Modeling
What is mathematical modeling?
The process of converting a real-world scenario into a math problem, solving it, converting the solution back to real life, and refining the model to repeat the cycle.
Express the nth term of this sequence recursively:
{1, 1/3, 1/9, 1/27, ...}
a(n) = a(n-1)/3; a(0) = 1
In the model a(n+1) = a(n) + 0.05*a(n), which part represents the change?
0.05*a(n)
High-order polynomials always make great models! (True or False)
FALSE! (say why)
Express the nth term of this sequence iteratively:
{1, 1/3, 1/9, 1/27, ...}
a(n) = 1/3^n
Write a difference equation to express the change during the nth interval as a function of the previous term in the sequence.
{1, 5, 25, 125, ...}
a(n+1) - a(n) = ?
4*a(n)
What are the main steps of the modeling process?
1. Identify the problem.
2. State your assumptions & define your variables.
3. Figure out how the variables are related (build the model).
4. Solve the model.
5. Verify the model with external information.
6. Put the model into use.
7. Maintain the model—back to step 1!
Calculate a(0), a(2), and a(5) for the sequence
a(n+1) = 0.8*a(n) - 1; a(0) = 0
0, -1.8, -3.3616
You have a savings bond worth $1000 that accumulates interest at a rate of 1% per month. How much will it be worth after 5 years?
$1816.70