Problem 13 - Calculus

Question

Take three steps of \(y(t+1)=2 y(t)\) from \(y_{0}=1\).

Step-by-Step Solution

Answer

After three steps, the value is 8.

Step 1: Calculate the first step

Starting with the initial value \(y_0 = 1\). We substitute it into the equation to find \(y_1\):\[y_1 = 2 imes y_0 = 2 imes 1 = 2\]Thus, \(y_1 = 2\).

Step 2: Calculate the second step

Now use the value of \(y_1\) to find \(y_2\):\[y_2 = 2 imes y_1 = 2 imes 2 = 4\]Thus, \(y_2 = 4\).

Step 3: Calculate the third step

Next, use the value of \(y_2\) to find \(y_3\):\[y_3 = 2 imes y_2 = 2 imes 4 = 8\]Thus, \(y_3 = 8\).

Key Concepts

Recurrence Relation

A recurrence relation is a mathematical way of defining a sequence of numbers. It describes each term in the sequence using previous terms. This approach is especially useful in modeling and solving discrete dynamical systems. In our exercise, the recurrence relation is given by:- \( y(t+1) = 2y(t) \)This relation means that every next value, \( y(t+1) \), is obtained by multiplying the current value, \( y(t) \), by 2. It acts as a rule for how the sequence evolves or "repeats" over time, hence the name "recurrence."

Recurrence relations are powerful tools because they help us predict future events or values without needing every single past observation. Instead, knowing just the last value is enough to move forward in most cases.

Initial Value Problem

An initial value problem in mathematics requires you to find a sequence or function based on initial conditions and a relation or rule. It asks for what happens next, given a starting point and some formula to follow. For example, in this exercise:- Initial value, \( y_0 = 1 \), is given- Recurrence relation \( y(t+1) = 2 y(t) \) is also provided.The initial value is like the starting point of a journey. From there, the recurrence relation guides how you proceed to discover subsequent steps. It is crucial as it sets the ball rolling for all future calculations.

• Step 1: Use the initial value to compute the next term.
• Step 2: Use the result of Step 1 to find the next term.
• Step 3: Continue using the results to move further along.

Initial value problems are foundational in mathematical modeling, offering a structured start to analyzing patterns and predicting results in sequences and functions.

Exponential Growth

Exponential growth describes a process that increases rapidly over time. In mathematical terms, it means a quantity increases by a consistent multiplicative factor, rather than an additive one. In our sequence:- Each term doubles the previous one following \( y(t+1) = 2y(t) \)- This pattern results in values going from 1 to 2 to 4 and, finally, to 8.Through these calculations, we see that the sequence isn't just growing—it's growing exponentially. That is because every step increases by a constant factor (in this case, 2).

Exponential growth generally illustrates how something like populations, investments, or biological processes can increase quickly over time as conditions allow that constant rate of growth.