find the slope to be Y=2x+.06. They then chose a random point on the plotted graph and used that information to find that Sara Lee would be their tester.

Farmer Dan and Farmer Billy Joe took off their hats to prepare for the ultimate math battle. Sara Lee gave them each a page of math problems. There was the derivative of f(x) = 10x^2+5x+C at x = 1 questions on the page. She stated the rules: "Alright boys, whoever can solve all of these math problems first gets to keep all of the goats." Farmer Billy Joe said, "Good thing I was a mathlete in high school." Then Farmer Dan snapped back, "I bet you didn't win math student of the year though."

At the end of the competition, once the timer had run out, both Farmer Dan and Farmer Billy Joe were feeling confident. Sara Lee said "Time is up boys hand me your papers." They went to give her their answer sheets and all of a sudden there was a big gust of wind with the speed of p(t) = d/dt (3t^2 +2t+5) at v(2). This wind completely startled the farmers! The papers flew out of their hands. As they watched the papers fly away, they also lost all hope of finding a winner in the competition. Both Farmer Dan and Farmer Billy Joe screamed, "Noooooo! I was totally going to win."

Although it seemed like there was no way to solve their problems, Sara Lee had an idea. She exclaimed, "Wait! We can do one final tie breaker to decide who wins this competition. I will ask one question and whoever answers correctly will win all of the goats!"

"That works for me! I've been a genius at math my entire life! There is no way Farmer Billy Joe can answer before me!" Farmer Dan said very confidently. 

"Yeah right! I have always been better than you in everything that we do! There is no way Farmer Dan could beat me in a math competition, let alone any competition!" Farmer Billy Joe exclaimed back. 

Sarah Lee stated the final question, "If the limit of x as it approaches 0 of 3x^2+4x-4 equals 8, what is the limit f(x) as it approaches 0 equal?"