The number of Internet host computers (computers connected directly to the Internet, for networks, bulletin boards, or online services) has been growing at the rate of f(x) = xe^(0.1x) million per year, where x is the number of years since 1990. Find the total number of Internet host computers added during the years 1990–2010.
I don't know how to start this problem. Do I plug the f(x) inside the graph and do I check the years from 1990-2010 in the table? I assume it would be from 0-20 on the table. Can anyone help me on this? I think that's what might happen, but I am not sure.
Thanks again!
sounds like you want
f'(x) = x e^(.01x)
since you say that's the growth rate.
So, the total number of computers for 20 years is the integral from 0 to 20 of f'(x) dx
To do the integral, use integration by parts.
To find the total number of Internet host computers added during the years 1990-2010, you need to calculate the definite integral of the function f(x) = xe^(0.1x) over the interval from 0 to 20.
The integral of f(x) with respect to x can be found using integration techniques. Applying the power rule and the integration by parts method, we have:
∫(xe^(0.1x)) dx = (10x^2 - 100x + 1000)e^(0.1x) + C
Now, evaluate the definite integral over the interval [0, 20]:
∫(from 0 to 20) (xe^(0.1x)) dx
= [(10x^2 - 100x + 1000)e^(0.1x)] (from 0 to 20)
Evaluating this expression, we get:
[(10(20)^2 - 100(20) + 1000)e^(0.1(20))] - [(10(0)^2 - 100(0) + 1000)e^(0.1(0))]
Simplifying further:
[(4000 - 2000 + 1000)e^(2)] - [1000e^0]
= [3000e^(2)] - [1000]
Use the fact that e^2 ≈ 7.389 to approximate the final result:
3000e^2 ≈ 3000 × 7.389 ≈ 22,167
So, the total number of Internet host computers added during the years 1990-2010 is approximately 22,167 million.
To find the total number of Internet host computers added during the years 1990-2010, you need to integrate the growth rate function f(x) = xe^(0.1x) from 1990 to 2010.
Here's how you can approach the problem step by step:
1. Determine the range of years you need to consider. In this case, you want to find the total number of computers added between 1990 and 2010, which corresponds to 20 years (2010 - 1990 = 20).
2. Set up the integral to represent the total number of computers added. Since the growth rate function f(x) is given in millions per year, you want to integrate it over the range of 20 years. The integral should be set up as follows:
∫[from 0 to 20] (xe^(0.1x)) dx
3. Evaluate the integral. The integral can be solved using various methods such as integration by parts or substitution. In this case, integration by parts is a good approach. You can use the formula:
∫uv dx = uv - ∫vu' dx
Let u = x and dv = e^(0.1x) dx. Differentiating u gives du = dx, and integrating dv gives v = (10e^(0.1x))/1.
Applying the formula, you can write the integral as:
∫(xe^(0.1x)) dx = x((10e^(0.1x))/1) - ∫((10e^(0.1x))/1) dx
Performing the integration and evaluating the integral limits, you get:
[(10e^(0.1x))x - 10e^(0.1x)] evaluated from 0 to 20
4. Compute the final answer. Plug in the upper limit (20) and subtract the result with the lower limit (0) to get the total number of Internet host computers added during the years 1990-2010.
[(10e^(0.1 * 20)) * 20 - 10e^(0.1 * 20)] - [(10e^(0.1 * 0)) * 0 - 10e^(0.1 * 0)]
Simplifying the expression, you can calculate the integral to find the answer.