Problem 47 - Calculus Early Transcendentals

Question

\(47-50\) Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. \(x^{4}+x-3=0\), \((1,2)\)

Step-by-Step Solution

Answer

There is a root in the interval \((1, 2)\) because \(f(x)\) changes sign from negative to positive.

Step 1: Understand the Intermediate Value Theorem

The Intermediate Value Theorem (IVT) states that if a continuous function, \(f(x)\), has values of opposite sign at two points \(a\) and \(b\), then there is at least one point \(c\) in the interval \((a, b)\) where \(f(c) = 0\). Essentially, if \(f(x)\) changes sign over \( (a, b) \), then it must cross the x-axis—indicating a root exists in that interval.

Step 2: Define the Function

Here, we are given the equation \(x^4 + x - 3 = 0\). Define \(f(x) = x^4 + x - 3\). This function is a polynomial, hence, continuous over all real numbers, including the interval \((1, 2)\).

Step 3: Evaluate the Function at the Endpoints

Calculate \(f(1)\) and \(f(2)\) to determine the sign of the function at these points. \[f(1) = 1^4 + 1 - 3 = -1\] and \[f(2) = 2^4 + 2 - 3 = 15\].

Step 4: Determine the Sign Change

From Step 3, \(f(1) = -1\) (negative) and \(f(2) = 15\) (positive). Since \(f(x)\) changes from negative to positive, it means there is at least one root between \(x = 1\) and \(x = 2\), according to the Intermediate Value Theorem.

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions involving a sum of terms, each consisting of a variable raised to a non-negative integer power, multiplied by a coefficient. In our exercise, the equation is given by \(x^4 + x - 3 = 0\). This is a polynomial equation of degree 4 because the highest power of the variable \(x\) is 4.

Polynomial equations are highly significant in mathematics because they model a wide range of real-world phenomena. Each term in a polynomial equation, such as \(x^4\), \(x\), and constants like \(-3\), plays an essential role in shaping the graph of the polynomial function.

• The term \(x^4\) has the most influence on the function's behavior as \(x\) becomes large.
• The linear term \(x\) and the constant \(-3\) have localized impacts on specific parts of the graph.

Understanding the structure of polynomials can help predict how their graphs behave over different intervals.

Root Finding

Root finding is the process of locating solutions, or "roots," for an equation where the function equals zero. For polynomial equations like \(x^4 + x - 3 = 0\), a root corresponds to a point, \(x = c\), where the polynomial equals zero, i.e., \(f(c) = 0\).

Various methods exist to find roots of polynomial equations:

• The graphical method involves plotting the function and finding the points where it crosses the x-axis.
• Analytical methods, such as factoring or using special formulas, work well for simpler polynomials.
• The Intermediate Value Theorem helps identify an interval where a root exists, particularly useful for complex polynomials.

In our exercise, we have used the Intermediate Value Theorem to confirm that a root exists between \(x = 1\) and \(x = 2\).

Continuous Functions

A function is continuous if its graph is unbroken or smooth, without any jumps or gaps. For our equation \(x^4 + x - 3 = 0\), the function \(f(x) = x^4 + x - 3\) is a polynomial, and polynomials are known to be continuous over all real numbers. This property is essential when applying the Intermediate Value Theorem.

Why is continuity important? Because a continuous function guarantees that in moving from one value, \(a\), on its domain to another value, \(b\), every intermediate value is hit, including zero if there is a sign change. Without continuity, we couldn't safely assert that a root exists between two points just because the function values at those points have opposite signs.

Sign Change

A sign change in a function happens when its value switches from positive to negative or vice versa over an interval. This concept is crucial for applying the Intermediate Value Theorem. In the exercise, we observe the values at the endpoints:

• \(f(1) = -1\), which is negative.
• \(f(2) = 15\), which is positive.

The function changes from negative at \(x = 1\) to positive at \(x = 2\), indicating a sign change has occurred. This change signifies that the function must cross the x-axis within the interval \((1,2)\), thus implying the presence of at least one root between these points.

X-axis Crossing

When a graph of a function crosses the x-axis, it means that the function has a root, or a zero, at that point. In mathematical terms, at this crossing point, the value of the function \(f(x)\) equals zero. This is the key indication of a root.

Using the Intermediate Value Theorem, we showed that a sign change occurred between \(x = 1\) and \(x = 2\). Therefore, the graph of \(f(x) = x^4 + x - 3\) must cross the x-axis somewhere between these values.

Understanding the concept of x-axis crossing is vital when dealing with polynomials, as it links the algebraic nature of equations to their graphical representation. Each crossing point of the x-axis corresponds to a solution of the equation, helping us visualize the roots.