Problem 70 - Algebra and Trigonometry

Question

\(65-70\) Draw the graph of the function in a suitable viewing rectangle and use it to find the domain, the asymptotes, and the local maximum and minimum values. $$ y=x \log _{10}(x+10) $$

Step-by-Step Solution

Answer

Domain: \( x > -10 \). Vertical asymptote at \( x = -10 \). Critical points lead to local extrema; solve derivation for specifics.

Step 1: Identify the Function and Its Properties

We are given the function \( y = x \log_{10}(x + 10) \). This is a combination of polynomial and logarithmic functions. Our aim is to draw its graph to determine the domain, asymptotes, and local extrema.

Step 2: Determine the Domain

The domain of the function \( y = x \log_{10}(x + 10) \) depends on the logarithmic part \( \log_{10}(x + 10) \), which is defined only for positive arguments. Therefore, \( x + 10 > 0 \), implying \( x > -10 \). Thus, the domain is \( x > -10 \).

Step 3: Graph the Function

To graph the function, choose a software tool or calculator with graphing capabilities. Set the viewing window to capture significant features of the function, ensuring it includes \(-10 < x < 10\) and an appropriate \(y\)-range.

Step 4: Identify Asymptotes

A vertical asymptote may occur where the argument of the logarithm is zero, i.e., as \( x \to -10^+ \). There are no horizontal or oblique asymptotes for this function because the term \( x \log_{10}(x+10) \) grows indefinitely as \( x \to \infty \).

Step 5: Calculate Derivative for Extrema

To find local extrema, we need the derivative of \( y = x \log_{10}(x+10) \), using the product and chain rule:\[ y' = \log_{10}(x+10) + \frac{x}{(x+10) \ln(10)} \]Set \( y' = 0 \) and solve for \( x \) to find critical points.

Step 6: Solve the Derivative Equation

Solving \( \log_{10}(x+10) + \frac{x}{(x+10) \ln(10)} = 0 \), gives critical points. Use numerical methods or graphing technology to approximate these solutions to find the \(x\)-coordinates of the local extrema.

Step 7: Evaluate the Critical Points

Substitute the critical points back into the original function to find the corresponding \( y \)-values, which will give us the local maximum and minimum.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (usually represented by the variable \( x \)) for which the function is defined. For the function \( y = x \log_{10}(x+10) \), the domain is particularly influenced by the logarithmic component. The expression \( \log_{10}(x+10) \) is only defined for positive values of its argument, which means we need \( x + 10 > 0 \). This results in \( x > -10 \). Therefore, the domain of the function is all real numbers greater than \(-10\). This means:

• The function exists for any value of \( x \) that is greater than \(-10\).
• Values such as \( x = -9, 0, 5, \) etc., are within the domain.

Understanding the domain helps address which inputs make the function valid, aiding in accurate graphing and analysis.

Asymptotes

Asymptotes are lines that a graph approaches as the input values become very large or very close to specific points. In this function \( y = x \log_{10}(x+10) \), a vertical asymptote occurs when the argument of the logarithm approaches zero since \( \log_{10}(0) \) leads to an undefined point or a value that extends towards negative infinity. This happens as \( x \) approaches \(-10\) from the right side. Thus, \( x = -10 \) is a vertical asymptote for the function. Horizontal or oblique asymptotes involve the behavior of the function as \( x \to \infty \) or \( x \to -\infty \). However, in this case:

• As \( x \to \infty \), \( x \log_{10}(x+10) \) grows indefinitely, indicating no horizontal asymptote.
• Asymptotic behavior is crucial for understanding long-term trends and limits in a function's graph.

Local Extrema

Local extrema consist of the highest or lowest points in a particular region of a graph. These are known as local maxima and minima, respectively. For \( y = x \log_{10}(x+10) \), finding these points involves using calculus to compute derivatives. The first derivative, \( y' = \log_{10}(x+10) + \frac{x}{(x+10) \ln(10)} \), finds where the slope equals zero, indicating potential extrema. Solving \( y' = 0 \) provides critical points, where numerical methods or graphing tools can approximate these values. Once the critical points are found, their \( y \)-values in the function indicate whether they are a local maximum or minimum. Identifying local extrema in functions helps understand how the graph behaves in different intervals, how it peaks, and where dips or valleys are located.

Logarithmic Functions

Logarithmic functions, often written as \( \log_{b}(x) \), represent the inverse of exponential functions. The function in our exercise, \( \log_{10}(x+10) \), is a logarithmic function base 10, shifted left along the \( x \)-axis by 10 units. Logarithmic functions are only defined for positive arguments, meaning \( x+10 > 0 \), resulting in the domain \( x > -10 \). They have unique properties:

• They grow logarithmically, meaning the output increases slowly as compared to polynomial functions.
• They have a vertical asymptote at the point where the argument is zero, in this case, \( x = -10 \).
• The derivative of \( \log(x) \) shows a decreasing rate of change, which is important for calculations of growth and decay within calculus.

Understanding logarithmic functions is pivotal for solving equations involving exponential growth, decibels, pH levels, and other scientific measurements.