Derivatives Question - temperature as a funtion of time

In summary, the equation for temperature as a function of time is T(t) = 540(1 – e–0.1t). When t = 10 minutes, the temperature is 540(1-1/e) and the rate of change at that time is dT/dt = 54e^-0.1 minutes^-1. To get the derivative, we used the chain rule and found the derivative of e^-0.1t to be -0.1e^-0.1t.
  • #1
livestrong136
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Derivatives Question -- temperature as a funtion of time

1. When a certain object is placed in an oven at 540°C, its temperature T(t) rises according to the equation T(t) = 540(1 – e–0.1t), where tis the elapsed time (in minutes). What is the temperature after 10 minutes and how quickly is it rising at this time?

My Work: Temp after t minutes is given. Just plug in t = 10 mins in the equation ie T = 540(1-1/e)
rate of change is dT/dt (at t = 10) = 54/e

I said the rate of change at t=10 is 54/e. 54/e is incorrect. not sure why I was dividing, in this question we need to show how you got the derivative to get full marks. Can someone help me with this.
 
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  • #2
hi livestrong136! :smile:

(try using the X2 button just above the Reply box :wink:)

T = 540(1 – e–0.1t)

so what is dT/dt ? :smile:

(use the chain rule)
 

1. How do you calculate the derivative of temperature as a function of time?

To calculate the derivative of temperature as a function of time, you would need to use the fundamental definition of derivative, which is the limit of the difference quotient. This involves taking the difference of the temperature values at two different time points and dividing it by the difference in time between those points. As the time interval approaches 0, the derivative will become more accurate.

2. What does the derivative of temperature as a function of time represent?

The derivative of temperature as a function of time represents the rate of change of temperature over time. This means it tells us how quickly the temperature is changing at a specific point in time. A positive derivative indicates an increasing temperature, while a negative derivative indicates a decreasing temperature.

3. How can derivatives of temperature as a function of time be used in real life?

There are many real-life applications of derivatives of temperature as a function of time. For example, it can be used in weather forecasting to predict how the temperature will change over time. It can also be used in thermodynamics to analyze the efficiency of heating or cooling systems.

4. What are the units of the derivative of temperature as a function of time?

The units of the derivative of temperature as a function of time will depend on the units used for temperature and time. For example, if temperature is measured in degrees Celsius and time is measured in seconds, then the derivative would have units of degrees Celsius per second.

5. How does the shape of a graph of temperature as a function of time relate to its derivative?

The shape of a graph of temperature as a function of time is directly related to its derivative. The derivative at a specific point on the graph represents the slope of the tangent line at that point. So, if the graph is steeply increasing, the derivative will be large and positive, and if the graph is gently decreasing, the derivative will be small and negative.

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