The I found that x=was equal to 1958.

The steps in calculating the first derivative are as follows. I first applied the sum rule to split the equation into two parts, so that I could differentiate each part. Then, I applied the constant multiple rule to the first part of the equation, which was a(x). I then, applied the chain rule along with a trig derivative also to a(x), and then applied power rule and constant rule which then gave me the final derivative equation as seen in figure 11.

              Therefore, this equation indicates what the temperature’s rate of change is between the years 1943 and 2016. Like I did with the previous equation I decided to calculate when the rate of change was at zero between this time period by hand.

 

              Solving this equation also proved to be very difficult. Again I used the same approach this time however, had to calculate what k had to find the solution to the equation during the time period I was looking at. Therefore, after solving for x and plugging in 27 for k I found that x=was equal to 1958. This indicates the year at which the temperature’s rate of change is at zero. Again, I decided that I would graph this equation’s first derivative equation on Desmos in order to check and see if my calculations were correct.The steps in calculating the first derivative are as follows. I first applied the sum rule to split the equation into two parts, so that I could differentiate each part. Then, I applied the constant multiple rule to the first part of the equation, which was a(x). I then, applied the chain rule along with a trig derivative also to a(x), and then applied power rule and constant rule which then gave me the final derivative equation as seen in figure 11.

              Therefore, this equation indicates what the temperature’s rate of change is between the years 1943 and 2016. Like I did with the previous equation I decided to calculate when the rate of change was at zero between this time period by hand.

 

              Solving this equation also proved to be very difficult. Again I used the same approach this time however, had to calculate what k had to find the solution to the equation during the time period I was looking at. Therefore, after solving for x and plugging in 27 for k I found that x=was equal to 1958. This indicates the year at which the temperature’s rate of change is at zero. Again, I decided that I would graph this equation’s first derivative equation on Desmos in order to check and see if my calculations were correct.