Question

The average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-days year. The equation that approximates the temperature on day 𝑥 is
𝑦 = 37𝑠𝑖𝑛 [(2𝜋/365)(𝑥 − 101)] + 25
i. On what day is the temperature increasing the fastest?
ii. About how many degrees per day is the temperature increasing when it is increasing at the fastest?

Answer 1

Think about the most basic concepts of Calculus here.
For some given function f(x), f ' (x) gives you the rate of change
If you want the maximum of the function, you would set f ' (x) = 0
So if you want the maximum of the rate of change, you would set f '' (x) = 0

So find the second derivative, set it equal to zero, and solve for x

for b) sub that value of x into the 2nd derivative

Here is a graph of the original function, you can use to see if you answers make sense
(I graphed beyond the one year of 365 days)

https://www.wolframalpha.com/input/?i=y+%3D+37sin%28%282%CF%80%2F365%29%28x+%E2%88%92+101%29%29+%2B+25+from+0+to+500