Problem 32 - Calculus Volume 1

Question

Given the function \(y=f(x)\), a. find the slope of the secant line \(P Q\) for each point \(Q(x, f(x))\) with \(x\) value given in the table. b. Use the answers from a. to estimate the value of the slope of the tangent line at \(P\). c. Use the answer from b. to find the equation of the tangent line to \(f\) at point \(P\). $$ f(x)=\frac{x+1}{x^{2}-1}, P(0,-1) $$ $$ \begin{array}{|l|l|l|l|} \hline \boldsymbol{x} & \begin{array}{l} \text { Slope } \\ \boldsymbol{m}_{P Q} \end{array} & \boldsymbol{x} & \begin{array}{c} \text { Slope } \\ \boldsymbol{m}_{P Q} \end{array} \\ \hline 0.1 & \text { (i) } & -0.1 & \text { (vii) } \\ \hline 0.01 & \text { (ii) } & -0.01 & \text { (viii) } \\ \hline 0.001 & \text { (iii) } & -0.001 & \text { (ix) } \\ \hline 0.0001 & \text { (iv) } & -0.0001 & \text { (x) } \\ \hline 0.00001 & \text { (v) } & -0.00001 & \text { (xi) } \\ \hline 0.000001 & \text { (vi) } & -0.000001 & \text { (xii) } \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Find secant slopes, estimate tangent slope, then calculate tangent line.

Step 1: Identify the function and points

We are given the function \( f(x)=\frac{x+1}{x^{2}-1} \) and the point \( P(0,-1) \). The task is to calculate the secant slopes for various points \( Q(x, f(x)) \).

Step 2: Find Slope Formula of Secant Line

The formula for the slope of the secant line \( m_{PQ} \) between the points \( P(0, -1) \) and \( Q(x, f(x)) \) is: \[ m_{PQ} = \frac{f(x) - (-1)}{x - 0} = \frac{f(x) + 1}{x} \].

Step 3: Calculate Function Values for Different x's

Substitute the corresponding \( x \) values from the table into \( f(x) \). We need to compute \( f(x) \) for each \( x \) value, e.g., for \( x = 0.1, f(0.1) = \frac{0.1+1}{(0.1)^{2}-1} \).

Step 4: Compute Slopes of Secant Lines

Plug the values of \( f(x) \) from Step 3 into the slope formula: \( m_{PQ} = \frac{f(x) + 1}{x} \), for each \( x \) from the table. This will give values for \( m_{PQ} \) corresponding to \( x = 0.1, 0.01, \ldots \), both positive and negative, so for both \( x=-0.1, -0.01, ... \).

Step 5: Estimate Slope of Tangent Line

Observe the calculated slopes \( m_{PQ} \) as \( x \) approaches \( 0 \) from both positive and negative sides. The tangent slope \( m \) at \( x=0 \) is the limit of these slopes as \( x \to 0 \).

Step 6: Calculate Tangent Line Equation

Use the estimated tangent slope \( m \) from Step 5 and the point-slope form of line equation, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (0, -1) \) to find the equation of the tangent line.

Key Concepts

Secant Line

A secant line is essentially a straight line connecting two points on a curve. Imagine drawing a line from one point to another on a curve. This line is our secant line.

The concept of a secant line is especially crucial when studying the average rate of change between two points on a graph. For the given function in the exercise, the secant line connects point \( P(0, -1) \) and \( Q(x, f(x)) \).

The slope of this line is determined by the formula:

• \( m_{PQ} = \frac{f(x) + 1}{x} \)

This formula helps us calculate how steep the secant line is by considering the changes in the \( y \)-values relative to the changes in the \( x \)-values. By evaluating the slope of the secant line for different \( x \)-values provided in a table, we start seeing an insight into how the curve behaves.

Limit of Function

In calculus, understanding the limit of a function is a core idea when approaching problems that involve secant and tangent lines. Limits help us predict the behavior of a function as it approaches a particular point.

In our exercise, we seek to determine the limit of the slopes of the secant lines as \( x \) gets infinitely close to zero.

This limit gives us a glimpse into the nature of the tangent line, which is the line that just "touches" the curve at a given point. Essentially, we are observing how the secant line slopes converge as the points get closer together. By examining the slopes

• for values like \( 0.1, 0.01, ... \)
• and their negative counterparts \(-0.1, -0.01, ...\)

we can recognize a pattern indicating how the slope of the tangent line is formed as \( x \to 0 \). The concept of a limit is key in comprehending instantaneous change—what calculus is all about.

Slope of a Line

The slope of a line is an essential concept in understanding curves and changes, especially in calculus. Slope describes how steep a line is, which can indicate how fast a function value changes for a given change in \( x \).

For the secant line connecting points \( P \) and \( Q \), the slope is calculated using

• \( m = \frac{\Delta y}{\Delta x} = \frac{f(x) + 1}{x} \).

This formula is a classic method of describing how much \( y \) increases or decreases as \( x \) shifts by a particular amount.

The slope helps in transitioning from an average change (secant) to understanding instantaneous changes (tangent)—a key leap in most calculus problems. Recognizing the role of slope in defining both the secant and tangent lines is vital in comprehending more advanced calculus concepts.

Calculus Problem Solving

Solving calculus problems often involves breaking down complex scenarios into simpler, more manageable parts. With calculus, especially, it's about understanding fundamental principles like limits and slopes to solve for things like tangent lines.

• First, identify the function and key points, like \( P(0, -1) \) in this task.
• Then, compute secant slopes for various \( x \)-values, and analyze these slopes as \( x \) narrows towards zero.

Through these steps, we make educated estimates of a tangent slope at a particular point.

After determining the tangent slope, you use simple equations to describe the tangent line itself.

• Employ the point-slope formula: \( y - y_1 = m(x - x_1) \).

This method is part of mastering calculus problem-solving techniques, often requiring repeated practice to fully grasp. Calculus encourages thinking incrementally and understanding gradual changes in functions—transforming how we view and solve problems.