Problem 70 - Calculus Single Variable

Question

A function \(f\) and four values \(c\) \(x_{1}, x_{2}\) and \(x_{3}\) are given. Using an appropriate viewing rectangle centered about the point \(P=(c, f(c)),\) graph \(f\) and the three secant lines passing through \(P\) that are determined by \(\left(x_{1}, f\left(x_{1}\right)\right),\left(x_{2}, f\left(x_{2}\right)\right),\) and \(\left(x_{3}, f\left(x_{3}\right)\right) .\) Use the secant line through \(\left(x_{3}, f\left(x_{3}\right)\right)\) to estimate \(f^{\prime}(c)\) $$ f(x)=\sqrt{x+1}, \quad c=0, x_{1}=0.5, x_{2}=0.3, x_{3}=0.1 $$

Step-by-Step Solution

Answer

The estimate for \( f'(0) \) is the slope \( m \approx \frac{\sqrt{1.1} - 1}{0.1} \).

Step 1: Define Function and Points

The function given is \(f(x) = \sqrt{x+1}\). We need to evaluate this function at the points \( c=0 \), \( x_1=0.5 \), \( x_2=0.3 \), and \( x_3=0.1 \). These points help us form pairs of \( (x, f(x)) \). The base point \( P \) is \((c, f(c)) = (0, \sqrt{0+1}) = (0, 1)\).

Step 2: Calculate Function Values at given Points

Calculate the function values: \( f(0.5) = \sqrt{0.5+1} = \sqrt{1.5} \), \( f(0.3) = \sqrt{0.3+1} = \sqrt{1.3} \), \( f(0.1) = \sqrt{0.1+1} = \sqrt{1.1} \). These results give points \( (0.5, \sqrt{1.5}) \), \( (0.3, \sqrt{1.3}) \), and \( (0.1, \sqrt{1.1}) \).

Step 3: Determine Secant Lines Equations

For each secant line, calculate the slope \( m \) using the formula \( m = \frac{f(x_i) - f(c)}{x_i - c} \). For \( x_3=0.1 \), the slope is \( m = \frac{\sqrt{1.1} - 1}{0.1} \). Repeat for \( x_1 \) and \( x_2 \).

Step 4: Estimate Derivative using Secant Line

Use the slope from the secant line through \((0.1, \sqrt{1.1})\) as an estimate for \( f'(c) \). Calculate the slope: \( m = \frac{\sqrt{1.1} - 1}{0.1} \). This slope approximates the derivative \( f'(0) \).

Step 5: Graph Function and Secant Lines

Graph \( f(x) \) and the secant lines on a coordinate system centered around \((c, f(c)) = (0,1)\). Ensure the viewing window is centered around \( (0, 1) \) to visualize the intersections effectively. Plot the function from a reasonable range, e.g., \([-1, 1]\), and adjust your window height accordingly for visibility.

Key Concepts

Graphing Functions

Graphing functions is a fundamental aspect of understanding their behavior visually. When you graph a function, like our given function, \( f(x) = \sqrt{x+1} \), you are plotting all the points where \( x \) and \( f(x) \) correspond. Here, we're focusing specifically on the function around the point \( P = (c, f(c)) = (0, 1) \).

To visualize this, think of your graph as a bird's-eye view of a landscape. The curve of the function will reveal how the values change as \( x \) varies. In our case, as \( x \) increases, \( f(x) \) increases since square root functions grow slowly.

When graphing, it's crucial to ensure your viewing window is appropriately set. Ideally, it should be centered around the point of interest and allow you to see how small changes around this point affect the function's behavior. For our example, you would graph \( f(x) \) using a window range of \([-1, 2]\) to capture essential features around \( x = 0 \), especially since we're interested in small increments such as \( x = 0.1 \), \( 0.3 \), and \( 0.5 \).

Remember, the purpose of graphing isn't just to plot points but to use these visual cues to understand the function's tendencies and predict behavior.

Secant Lines

Secant lines are straight lines that intersect a curve at two or more points. They are incredibly beneficial for approximating the behavior of functions over an interval. In our exercise, we use secant lines determined by the points \((x_i, f(x_i))\), where \(x_i\) are the given values \(0.1\), \(0.3\), and \(0.5\) for the function \(f(x) = \sqrt{x+1}\).

The slope of a secant line can be found using the formula:

• \( m = \frac{f(x_i) - f(c)}{x_i - c} \)

This formula essentially carries out the difference quotient, measuring the average rate of change of the function over the interval from \(c\) to \(x_i\).

In practical terms, the secant line provides a way to "connect" two distinct points on the curve, offering insight into the overall trend in the segment. Importantly, as the points used for the secant get closer to the point \(c\), the secant line becomes a better approximation of the function's behavior at \(c\). It's like taking smaller and smaller sections of the curve, so the straight line mimics the curve more closely.

Derivative Estimation

Estimating derivatives using secant lines is a crucial concept in differential calculus. The derivative of a function at a specific point tells us the instantaneous rate of change, mimicking the curve's behavior at that spot with a tangent line.

To estimate the derivative at \(c\), we take the slope of a secant line that's incredibly close to \(c\). In our example, the slope of the secant line connecting \((0, f(0))\) and \((0.1, f(0.1))\) is used to approximate \(f'(0)\). The formula used here:

• \( f'(0) \approx \frac{\sqrt{1.1} - 1}{0.1} \)

Essentially, this slope signifies how much \(f(x)\) changes near \(x = 0\).

As \( x \) values approach \(c\), the secant line becomes nearly identical to a tangent line. Hence, the slope obtained provides an increasingly accurate and useful approximation of the derivative. This method highlights the core principle of calculus, which examines infinitesimal changes to comprehend a broader spectrum of mathematical phenomena.