Problem 2 - Differential Equations and Linear Algebra

Question

Determine \(f * g\) $$f(t)=6 t^{2}, \quad g(t)=5 t^{3}$$

Step-by-Step Solution

Answer

The product of the functions \(f(t) = 6t^2\) and \(g(t) = 5t^3\), denoted as \(f * g\), is equal to \(30t^5\).

Step 1: Write Down the Functions

We are given the following functions: \(f(t) = 6t^2\) \(g(t) = 5t^3\)

Step 2: Multiply the Functions

To find the product of the two functions, we must multiply them together: \[(f * g)(t) = f(t) \cdot g(t) = (6t^2)(5t^3)\]

Step 3: Simplify the Expression

When multiplying the two expressions, we multiply the coefficients and add the exponents of the corresponding variables: \[(f * g)(t) = 30t^{(2+3)}\]

Step 4: Write the Final Result

After simplifying the expression, we get the final product: \((f * g)(t) = 30t^5\) The product of the functions \(f(t)\) and \(g(t)\), denoted as \(f * g\), is equal to \(30t^5\).

Key Concepts

Multiplication of Functions

When we talk about multiplication of functions, we are essentially combining two functions into one by multiplying their outputs. Suppose we have two functions, like in the exercise:

• \( f(t) = 6t^2 \)
• \( g(t) = 5t^3 \)

This means if we want to find the result of \( (f * g)(t) \), we multiply the outputs of these functions at each point \( t \). It becomes a simple product of the two given expressions: \[ (f * g)(t) = f(t) \cdot g(t) = (6t^2) \cdot (5t^3) \] This is just like multiplying numbers, where you multiply everything inside the parentheses by everything in the other parentheses.

Simplification of Expressions

Simplifying expressions involves reducing them into their simplest or most compact form. After multiplying the functions, as in our exercise example,we got:\[ (f * g)(t) = (6t^2) \cdot (5t^3) \] The next step is to simplify this expression by:

• Multiplying the coefficients: \(6\times 5\).
• Applying rules of exponents to combine the \(t\) terms.

So, you end up with:\[ (f * g)(t) = 30t^{(2+3)} \].Breaking it down helps keep complex multiplication easy to manage.

Exponent Rules

Exponent rules are crucial when working with expressions involving powers. These rules help us work with terms like \( t^2 \) and \( t^3 \) in the multiplication of functions. The basic exponent rule applied here is: when you multiply powers with the same base, you add the exponents. So for our functions:

• Multiplying \( t^2 \) and \( t^3 \) results in \( t^{2+3} \).
• This simplifies to \( t^5 \), meaning the base stays the same and the exponents are added together.

This rule simplifies our expression from \((6t^2)(5t^3)\) to \(30t^5\).Utilizing exponent rules makes it easier to handle and reduce expressions that seem complex at first glance.