Problem 78 - Calculus Single Variable

Question

Plot the given curve in a viewing window containing the given point \(P\). Zoom in on the point \(P\) until the graph of the curve appears to be a straight line segment. Compute the slope of the line segment: It is an approximation to the slope of the curve at \(P\). $$ y=2 x /\left(x^{2}+1\right) ; P=(1,1) $$

Step-by-Step Solution

Answer

The slope of the curve at \( P = (1, 1) \) is approximately 0.

Step 1: Understand the Curve Equation

The curve given is described by the function \( y = \frac{2x}{x^2 + 1} \). Our goal is to plot this curve and explore its behavior around the point \( P = (1, 1) \).

Step 2: Verify Point P Lies on Curve

To confirm that point \( P = (1, 1) \) is on the curve, substitute \( x = 1 \) into the function: \( y = \frac{2(1)}{1^2 + 1} = \frac{2}{2} = 1 \). This verifies that \( P \) lies on the curve since \( y = 1 \).

Step 3: Plot the Curve

Plot the function \( y = \frac{2x}{x^2 + 1} \) over a reasonable domain, say from \( x = -3 \) to \( x = 3 \), and a range from \( y = -1 \) to \( y = 1.5 \).

Step 4: Zoom in Around Point P

Zoom closer to the point \( P = (1, 1) \). Start with a smaller window, such as \( 0.75 \leq x \leq 1.25 \) and \( 0.9 \leq y \leq 1.1 \), to see if the curve appears as a straight line segment.

Step 5: Adjust Viewing Window

If the curve doesn't appear linear yet, further zoom in to an even smaller window such as \( 0.9 \leq x \leq 1.1 \) and \( 0.98 \leq y \leq 1.02 \) to ensure the graph looks like a straight line.

Step 6: Determine Slope of the Line Segment

Find two points close to \( P = (1, 1) \) on the zoomed-in straight line segment, say \( (1.01, y_1) \) and \( (0.99, y_2) \). Compute \( y_1 = \frac{2(1.01)}{(1.01)^2 + 1} \) and \( y_2 = \frac{2(0.99)}{(0.99)^2 + 1} \).

Step 7: Compute Slope

Calculate the slope \( m \) of the line segment using the formula for slope: \( m = \frac{y_1 - y_2}{1.01 - 0.99} \). This gives an approximate slope at point \( P = (1, 1) \) on the curve.

Key Concepts

Curve Plotting

Curve plotting is a technique in calculus that involves drawing a function on a coordinate plane. It helps us visualize the behavior of mathematical equations. The given exercise requests plotting the curve representing the function \( y = \frac{2x}{x^2 + 1} \). We aim to see how this curve behaves around a specific point, \( P = (1, 1) \).

To plot the curve, choose a domain, which is the range of \( x \) values, such as \( -3 \) to \( 3 \). Correspondingly, choose a suitable range for \( y \), perhaps \( -1 \) to \( 1.5 \). This setting helps in getting a clear picture of the function's behavior across this interval. Use graphing software or manual plotting to draw the function.

By confirming that \( P \) lies on the curve and then plotting, we form a basis for further detailed exploration such as slope approximation and zooming. Understanding curve plotting is essential for breaking down complex equations visually and helps us in interpreting the behavior at specific points.

Slope Approximation

The point of slope approximation lies in estimating the steepness or angle of the curve at a specific point. This exercise calls for us to approximate the slope near point \( P = (1, 1) \) on our plotted curve.

Slope approximation is valuable since exact calculations at specific curve points can be complex. If the curve appears linear near a point upon zooming in, we can find points very close to \( P \), such as \( (0.99, y_2) \) and \( (1.01, y_1) \).

Use these to calculate the slope of the line segment between them with the formula:

• Find \( y_1 = \frac{2(1.01)}{(1.01)^2 + 1} \)
• Find \( y_2 = \frac{2(0.99)}{(0.99)^2 + 1} \)
• Compute the slope \( m = \frac{y_1 - y_2}{1.01 - 0.99} \)

This slope gives a strong approximation of the curve's slope at \( P \). Such understanding is crucial when exact analytical differentiation may not be accessible or straightforward.

Zooming Technique

The zooming technique is a powerful method in calculus for gaining insights into a curve's local behavior around a specific point. We achieve this by progressively narrowing our focus to a smaller region around the point of interest, \( P = (1, 1) \).

Initially, start with a moderate viewing window. If the curve isn't straight yet, continue to zoom in. A recommended process is to iterate: start with, say, \( 0.75 \leq x \leq 1.25 \) and \( 0.9 \leq y \leq 1.1 \), then further tighten to \( 0.9 \leq x \leq 1.1 \) and \( 0.98 \leq y \leq 1.02 \).

Zooming in helps the curve appear as a straight line segment. This resemblance to a straight line allows us to better estimate the slope at the point \( P \). This technique reveals smaller-scale behaviors otherwise invisible on larger plots, offering a practical approach to measuring instantaneous rates of change.

Calculus Problem Solving

Calculus problem solving involves applying various strategies and methods to break down and resolve calculus exercises. This particular task involves multiple concepts like curve plotting, slope approximation, and using the zooming technique effectively.

In solving such calculus problems, ensure you:

• Understand what the function represents and accurately plot it using a helpful window to see the entire behavior around points of interest.
• Utilize analytical techniques, such as deriving the function when possible, to grasp changes and slopes precisely.
• Employ graphical methods, like the zooming technique, to get a closer view facilitating the approximation.
• Integrate discrete measurements and calculations, especially when direct computation becomes complex or unavailable.

Proper application of these methods leads to a deeper understanding of calculus concepts and accelerates problem-solving efficiency, preparing you for more advanced mathematics challenges.