Problem 16 - Calculus

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(2zh - h^2\).

Step 1: Apply the function to each term

First, we apply the given function \( m(z) = z^2 \) to each part of the expression \( m(z) - m(z-h) \). This results in two expressions: \( m(z) = z^2 \) and \( m(z-h) = (z-h)^2 \). Thus, our expression becomes \( z^2 - (z-h)^2 \).

Step 2: Expand the Square of the Binomial

Next, we expand the expression \((z-h)^2\) using the formula \((a-b)^2 = a^2 - 2ab + b^2\). Applying this, we have \((z-h)^2 = z^2 - 2zh + h^2\).

Step 3: Simplify the Expression

Substitute the expanded form \((z-h)^2 = z^2 - 2zh + h^2\) back into the original expression \(z^2 - (z-h)^2\). This gives \[ z^2 - (z^2 - 2zh + h^2) \]. Simplify the expression by distributing the negative sign: \[ z^2 - z^2 + 2zh - h^2 \]. The \(z^2\) terms cancel out, leaving \[ 2zh - h^2 \].

Key Concepts

Binomial Expansion

In calculus, binomial expansion plays a key role when we need to expand expressions involving terms raised to a power. An expression of the form

• \((a-b)^2\)

can be expanded using the formula:

• \((a-b)^2 = a^2 - 2ab + b^2\)

This formula breaks down the square of a binomial into separate terms, making it easier to deal with. Expanding binomials is a fundamental skill in algebra that simplifies complex expressions, enabling mathematicians and students alike to work through complex calculations with ease. When you apply this concept to the problem at hand, like the expression \((z-h)^2\), it helps to convert it into \(z^2 - 2zh + h^2\). This step is essential as it paves the way for further simplification and understanding of the expression's structure. Expanding binomials reduces errors when dealing with equations involving multiple terms and is a critical step in algebraic manipulations.
Remember, practicing these expansions with a variety of binomial combinations sharpens your skills and aids in more advanced calculus concepts.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. It is an integral part of calculus, as it allows complex formulas to be expressed more conveniently. In the provided exercise, once we expanded the binomial expression, our task was to simplify the overall equation by eliminating unnecessary terms. The expanded form of the expression was

• \(z^2 - (z^2 - 2zh + h^2)\)

The simplification step requires you to distribute any negative signs across the terms and combine like terms together. This transforms the expression into a more manageable form, in this case, yielding

• \(2zh - h^2\)

Simplifying expressions by cancelling out similar terms provides clarity, which is crucial for problem-solving in calculus. By thoroughly understanding and practicing these simplifications, students develop a better grasp of algebraic manipulation, making it easier to tackle even more complex calculus problems in the future.

Function Transformation

Function transformation is the alteration of a function's formula to achieve a different graphical representation or to simplify an expression. It often involves shifting, stretching, or compressing a function's graph. In this particular exercise, we encountered a basic transformation where the function \(m(z) = z^2\) was applied to create new expressions, specifically \(m(z-h) = (z-h)^2\). Function transformation enables us to look at functions and their output from different perspectives. By understanding how transformations work, like shifting or compressing, it becomes easier to predict changes and adapt equations.

• Shifting involves moving a graph horizontally or vertically, as seen with \((z-h)^2\).
• In calculus, working with transformations is crucial for visualizing and understanding how functions behave under various conditions.

Though our exercise involved relatively simple quadratic transformations, these concepts apply broadly across more advanced calculus applications, including integration and derivative functions. Understanding transformations deepens your comprehension of mathematical relationships and enhances problem-solving skills.