Problem 58 - Calculus Single Variable

Question

Find an explicit formula for the polynomial \(p(x)\) of degree 2 such that \(p(0)=2, p^{\prime}(0)=4,\) and \(p^{\prime \prime}(0)=-7\).

Step-by-Step Solution

Answer

The explicit formula is \(p(x) = -\frac{7}{2}x^2 + 4x + 2\).

Step 1: Define the General Formula for a Quadratic Polynomial

A quadratic polynomial of degree 2 can be expressed generally as: \[ p(x) = ax^2 + bx + c \] We need to find the coefficients \(a\), \(b\), and \(c\) that satisfy the given conditions.

Step 2: Apply the Initial Condition \(p(0) = 2\)

Substitute \(x = 0\) into the general formula: \[ p(0) = a(0)^2 + b(0) + c = c = 2 \]Thus, \(c = 2\).

Step 3: Derive the First Derivative and Apply \(p^{\prime}(0) = 4\)

First, compute the first derivative of \(p(x)\): \[ p^{\prime}(x) = 2ax + b \]Apply the condition \(p^{\prime}(0) = 4\): \[ p^{\prime}(0) = 2a(0) + b = b = 4 \]Thus, \(b = 4\).

Step 4: Derive the Second Derivative and Apply \(p^{\prime \prime}(0) = -7\)

Compute the second derivative: \[ p^{\prime \prime}(x) = 2a \]Apply the condition \(p^{\prime \prime}(0) = -7\): \[ 2a = -7 \]Solve for \(a\): \[ a = -\frac{7}{2} \].

Step 5: Write the Explicit Formula

Substitute \(a = -\frac{7}{2}\), \(b = 4\), and \(c = 2\) back into the general form of the polynomial:\[ p(x) = -\frac{7}{2}x^2 + 4x + 2 \]This is the explicit formula for the polynomial \(p(x)\).

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a polynomial expression of degree 2. It typically takes the form \( p(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The expression represents a parabola when graphed. Quadratic polynomials are essential in many branches of mathematics because they model simple relationships and serve as the building blocks for more complex polynomial functions.
Understanding the basic structure helps students tackle various mathematical problems:

• The coefficient \( a \) determines the concavity of the parabola.
• The term \( bx + c \) affects the position and direction of the graph.
• The constant term \( c \) represents the y-intercept, which is where the graph crosses the y-axis.

By applying these insights, students can predict and explain various properties of quadratic functions.

Polynomial Derivatives

Derivatives of polynomials are crucial in understanding how functions behave. For polynomial functions, derivatives help us determine the slope of the tangent lines at any given point. This means you can find how quickly or slowly a function changes at a specific location. For quadratic polynomials:

• The first derivative \( p'(x) \) tells you the slope of the function and its direction. For \( ax^2 + bx + c \), the derivative is \( 2ax + b \).
• The second derivative \( p''(x) \) gives you information about the concavity of the function. A positive \( p''(x) \) indicates a concave up, and a negative value indicates a concave down. For our quadratic, it is simply \( 2a \).

Applying derivatives allows you to explore the function's critical points, understanding where the maximum and minimum values occur. In this way, derivatives provide a deeper insight into the polynomial's behavior.

Explicit Formula

An explicit formula in the context of polynomial functions is a precise expression of the polynomial in terms of its variables and coefficients. In our exercise, the final explicit formula for a given quadratic polynomial involves substituting the specific determined values of \( a \), \( b \), and \( c \) into the general quadratic form \( ax^2 + bx + c \) to obtain a specific expression:

• The explicit formula for our quadratic polynomial \( p(x) \) is \(-\frac{7}{2}x^2 + 4x + 2 \).

Knowing the explicit formula allows students to evaluate the polynomial at any point simply by substituting the value of \( x \) into the equation. It's a powerful tool for predicting outcomes and analyzing the function's behavior across its domain.

Polynomial Coefficients

Polynomial coefficients are the multipliers in front of each term in a polynomial. Understanding the role these coefficients play is critical in polynomial analysis. In a quadratic polynomial \( ax^2 + bx + c \), each coefficient affects the function:

• \( a \) – Affects the width and direction of the parabola. A larger absolute value means a narrower parabola. If \( a \) is positive, the parabola opens upwards, if negative, it opens downwards.
• \( b \) – Affects the symmetry and tilt of the graph. The role of \( b \) is more subtle, influencing the direction in which the graph shifts.
• \( c \) – The y-intercept where the graph crosses the y-axis. It's the simplest to find when substituting \( x = 0 \).

These coefficients can be identified and utilized in various calculations, such as finding the vertex or factoring the polynomial.