Problem 16 - Applied Calculus

Question

Simplify the quantities in Problems \(16-19\) using \(m(z)=z^{2}\) $$m(z+1)-m(z)$$

Step-by-Step Solution

Answer

The simplified expression is \(2z + 1\).

Step 1: Understand the Function

The function given is \( m(z) = z^2 \). This implies for any input \( z \), the output is the square of \( z \).

Step 2: Substitute \( z+1 \) into the Function

We need to find \( m(z+1) \). Substitute \( z+1 \) into the function: \[ m(z+1) = (z+1)^2 \].

Step 3: Expand the Expression

Expand the expression \((z+1)^2\). \[(z+1)^2 = z^2 + 2z + 1 \].

Step 4: Write the Expression for \(m(z+1) - m(z)\)

Now, find \( m(z+1) - m(z) \). This requires substituting the expanded form and original function: \[ m(z+1) - m(z) = (z^2 + 2z + 1) - z^2 \].

Step 5: Simplify the Expression

Simplify the expression by canceling common terms:\[ m(z+1) - m(z) = z^2 + 2z + 1 - z^2 = 2z + 1 \].

Key Concepts

Understanding Function Substitution

Function substitution is a powerful tool in calculus and algebra. It involves replacing a variable or an entire expression within a function with another value or expression. This technique can make complex problems easier to solve by breaking them down into smaller, more manageable parts.

In the exercise, the function given is \( m(z) = z^2 \). To solve the problem, we needed to substitute \( z+1 \) into the function. This required us to replace every instance of \( z \) with \( z+1 \) to find \( m(z+1) \). Remember, substitution helps us understand how changes in variables affect the function's value. It's like looking through a new lens to see how the function behaves differently.

• Identify the function and its components.
• Substitute new expressions systematically.
• Rewrite the function to reflect the substitution.

By mastering this technique, you can handle increasingly complex functions with confidence.

Exploring Algebraic Expansion

Algebraic expansion is the process of multiplying out brackets to simplify expressions and combine like terms. It often requires distributing terms and arranging them systematically. The goal is to rewrite expressions without changing their value, making them easier to work with for further operations.

In our problem, we expanded \((z+1)^2\). This involved straightforward algebra. Think of \((z+1)(z+1)\) as multiplying each term in the first bracket with every term in the second.
First, multiply \(z\) by \(z\) to get \(z^2\). Next, multiply \(z\) by \(1\) to get \(z\). Do the same for the second \(1\), resulting in another \(z\). Finally, multiply \(1\) by \(1\) to get \(1\). When combined, these products form \(z^2 + 2z + 1\).

• Ensure every term is multiplied correctly.
• Look out for common terms to combine.
• Double-check for any missed terms.

This method builds a solid foundation for tackling more complex algebraic expressions.

Mastering Expression Simplification

Expression simplification combines several techniques to represent an expression in its simplest form. Simplifying expressions is essentially cleaning up your work to reveal its core components. By doing this, we make complex expressions more intuitive to understand.

In our example, once we expanded \( m(z+1) = z^2 + 2z + 1 \), we moved to simplify \( m(z+1) - m(z) \). This step required us to recognize and cancel out the common terms \(z^2\) from each part: \((z^2 + 2z + 1) - z^2\). The \(z^2\) terms cancel each other, leaving us with \(2z + 1\).

• Identify and remove terms that appear on both sides of the expression.
• Combine like terms to make the expression as concise as possible.
• Simplifying helps in understanding and solving further parts of complex problems.

Reducing expressions makes it easier for us to work with them later, whether evaluating, comparing, or further manipulating them.