Problem 8 - A Brief Introduction to Engineering Computation with MATLAB

Question

Determine the value of each of the following. Evaluate \(\mathrm{y}\) at 5 . \(y=4 x^{4}+3 x^{2}-x\)

Step-by-Step Solution

Answer

When x = 5, the value of the function y is 2570.

Step 1: Write down the given function

Here, we have the function y of x given by: \[y = 4x^4 + 3x^2 - x\]

Step 2: Substitute x with 5

In this step, we will replace x with 5 in the given function: \[y = 4(5)^4 + 3(5)^2 - (5)\]

Step 3: Simplify the expression

Now, we will simplify the given expression by calculating the values of the terms with an exponent and then add or subtract these terms to find the final value of y: \[y = 4(625) + 3(25) - 5\]

Step 4: Perform the operations

Finally, we perform the multiplications and additions/subtractions to find the value of y: \[y = 2500 + 75 - 5\] \[y = 2570\]

Step 5: State the result

Therefore, when x = 5, the value of the function y is 2570.

Key Concepts

Mathematical Evaluation

Mathematical evaluation is the process of finding the value of a mathematical expression. Think of it as solving a small puzzle using numbers and operations like addition, subtraction, multiplication, or exponentiation. It’s important to follow the order of operations to evaluate expressions accurately. In mathematical terms, this is referred to as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This hierarchy ensures that calculations are carried out in the correct sequence.
For example, when you evaluate the expression \(4x^4 + 3x^2 - x\) at \(x = 5\), you need to:

• Calculate the powers first, such as \(5^4\) and \(5^2\).
• Then, multiply these powers by the coefficients in front of them.
• Finally, sum up or subtract the resulting values.

This clear sequence of operations leads to the correct evaluation of the expression.

Substitution Method

The substitution method is a fundamental technique in algebra. It involves replacing variables in an equation or expression with their equivalent values. This is particularly useful for evaluating expressions at specific points.
To use the substitution method effectively, follow these steps:

• Identify the variable to be substituted and note its given value.
• Replace every instance of this variable in the expression with the given value.
• Ensure that the substitution is done carefully to avoid any errors.

In our example, we chose to evaluate the function \(y = 4x^4 + 3x^2 - x\) by substituting \(x = 5\). Replacing \(x\) with 5 in the equation, each part of the expression is then calculated step by step. This straightforward method helps simplify complex expressions into manageable numerical calculations.

Algebraic Expressions

An algebraic expression comprises constants, variables, and operators such as \(+, - , imes, ext{ or } /\). Unlike equations, expressions do not contain an equals sign but instead represent a mathematical phrase that can be simplified or evaluated.
Algebraic expressions can be as simple as \(x + 2\) or as complex as \(4x^4 + 3x^2 - x\). Each term in the expression can include a coefficient (the number multiplied by the variable), a variable (like x), and an exponent that indicates how many times the variable is multiplied by itself.
Understanding how to manipulate and evaluate algebraic expressions is key to solving algebra problems. This involves identifying like terms that can be combined, understanding how different operations impact the expression, and correctly applying algebraic rules.

• Using substitution in algebraic expressions often requires evaluating powers and performing arithmetic operations.
• The overall goal is to simplify the expression to find the desired value.

Grasping these concepts paves the way to more complex algebraic problem-solving.