Problem 37 - Calculus Single Variable

Question

Find a polynomial function \(p(x)\) of degree 1 such that \(p(2)=6\) and \(p^{\prime}(4)=-5\).

Step-by-Step Solution

Answer

The polynomial function is \( p(x) = -5x + 16 \).

Step 1: Understand the Problem

We need to find a linear polynomial function, which is a first-degree polynomial. The general form of a first-degree polynomial is \( p(x) = ax + b \), where \( a \) and \( b \) are constants.

Step 2: Use the First Condition

The first condition given is \( p(2) = 6 \). Substituting \( x = 2 \) into \( p(x) = ax + b \), we get \( 2a + b = 6 \). This equation will help us find the values of \( a \) and \( b \).

Step 3: Use the Second Condition

The second condition is \( p^{\prime}(4) = -5 \). The derivative of \( p(x) = ax + b \) is \( p^{\prime}(x) = a \), since the derivative of a constant is zero. Thus, \( p^{\prime}(4) = a = -5 \).

Step 4: Solve for Constants

From Step 3, we have \( a = -5 \). Substitute \( a = -5 \) into the equation from Step 2: \( 2(-5) + b = 6 \). Simplifying gives \( -10 + b = 6 \), resulting in \( b = 16 \).

Step 5: Write the Polynomial Function

With \( a = -5 \) and \( b = 16 \), the polynomial function is \( p(x) = -5x + 16 \).

Key Concepts

Linear Polynomial

A linear polynomial is one of the simplest forms of polynomial functions. It is called "linear" because when graphed it forms a straight line. The general format of a linear polynomial is expressed as \( p(x) = ax + b \). Here:

• \( a \) represents the slope of the line, indicating how steep or flat the line is.
• \( b \) is the y-intercept, which tells you where the line crosses the y-axis.

Linear polynomials are first-degree polynomials, as the highest power of the variable \( x \) is one. Understanding this simple structure is essential, as it forms the foundation for more complex polynomial functions. In essence, every straight line you see on a coordinate plane can be defined by a linear polynomial.

First-Degree Polynomial

A first-degree polynomial is essentially another term for a linear polynomial. The part "first-degree" refers to the degree of the polynomial, which is the highest power of the variable \( x \) present in the equation. For first-degree polynomials, this power is one, represented by the equation \( ax + b \).
First-degree polynomials are central in algebra due to their simplicity and foundational properties:

• They are easy to solve and graph.
• They provide the simplest form of linear relationships between variables.

These are often the first type of polynomial equations students will master, forming a basis for understanding multidimensional polynomials and equations in further mathematical studies.

Derivative of a Function

The concept of the derivative is a fundamental aspect of calculus, representing how a function changes as its input changes. For a linear polynomial \( p(x) = ax + b \), finding its derivative is straightforward due to its simplicity.

• The derivative of \( ax + b \) is \( a \), because the rate of change of \( ax \) is constant.
• The derivative of \( b \), a constant, is zero because constants do not change with respect to \( x \).

Thus, the derivative of \( p(x) = ax + b \) is \( p'(x) = a \), reflecting the fact that the slope of the line is unchanging. In our specific example, \( p'(4) = -5 \) indicates that the line is decreasing, with a consistent steep slope of \(-5\), confirming that the function is straightforward, yet responsive to changes in \( x \). Understanding the derivative helps to illuminate the behavior of functions and is a powerful tool in analyzing complex mathematical models.