Problem 17 - Applied Calculus

Question

Simplify the quantities using \(m(z)=z^{2}\). $$m(z)-m(z-h)$$

Step-by-Step Solution

Answer

The expression simplifies to \(2zh - h^2\).

Step 1: Understand the Function

The function given is \(m(z) = z^2\), which means it takes any input \(z\) and squares it. We need to simplify the expression \(m(z) - m(z-h)\).

Step 2: Substitute the Function into the Expression

Replace \(m(z)\) with \(z^2\) and \(m(z-h)\) with \((z-h)^2\) in the expression. This gives us:\[m(z) - m(z-h) = z^2 - (z-h)^2\]

Step 3: Expand the Expression

Expand \((z-h)^2\) to make it easier to simplify:\[(z-h)^2 = z^2 - 2zh + h^2\]

Step 4: Substitute and Simplify

Substitute \((z-h)^2\) back into the expression and simplify:\[z^2 - (z^2 - 2zh + h^2)\]This results in:\[z^2 - z^2 + 2zh - h^2\]

Step 5: Cancel Out Terms

Cancel like terms in the expression:\[z^2 - z^2 = 0\]Leaving us with:\[2zh - h^2\]

Key Concepts

Understanding the Difference Quotient

The difference quotient is a fundamental concept in calculus used to determine the rate of change of a function. In simple terms, it helps us understand how changing the input of a function slightly affects its output. This concept is often represented using the formula \[ f'(z) = \frac{f(z) - f(z - h)}{h} \]where \( f \) is the function, and \( h \) represents a small change in the input \( z \).
When we apply the difference quotient to our function \( m(z) = z^2 \), we use it to simplify the expression \( m(z) - m(z-h) \). Here, we do not divide by \( h \), but we do simplify the difference between two function values. This is like the first step in many calculus problems where we find derivatives. Understanding this idea is invaluable because it sets the groundwork for analyzing more complex functions later on in mathematics.

Expanding Squared Functions for Simplification

A squared function, specifically in our exercise, involves expressions like \( (z-h)^2 \). These require expansion to simplify. When you see an expression squared, like \((z-h)^2\), it's really about breaking it down into easier parts.
By expanding \((z-h)^2\), you open it up into \( z^2 - 2zh + h^2 \). - **`Expand Each Part`**: \( z^2 \) and \(-2zh\) complement each other by indicating how both \(z\) and \(h\) interact.- **`Resulting Simplicity`**: The individual parts become manageable, helping us identify terms that can be cancelled or simplified.This expansion plays a crucial role in how we can manipulate the function algebraically. It allows us to later cancel terms as needed, simplifying the function step by step.

Algebraic Manipulation for Simplification

Algebraic manipulation involves using basic algebra principles to simplify expressions. In our example, it is integral to transforming our function difference into a more straightforward expression.
To achieve this:- **`Substitute`**: Replace expressions based on their expanded form; for example, substitute \((z-h)^2\) with \(z^2 - 2zh + h^2\).- **`Cancel Out`**: Eliminate terms that appear in opposite forms, like \(z^2\), which cancel each other out: \( z^2 - z^2 = 0 \).- **`Simplify Further`**: Focus on combining or reducing remaining terms such as \(2zh - h^2\).This strategic rearranging underscores the value of breaking expressions into their components to see where simplifications can occur. Through such manipulation, complex expressions become elegant and easy to handle — turning a problem on its head from seemingly complex to very simple.