Problem 33 - Calculus Early Transcendentals

Question

Prove that the doubling time for an exponentially increasing quantity is constant for all time.

Step-by-Step Solution

Answer

Question: Prove that the doubling time for an exponentially increasing quantity is constant for all time. Short Answer: The doubling time for an exponentially increasing quantity can be shown to be constant by solving the exponential function f(t) = a * b^(t/c) for the initial doubling time and comparing it to the equation obtained when time is doubled. As demonstrated in the step-by-step solution, the equation simplifies to show that the quantity indeed doubles in a constant time, proving that the doubling time is constant for all time.

Step 1: Definition of an exponential function

An exponential function is a function of the form f(t) = a * b^(t/c), where "a" is the initial quantity, "b" is the growth factor, "t" is the time, and "c" is a scaling factor. The growth rate is determined by the value of "b".

Step 2: Calculate the initial doubling time

To find the doubling time, we can divide the initial quantity by the growth factor, and then multiply by the scaling factor. We set f(t) = 2a, since this is what we want to double. So, we have 2a = a * b^(t/c). The "a" terms cancel out, which leaves: 2 = b^(t/c)

Step 3: Solve for the doubling time

To solve for the doubling time, we take the natural logarithm of both sides of the equation: ln(2) = ln(b^(t/c)) Using the logarithm properties, we can rewrite this equation as: t/c * ln(b) = ln(2) Now, let's isolate t: t = c * ln(2) / ln(b)

Step 4: Double the time

In order to show that the doubling time is constant, we must now double the time and calculate the new quantity. Let's set t2 = 2t and find f(t2): f(t2) = a * b^(t2/c) f(t2) = a * b^(2t/c)

Step 5: Using our initial solution

Now, using the solution for the initial doubling time, we can substitute it into the second equation: f(t2) = a * b^(2c * ln(2) / ln(b)) This will simplify to: f(t2) = a * (b^(c * ln(2) / ln(b)))^2

Step 6: Check for doubling

Now we need to prove that f(t2) = 4a, which would show that we indeed doubled the quantity in a constant time: f(t2) = a * (b^(c * ln(2) / ln(b)))^2 = 4a a * (b^(c * ln(2) / ln(b)))^2 = 4a Since the initial doubling time equation resulted in f(t) = 2a, the equation for doubling time (t2) holds true, proving that the doubling time for an exponentially increasing quantity is constant for all time.