Problem 16 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Evaluate the Original Function

First, let's write down the given function: \( m(z) = z^2 \). This is a basic quadratic function.

Step 2: Evaluate the Function at \( z + h \)

Next, we need to substitute \( z + h \) into the function. Compute \( m(z + h) \):\[ m(z + h) = (z + h)^2 \]

Step 3: Expand \((z+h)^2\)

Now, expand \((z + h)^2\) using the binomial theorem:\[ (z + h)^2 = z^2 + 2zh + h^2 \]

Step 4: Compute \( m(z + h) - m(z) \)

Substitute the expressions for \( m(z + h) \) and \( m(z) \) into the original difference expression:\[ m(z + h) - m(z) = (z^2 + 2zh + h^2) - z^2 \]

Step 5: Simplify the Expression

Cancel out \( z^2 \) terms from the expression:\[ m(z + h) - m(z) = z^2 + 2zh + h^2 - z^2 = 2zh + h^2 \] This is the simplified form of the expression.

Key Concepts

Functions

A function is a special relationship between sets of numbers or objects, where each input has exactly one output. Think of a function like a machine: you put in an input, the machine processes it, and you get an output. A function can be represented by equations, tables, or graphs. In this context, the function is represented by an equation.

• An example of a function: Let’s say a function is defined by the equation \( y = f(x) \). Here, \( x \) is the input and \( y \) is the output.
• The equation specifies how to calculate the output for any given input.
• For our example in the exercise, the function \( m(z) = z^2 \) means every input \( z \) is squared to generate the output \( m(z) \).

Functions play a fundamental role in calculus and mathematics, allowing for the modeling of relationships and changes that occur in different settings. Understanding how to manipulate and evaluate functions is a key skill in solving equations and inequalities.

Quadratic Functions

Quadratic functions are a type of polynomial function where the highest degree of the variable is 2. In simpler terms, a quadratic function can be described by the formula \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).

• The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \).
• Quadratic functions are common as they model many real-world phenomena such as projectile motion and areas.
• In the exercise, the function is a simplified quadratic function \( m(z) = z^2 \). This tells us the parabola opens upwards since the coefficient is positive.

To analyze or simplify expressions involving quadratic functions, you might need to apply algebraic rules like expanding brackets, simplifying terms, or completing the square. The exercise involved evaluating the function at different points and simplifying to find the rate of change when moving from \( z \) to \( z+h \). This is useful in understanding how the output changes relative to small changes in the input.

Binomial Theorem

The binomial theorem is a powerful tool in algebra and calculus. It provides a way to expand expressions that are raised to a power, such as \((x + y)^n\). This theorem is especially useful for handling quadratic and higher polynomial expressions.

• The binomial theorem states: \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \)
• Each term in the expansion is a product of a binomial coefficient, \( \binom{n}{k} \), and the variables raised to powers that add up to \( n \).
• In our exercise, we only use the quadratic case, \((z+h)^2\), which expands to \(z^2 + 2zh + h^2 \).

This expansion is vital for solving expressions such as the one in the exercise. By expanding \((z+h)^2\), you can clearly see and subtract terms, simplifying the expression. It also highlights how each part of the expression changes relative to the others when different inputs are used. Understanding the binomial theorem not only helps in algebraic manipulations but also prepares you for calculus concepts involving rates of change and series expansions.