Problem 15 - Applied Calculus

Question

Simplify the quantities using \(m(z)=z^{2}\). $$m(z+1)-m(z)$$

Step-by-Step Solution

Answer

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

Step 1: Apply the function to the first term

First, we need to understand that the function given is \(m(z) = z^2\). So, we apply the function to \(z+1\), which gives us \(m(z+1) = (z+1)^2\).

Step 2: Expand the expression \((z+1)^2\)

Now we expand \((z+1)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\). Here, \((z+1)^2 = z^2 + 2z + 1\).

Step 3: Write the function value for \(m(z)\)

We already know that \(m(z) = z^2\). So, this value needs no further simplification.

Step 4: Calculate the difference

The expression given is \(m(z+1) - m(z)\). Substituting our previous results, this becomes \((z^2 + 2z + 1) - z^2\).

Step 5: Simplify the expression

Now simplify \((z^2 + 2z + 1) - z^2\) by canceling out \(z^2\) from both terms. This leaves us with \(2z + 1\).

Key Concepts

Functions

When we talk about functions in mathematics, we are referring to a special relationship where each input is related to exactly one output. A function can be thought of as a machine that processes numbers fed into it, known as inputs, and churns out numbers, called outputs, based on specific rules.
For example, in our given exercise, the function provided is expressed as:

• \(m(z) = z^2\)

Here, the input is \(z\), and the output is \(z^2\). This means whatever number we plug into \(z\), we will square it to get the result. Functions are everywhere in calculus, providing a way to describe how values change and behave under certain transformations. They can range from simple equations like this one to more complex expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They serve as the building blocks of algebra and provide a way to encapsulate mathematical relationships.
In our problem, the expression \((z+1)^2\) is an algebraic expression. It includes:

• Two terms: \(z\) and \(1\)
• An operation: addition
• A power: squaring

Understanding how to manipulate and interpret these expressions is fundamental in algebra. It allows us to simplify complex problems, solve equations, and explore mathematical relationships.

Simplification

Simplification is a crucial concept in algebra. It involves rewriting an expression in its simplest form, making it easier to work with. In our exercise, we moved from a more complex expression to a more straightforward one by systematically reducing terms.
Here's a quick walk-through:

• Begin with the expression we got: \((z^2 + 2z + 1) - z^2\)
• Identify like terms. Here, \(z^2\) is present in both parts.
• Cancel out \(z^2\) since it appears with both a positive and a negative sign.

This simplification reduced our original problem to \(2z + 1\). Simplification not only makes solving equations easier but also helps in revealing any underlying relationships in algebraic expressions.

Quadratics

Quadratics are a type of polynomial that is characterized by the presence of an \(x^2\) term, often written in the form \(ax^2 + bx + c\). These equations, recognizable by their distinctive U-shaped graphs called parabolas, are central in algebra and calculus learning.
In our example, the quadratic part appears when we expanded \((z+1)^2\), yielding \(z^2 + 2z + 1\). This quadratic expression includes:

• The square term \(z^2\) which influences the 'opening' of the parabola
• The linear term \(2z\) which shifts the parabola along the graph
• The constant \(1\) which moves it vertically

Understanding how these components affect the shape and position of parabolas helps in visualizing and solving quadratic equations, whether they appear in simple forms or as part of more complex expressions.