Problem 87 - Precalculus with Limits

Question

In Exercises 85-90, evaluate the determinant in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. \(\left| \begin{array}{r} e^{2x} & e^{3x} \\ 2e^{2x} & 3e^{3x} \end{array} \right|\)

Step-by-Step Solution

Answer

The determinant of the given matrix is \(e^{5x}\).

Step 1: Identify the elements of the matrix

Since we are given the matrix \[ \left| \begin{array}{r} e^{2x} & e^{3x} \ 2e^{2x} & 3e^{3x} \end{array} \right|\], the elements of the matrix are identified as follows: \(a = e^{2x}\), \(b = e^{3x}\), \(c = 2e^{2x}\), and \(d = 3e^{3x}\)

Step 2: Apply the formula for the determinant

Apply the formula for the determinant to the given matrix. The determinant is computed as \(ad - bc\). Substituting the elements from Step 1, we get: Det = \(e^{2x}*3e^{3x} - e^{3x}*2e^{2x}\)

Step 3: Simplify the expression

Use the property of exponents where \(e^{a}*e^{b} = e^{a+b}\) and the property of real number multiplication (commutativity) where \(a*b = b*a\) to simplify the expression: Det = \(3e^{2x+3x} - 2e^{2x+3x} = e^{5x}\)

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when working with exponential expressions, particularly in evaluating determinants of matrices with exponential functions as elements. One key property is the multiplication of like bases, which states that when multiplying exponents with the same base, you add the exponents: \( e^{a} \cdot e^{b} = e^{a+b} \). This rule helps simplify the products of exponential terms.

Another important property is the power of a power rule, which states that \( (e^{a})^{b} = e^{a \cdot b} \). By mastering these rules, students can confidently simplify complex exponential expressions and accurately evaluate determinants and other mathematical problems involving exponents.

Matrix Determinants

The determinant of a matrix is a special number that can tell us many properties of the matrix. For a 2x2 matrix, the determinant is calculated with the formula \( ad - bc \) where \( a, b, c, \) and \( d \) are the elements of the matrix.

Evaluating the determinant is not only a fundamental operation in linear algebra but also plays a significant role in calculus, particularly when dealing with changes of variables and solving systems of equations. The value of a determinant can indicate whether the matrix is invertible or not—an essential aspect considering the determinant must be non-zero for the matrix to have an inverse.

Change of Variables in Calculus

In calculus, changing variables is a technique often used to simplify complex problems, making them more manageable to solve. This method is typically seen in integration, where a substitution is made to transform the integral into a more familiar or easier form.

The change of variables technique is closely related to the determinant when it comes to transformations in higher dimensions. For instance, when transforming from one coordinate space to another, the Jacobian determinant provides a measure of how much volume changes under that transformation, which is essential in evaluating integrals over a transformed region.

Simplifying Exponential Expressions

Simplifying exponential expressions involves using exponent rules to condense or reduce expressions into their simplest form. This process involves identifying bases that are the same and combining terms using the properties of exponents.

For example, in the exercise provided, the process of simplifying the determinant involves combining the exponential terms \( 3e^{5x} \) and \( -2e^{5x} \) to obtain \( e^{5x} \), by recognizing that they have the same base and exponent, which allows them to be subtracted directly. Simplification is a significant step in solving a variety of mathematical problems and helps in obtaining a more elegant and often more useful form of the expression.