The average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-days year. The equation that approximates the temperature on day π₯ is
π¦ = 37π ππ [(2π/365)(π₯ β 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?
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
To find when the temperature is increasing the fastest, we need to find the maximum value of the derivative of the equation with respect to x.
Let's start by finding the derivative of the equation:
π = 37sin[(2π/365)(π β 101)] + 25
We can use the chain rule to find the derivative of the equation:
π' = 37 * cos[(2π/365)(π β 101)] * (2π/365)
Now, let's set this derivative equal to zero and solve for x to find the critical points:
π' = 37 * cos[(2π/365)(π β 101)] * (2π/365) = 0
To find when the temperature is increasing the fastest, we need to find the value of π that maximizes the derivative. Since the derivative is equal to zero, we need to find the points where the derivative changes sign.
To find these points, we can solve the equation:
cos[(2π/365)(π β 101)] = 0
To find the solutions, we need to find the values of x that make the cosine of the expression equal to zero. This occurs when π β 101 is equal to either (365/2) or (3*365/2), since cosine is equal to zero at these values.
π β 101 = (365/2) or π β 101 = (3*365/2)
Solving each equation for x, we get:
π = (365/2) + 101 or π = (3*365/2) + 101
Simplifying these equations, we find:
π = 182.5 + 101 or π = 547.5 + 101
π = 283.5 or π = 648.5
Therefore, the critical points occur at π = 283.5 and π = 648.5.
Now that we have found the critical points, we need to determine which one corresponds to the maximum value of the derivative. To do this, we can evaluate the second derivative of the equation at each critical point.
The second derivative can be found by taking the derivative of the first derivative:
π'' = -((2π/365)^2) * 37 * sin[(2π/365)(π β 101)]
Now, we can evaluate the second derivative at π = 283.5 and π = 648.5:
π''(π = 283.5) = -((2π/365)^2) * 37 * sin[(2π/365)(283.5 - 101)]
π''(π = 648.5) = -((2π/365)^2) * 37 * sin[(2π/365)(648.5 - 101)]
We can evaluate these equations to find the second derivative at each critical point. The point where the second derivative is negative corresponds to the maximum value of the derivative, indicating the point where the temperature is increasing the fastest.
Finally, to find how many degrees per day the temperature is increasing when it is increasing at the fastest, we can substitute the value of π from the previous step into the equation for the derivative:
π' = 37 * cos[(2π/365)(π β 101)] * (2π/365)
Substituting π = π_value into this equation will give us the rate of change of temperature when it is increasing the fastest.
Please note that the calculations will depend on the numerical values you use for Ο and the temperature equation constants.