Problem 59 - Calculus Single Variable

Question

Find an explicit formula for the polynomial \(p(x)\) of degree 3 such that \(p(1)=1, p^{\prime}(2)=-2, p^{\prime \prime}(-1)=-14,\) and \(p^{\prime \prime \prime}(5)=6\).

Step-by-Step Solution

Answer

The polynomial is \(p(x) = x^3 - 4x^2 + 2x + 2\).

Step 1: Define the general form of the polynomial

Since the polynomial \(p(x)\) is of degree 3, it can be expressed in the general form: \[ p(x) = ax^3 + bx^2 + cx + d \]where \(a, b, c,\) and \(d\) are constants to be determined using the given conditions.

Step 2: Determine the third derivative

The third derivative of \(p(x)\) is:\[ p'''(x) = 6a \]We know from the problem that \(p'''(5) = 6\). Thus:\[ 6a = 6 \]Solving for \(a\), we get \(a = 1\).

Step 3: Determine the second derivative

Given \(a = 1\), the second derivative \(p''(x)\) is:\[ p''(x) = 6x + 2b \]Using the condition \(p''(-1) = -14\), substitute \(x = -1\):\[ 6(-1) + 2b = -14 \]This simplifies to \(-6 + 2b = -14\). Solving for \(b\), we find \(b = -4\).

Step 4: Determine the first derivative

With \(b = -4\), the first derivative \(p'(x)\) is:\[ p'(x) = 3x^2 - 8x + c \]Given \(p'(2) = -2\), substitute \(x = 2\):\[ 3(2)^2 - 8(2) + c = -2 \]Simplifying, \[12 - 16 + c = -2\]. Solving for \(c\), we find \(c = 2\).

Step 5: Determine the function itself

Given the values of \(a\), \(b\), and \(c\), the polynomial is:\[ p(x) = x^3 - 4x^2 + 2x + d \]Using \(p(1) = 1\), substitute \(x = 1\):\[ 1^3 - 4(1)^2 + 2(1) + d = 1 \]Simplifying, \[1 - 4 + 2 + d = 1\]. Solving for \(d\), we find \(d = 2\).

Step 6: Write the explicit formula

Now that all coefficients are known, the explicit formula for the polynomial \(p(x)\) is:\[ p(x) = x^3 - 4x^2 + 2x + 2 \]

Key Concepts

Derivative

A derivative of a function is essentially the rate at which the function's value changes with respect to a change in its input value. For polynomial functions, derivatives are calculated by applying the power rule, which involves multiplying each term by its respective exponent and subtracting one from each exponent.

• First Derivative: Taking the first derivative of a polynomial like \( ax^3 + bx^2 + cx + d \) results in \( 3ax^2 + 2bx + c \). This represents the slope of the tangent line to the polynomial at any point \( x \).
• Second Derivative: A second derivative, such as \( 6ax + 2b \), indicates the curvature or concavity of the function. It's derived by differentiating the first derivative.
• Third Derivative: The third derivative, \( 6a \) in this case, shows how the curvature itself is changing. This derivative is linear for our cubic polynomial.

Understanding derivatives is crucial for analyzing and predicting the behavior of polynomial functions over their domain.

Third Derivative

In our exercise, the third derivative of the polynomial was crucial for determining the constant \( a \) in the function's explicit formula. The third derivative of a degree 3 polynomial is a constant, since further differentiation of any cubic term results in a degree zero term.

When we computed the third derivative of \( p(x) = ax^3 + bx^2 + cx + d \), we got \( p'''(x) = 6a \). The given condition \( p'''(5) = 6 \) helped us find that \( a = 1 \).

It underscores the fact that for cubic polynomials, the third derivative provides a fixed rate of change in the curvature, which in this case confirmed our leading term coefficient. Always confirm third derivatives when dealing with polynomials of degree three; they can greatly simplify the problem.

Explicit Formula

An explicit formula is a straightforward expression that directly defines a function, as opposed to having to make recursive calculations or use equations. For polynomial functions, an explicit expression like \( p(x) = x^3 - 4x^2 + 2x + 2 \) gives a complete representation of the relationship between \( x \) and \( y \).

To derive the explicit formula for our cubic polynomial:

• We identified the degree of the polynomial and its general form.
• Using given derivative information at specific points, we solved for each constant \( a, b, c, d \).

These steps allow a plug-and-play approach to predict outputs for any input \( x \). This formula is not influenced by external factors in calculations and is very efficient for both computational tasks and conceptual analysis.

Polynomial Degree

The degree of a polynomial is defined by the highest power of \( x \) in the polynomial expression. In our exercise, we dealt with a polynomial of degree 3, meaning it has a highest term of \( x^3 \). The degree of a polynomial:

• Determines its overall shape and the number of possible real roots.
• A cubic polynomial (degree 3) is the simplest form to show inflection points.
• Ensures that the third derivative will be non-zero, providing valuable information about the function's fundamental behavior.

Understanding polynomial degrees helps in solving equations and predicting the behavior of graphs. Degree determines complexity and provides critical initial guidelines when attempting to sketch polynomial functions or when calculating limits and asymptotic behavior.