Constant Doubling Time Functions: Exponential vs. Linear | Homework Help

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In summary: I see now how the question is asking for the doubling time to be t-t0, rather than just t. So, in summary, the doubling time for linear function f(t) = a(t) + b at time t0 is t-t0 = b/a.
  • #1
physicsernaw
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Homework Statement


The only functions with a constant doubling time are the exponential functions P0ekt with k > 0. Show that the doubling time of linear function f(t) = a(t) + b at time t0 is t0 + b/a.

Homework Equations


n/a

The Attempt at a Solution



With initial time t0, P = at0 + b

At some time t, P is doubled: at + b = 2P

t = (2P - b)/a

Plug in P and simplify:

t = (2at0 + b)/a
t = 2t0 + b/a

What am I doing wrong?
 
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  • #2
physicsernaw said:

Homework Statement


The only functions with a constant doubling time are the exponential functions P0ekt with k > 0. Show that the doubling time of linear function f(t) = a(t) + b at time t0 is t0 + b/a.

Homework Equations


n/a

The Attempt at a Solution



With initial time t0, P = at0 + b

At some time t, P is doubled: at + b = 2P

t = (2P - b)/a

Plug in P and simplify:

t = (2at0 + b)/a
t = 2t0 + b/a

What am I doing wrong?

You haven't done anything wrong. But I think they mean the doubling time to be t-t0. That's the length of time after t0 that the function will double. Solve for that.
 
  • #3
Dick said:
You haven't done anything wrong. But I think they mean the doubling time to be t-t0. That's the length of time after t0 that the function will double. Solve for that.

Ahhh, thank you so much.
 

What is the difference between exponential and linear functions?

Exponential functions have a constant rate of growth, meaning they increase at a faster and faster rate over time. Linear functions, on the other hand, have a constant rate of change and increase at a steady rate over time.

How do you determine the doubling time of an exponential function?

The doubling time of an exponential function can be determined by dividing the natural logarithm of 2 by the growth rate of the function. This will give you the number of time periods it takes for the function to double in value.

Can a linear function have a doubling time?

No, a linear function does not have a doubling time because it increases at a constant rate and does not have a point where it doubles in value.

Why is it important to understand the difference between exponential and linear functions?

Understanding the difference between exponential and linear functions can help in predicting future trends and making informed decisions. Exponential functions can represent rapid growth or decay, while linear functions can represent steady growth or decline.

How can constant doubling time functions be applied in real life?

Constant doubling time functions can be used to model population growth, financial investments, and the spread of diseases. They can also be used to make predictions about future trends and make decisions based on those predictions.

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