Problem 76 - Precalculus: Functions and Graphs

Question

Graph the equation, and estimate the values of \(x\) in the specified interval that correspond to the given value of \(y\) $$y=\tan (\sqrt{x}), \quad[0,25] ; \quad y=5$$

Step-by-Step Solution

Answer

Due to periodicity of the tan function in this transformed way, approximate solutions are at \(x \approx 7.5, 15.5, 25\).

Step 1: Understand the Problem

We need to graph the equation \(y = \tan(\sqrt{x})\) and find the values of \(x\) where \(y = 5\) within the interval \([0, 25]\). Understanding how the transformation and range affect the graph is important for estimating values.

Step 2: Graph the Function

Sketch the graph of the function \(y = \tan(\sqrt{x})\) over the interval \([0, 25]\). The tangent function repeats every \(\pi\) radians, but this time the argument is \(\sqrt{x}\), which means the graph will show increasing frequency as \(x\) values grow larger.

Step 3: Identify Relevant Points

Note where the graph of \(y = \tan(\sqrt{x})\) intersects with \(y = 5\). These occur near points where the tangent function has particular angles \(x\) that make \(\tan(\sqrt{x}) = 5\).

Step 4: Estimate Values of \(x\)

By checking the periodic nature of \(\tan( heta)\) and its values per quarter period, estimate at which \(x\) values within the given range the graph hits \(y = 5\). Repeated checking of \(x = 7.5, 15.5, 25\) should yield values that are close to \(y = 5\).

Step 5: Verify Solutions

Plug the estimated values back into the equation \(y = \tan(\sqrt{x})\) to verify if \(y\) is approximately equal to 5. Adjust if necessary for better approximation.

Key Concepts

Graphing Equations

The process of graphing equations involves plotting a mathematical statement on an axis system. In our exercise, we are plotting the graph of the function \(y = \tan(\sqrt{x})\) over a specified range, which is from 0 to 25. Knowing how to represent the graph visually helps us see the behavior of the function across different values of \(x\). For the tangent function, it's important to remember the following:

• The tangent function is periodic, repeating every interval of \(\pi\) radians.
• However, replacing the variable with \(\sqrt{x}\) means the graph's nature changes, increasing in frequency as \(x\) increases.

Visualizing the graph helps in identifying where it intersects with a specific value, such as \(y = 5\). This visual representation is key before moving on to the next steps of calculating estimates.

Tangent Function

The tangent function, \(\tan(\theta)\), is one of the basic trigonometric functions. It is derived from the sine and cosine functions and is defined as \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). Here are some essential points to consider about the tangent function for this exercise:

• It has vertical asymptotes wherever \(\cos(\theta) = 0\), which occur at \(\theta = \frac{\pi}{2} + n\pi\), where \(n\) is an integer.
• The function repeats its pattern every \(\pi\) radians.
• Because our function uses \(\sqrt{x}\) instead of \(x\), each increase in \(x\) results in more rapid oscillation.

This transformation affects how quickly the tangent reaches its extrema, helping us see more clearly where it might intersect the line \(y = 5\). The tangent's unbounded nature in its range makes it particularly useful in such problems.

Estimating Values

When it comes to estimating values, we utilize the graph of \(y = \tan(\sqrt{x})\) to find approximate solutions. In this context, the goal is to find \(x\) when \(y\) equals 5, within the interval \([0, 25]\). Here’s how we do it:

• By observing the graph, look for points where it intersects \(y = 5\).
• Notice that as \(x\) increases, the function \(\tan(\sqrt{x})\) shows multiple peaks and waves.
• Specific values such as \(x = 7.5, 15.5,\) and \(25\) should be considered as they align closely with points on the graph where \(y\) could be near 5.

Once these values are identified, we substitute back into the function to check how close \(y\) is to 5. This step might require adjusting the estimates slightly to achieve a precise enough match.