These two q's for my homework I am very confused on how to do:
First Question:
Using Newton’s method, approximate the value of √5 up to 2 decimal points
starting with x1 = 3.
2nd Question:
Thomas Malthus was an economist that predicted that the population grows exponentially while the food supply grows linearly. If p(t) gives the population and f(t) gives the number of people that can be supported by the amount of food produced at t years, we would be concerned about when p(t) = f(t). After this point, the pop. would be beyond capacity that the environment can support. Suppose p(t)= 1000e^(0.02t) and f(t)= 30t + 2000. Using Newton’s method with x1 = 60, find x2 to approximate when the functions intersect.
so, what is confusing you? Newton's method says that, if you have an estimate a root of f(x)=0, you can get a better estimate by using the formula
f(x) -> f(x) - f(x)/f'(x)
You have the function f(x) = x^2-5, so that f(√5) = 0. f'(x) = 2x.
So, you start off with guerssing that x=3. So, the next guess is
3 - f(3)/f'(3) = 3-(9-5)/(2*3) = 2.333
now just keep iterating. A nice web site which shows the steps is
http://keisan.casio.com/exec/system/1244946907
Sure! I'll explain how to approach both of these questions step-by-step.
First Question:
To approximate the value of √5 up to 2 decimal points using Newton's method, we need to find the root of the equation f(x) = x² - 5 = 0.
Newton's method is an iterative process that involves updating an initial guess for the root until the desired accuracy is achieved.
Here's how you can use Newton's method to approximate √5:
1. Define the function f(x) = x² - 5.
2. Compute the derivative of f(x) with respect to x. In this case, f'(x) = 2x.
3. Choose an initial guess for the root, let's say x1 = 3.
4. Apply Newton's method formula to update the guess for the root:
x2 = x1 - (f(x1) / f'(x1))
5. Repeat step 4 until the desired accuracy is achieved. In this case, we want 2 decimal points, so repeat until the absolute difference between consecutive approximations is less than 0.01.
Let's calculate the approximations using this method:
- Initial guess: x1 = 3
- Newton's formula: x2 = x1 - (f(x1) / f'(x1)) = 3 - ((3² - 5) / (2 * 3)) = 3 - (4 / 6) = 3 - 0.6667 = 2.3333
- Repeat the process with the updated guess until the desired accuracy is achieved.
Keep applying the Newton's formula until the absolute difference between consecutive approximations is less than 0.01. Since Newton's method is an iterative process, you'll notice that the approximations get closer and closer to the actual root of the equation.
Now let's move on to the second question.
Second Question:
To use Newton's method to find the point of intersection between the functions p(t) = 1000e^(0.02t) and f(t) = 30t + 2000, we need to find the value of t where p(t) - f(t) = 0.
Here's how you can use Newton's method to approximate the point of intersection:
1. Define the function g(t) = p(t) - f(t) = 1000e^(0.02t) - (30t + 2000).
2. Compute the derivative of g(t) with respect to t. In this case, g'(t) = 20e^(0.02t) - 30.
3. Choose an initial guess for the point of intersection, let's say x1 = 60.
4. Apply Newton's method formula to update the guess for the root:
x2 = x1 - (g(x1) / g'(x1))
5. Repeat step 4 until the desired accuracy is achieved. In this case, follow the same procedure as in the first question.
By applying Newton's method iteratively, you'll find the approximate value of t where the functions p(t) and f(t) intersect.
I hope this helps! Let me know if you have any further questions.