Derivatives

[john3j]

I am not asking anyone to do this for me, just to point me in the right direction or tell me what steps I need to take to solve this.

Meteorology records for a certain city suggest that for the month of June, the daily temperature between midnight and 6:00 P.M. can be approximated by T(t)=-0.04t^3+1.14t^2-7.2t+66 degrees, where t is the number of hours after midnight and 0<=t<=18.

Find the maximum and minimum temperatures.
Find the maximum rate of increase in the temperature.

I know how to find the local minimum and maximums, but I am not sure if that is what is being asked here because I am horrible with word problems. Any help would be greatly appreciated.

Thanks,
John

 

[JeffM]

I am not asking anyone to do this for me, just to point me in the right direction or tell me what steps I need to take to solve this.

Meteorology records for a certain city suggest that for the month of June, the daily temperature between midnight and 6:00 P.M. can be approximated by T(t)=-0.04t^3+1.14t^2-7.2t+66 degrees, where t is the number of hours after midnight and 0<=t<=18.

Find the maximum and minimum temperatures.
Find the maximum rate of increase in the temperature.

I know how to find the local minimum and maximums, but I am not sure if that is what is being asked here because I am horrible with word problems. Any help would be greatly appreciated.

Thanks,
John

You ARE being asked to find the local minimum and local maximum of temperature.

You are also being asked to find the maximum rate of increase in temperature. You are not given directly a function for the rate of change in temperature, but you are given a function giving temperature T in terms of time t. What next?

PS The only practical reason in the world to learn mathematical techniques is to apply them to problems. The way to start with a word problem is to define the relevant variables in writing and assign a unique letter to each variable in writing. It eases the load on your memory and gives you a clue on what to think about.

\(\displaystyle T = temperature\ in\ degrees\ F.\)

\(\displaystyle t = time\ in\ hours\ from\ midnight.\)

The second step is to translate the information given in words into mathematical form.

You are given \(\displaystyle T = f(t) = -.04t^3 +1.14t^2 - 7.2t + 66.\)

If you were given that and asked to find the local maximum and minimum, would you even blink an eye?

 

[john3j]

You ARE being asked to find the local minimum and local maximum of temperature.

You are also being asked to find the maximum rate of increase in temperature. You are not given directly a function for the rate of change in temperature, but you are given a function giving temperature T in terms of time t. What next?

PS The only practical reason in the world to learn mathematical techniques is to apply them to problems. The way to start with a word problem is to define the relevant variables in writing and assign a unique letter to each variable in writing. It eases the load on your memory and gives you a clue on what to think about.

\(\displaystyle T = temperature\ in\ degrees\ F.\)

\(\displaystyle t = time\ in\ hours\ from\ midnight.\)

The second step is to translate the information given in words into mathematical form.

You are given \(\displaystyle T = f(t) = -.04t^3 +1.14t^2 - 7.2t + 66.\)

If you were given that and asked to find the local maximum and minimum, would you even blink an eye?

Jeff,

Thank you confirming what steps I need to do here. I guess the only question left I have about solving this would be what do I do with the <=t<=18? Wouldnt I plug 18 into the equation to solve? would I take the derivative of this function and then plug in 18 to find the maximum rate of change? I was fine with applying techniquies when I was in algebra 1 and 2 courses, but it has been a long time since I have been in a math course. Again, thank you for your help!

Thanks,
John

 

[JeffM]

Jeff,

Thank you confirming what steps I need to do here. I guess the only question left I have about solving this would be what do I do with the <=t<=18? Wouldnt I plug 18 into the equation to solve? would I take the derivative of this function and then plug in 18 to find the maximum rate of change? I was fine with applying techniquies when I was in algebra 1 and 2 courses, but it has been a long time since I have been in a math course. Again, thank you for your help!

Thanks,
John

You have a function that is defined over a bounded domain. This bounded function is only applicable between midnight and 6 PM, a period of 18 hours. Remember I told you to write down how t is defined. It is the time in hours since midnight. That means the function is only relevant (defined) if

\(\displaystyle 0 \le t \le 18.\)

Whenever you have a bounded function, a local minimum or local maximum may lie on the boundary without having a derivative of zero. Think of the function

f(x) = 2 + 3x\ for\ 1 \le x \le 5.[/tex]

In that example, the function nowhere has a first derivative of 0. But it has a local minimum at x = 1 where f(x) = 5 and a local maximum
x =5 where f(x) = 17. If x > 1 then 2 + 3x > 5. If x < 5 then f(x) < 17.

So when you are asked to find the minimum and maximum temperatures, you must not only look at the temperatures where the first derivative is zero, but also at the end points.

In most applications of calculus, this looking at boundary conditions is important.

Does this help?