Problem 30 - Calculus with Concepts in Calculus

Question

Have a computer or graphics calculator plot the graph of the function on the given \(x\) window. Alter the \(y\) window as needed in order to be able to discern the major features of the graph. By reading off the \(y\) coordinate of the lowest point on the graph, approximate the smallest value assumed by the function on the given interval. It may be helpful to adjust the window. $$ f(x)=x^{2}(x-5)^{2} ;[-3,7] $$

Step-by-Step Solution

Answer

The lowest point approximates to be around \(y=0\) at \(x=5\).

Step 1: Understand the Function

The function provided is \( f(x) = x^2 (x-5)^2 \). This is a polynomial function, which will have important features such as intercepts and turning points that can be visualized on a graph.

Step 2: Set the x-Window

The problem specifies to use the window \([-3, 7]\) for the \(x\)-axis. This means the graph should be analyzed for values from \(x = -3\) to \(x = 7\).

Step 3: Plot the Function

Use a computer or a graphing calculator to plot the function \( f(x) = x^2 (x-5)^2 \) over the \(x\)-interval \([-3, 7]\). Observe the general shape and identify the major features of the graph.

Step 4: Adjust the y-Window

Initially choose a standard \(y\)-axis window, such as \([-10, 10]\). Observe the graph, and adjust the \(y\)-axis range accordingly to capture the lowest point clearly. You may need to extend this range if the graph appears compressed.

Step 5: Identify the Lowest Point

By visual inspection of the graph in your adjusted window, locate the lowest point of the function on the interval \([-3, 7]\). This is usually where the curve reaches its minimum value.

Step 6: Approximate the Minimum Value

Read off the \(y\)-coordinate of the lowest point found in the previous step. This \(y\)-value approximates the smallest value that the function assumes on the interval \([-3, 7]\).

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool for visualizing mathematical functions, especially polynomial functions like the one given in the exercise. This technology allows you to plot complex functions with ease and helps you understand their behavior over a specified interval. When using a graphing calculator, you'll enter the function equation—here, that's \[ f(x) = x^2(x-5)^2 \]into the calculator. Once entered, the graphing calculator will display the curve on a coordinate plane, providing valuable insights into the shape and structure of the function. This not only aids in identifying key features like intercepts and turning points, but also in understanding the general behavior of the polynomial over the range of interest.

Graphs drawn by graphing calculators are very useful in finding approximate solutions to equations graphically and for analyzing the function at a glance without the need to solve the equation manually.

Plotting Functions

Plotting functions is a crucial skill in mathematics that involves displaying a visual representation of an equation on a coordinate plane. For the function in question, \[ f(x) = x^2(x-5)^2 \],you need to utilize the graphing capabilities of your calculator or software to generate a graph.

Here's how you can plot a function:

• Enter the function into the calculator or software tool.
• Select the correct viewing window for both the x and y axes to ensure that all critical points are visible.
• Initiate the plot and observe the graph's shape and notable features.

Functions can have different shapes, depending on their degree and terms. For polynomials like ours, look out for curves, peaks, and troughs that signify turning points and intercepts. Plotting functions is important because it provides a spatial understanding of mathematical relationships, enabling easier interpretation and analysis.

X and Y Windows

Choosing the correct x and y window settings is key in accurately displaying a graph. The 'window' in terms of graphing refers to the range of values for which the graph is visible on the screen. For the given exercise, your x-window is set to the interval \([-3, 7]\).This means you are viewing the function from when x equals -3 up to when x equals 7.

For the y-window, begin with a typical range, such as \([-10, 10]\),and adjust based on what you see. If the graph appears squished or doesn't show all important points clearly, expand the y range to better visualize the peaks and valleys.

Adjustments may be necessary because polynomial functions can have values that exceed basic windows due to large outputs. By customizing your windows, you can ensure that you capture all significant aspects of the graph, making accurate readings possible.

Minimum Value Approximation

After plotting your function within the correct windows, the next step is to approximate the minimum value of the function over the specified interval. With the function,\[ f(x) = x^2(x-5)^2 \],a graph typically features a curve that reaches its lowest point.

To approximate the minimum value:

• Inspect the graph within your adjusted x and y windows.
• Look for the lowest point on the graph, which is where the curve dips the furthest down.
• Record the y-coordinate at this point, as it reflects the minimum value of the function on the interval.

This process of locating the minimum value provides an estimate, not an exact number, unless performed analytically. It's a powerful visual method for understanding how a function behaves and identifying its extreme values quickly.