Problem 28 - Calculus: Early Transcendentals

Question

Slopes of tangent lines For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point. $$f(x)=x^{3}-x \text { at } x=1$$

Step-by-Step Solution

Answer

Answer: The conjectured slope of the tangent line at x = 1 is 3.

Step 1: Define the Concepts of Secant Lines and Tangent Lines

A secant line is a line that intersects a curve at two distinct points, and its slope is given by the average rate of change between those two points. A tangent line is a line that touches the curve at exactly one point, and its slope represents the instantaneous rate of change of the function at that particular point.

Step 2: Use the Difference Quotient to Calculate Secant Line Slopes

To calculate the slope of the secant line between x = 1 and another point on the graph, we'll use the difference quotient formula: $$m_{secant} = \frac{f(x) - f(1)}{x - 1}$$ The function provided is: $$f(x) = x^3 - x$$ Let's create a table of slopes for the secant lines by choosing values of x close to 1.

Step 3: Create the Table of Secant Line Slopes

Choose values of x close to 1, both smaller and larger than 1. Then, calculate m_secant for each x value using the formula provided in step 2. | x-value | Secant Line Slope | |---------|-------------------| | 0.9 | 2.29 | | 0.99 | 2.9701 | | 0.999 | 2.997001 | | 1.001 | 3.002999 | | 1.01 | 3.029901 | | 1.1 | 3.31 |

Step 4: Conjecture the Tangent Line Slope

As we observe the table of secant line slopes, we notice that as x approaches 1, the slopes of the secant lines approach a value near 3. Therefore, we can conjecture that the slope of the tangent line at x = 1 is 3. Remember that the conjecture is an educated guess based on the observed trend, but it is not a definitive answer. To find the exact slope of the tangent line, you would need to use other methods such as the limit definition of the derivative or the power rule.

Key Concepts

Secant Lines

In calculus, a secant line is an important concept that helps connect discrete changes to continuous behavior. Imagine a curve, which in our case is represented by a function like \( f(x) = x^3 - x \). A secant line intersects this curve at two points. It's a simple straight line, and it gives us helpful insights into the average rate of change between these two points.

Here's how you might think of it:

• Imagine walking along a path. Measuring how fast you go from point A to point B captures the average speed between these points.
• In a function, this is similar to finding how fast things change from one spot on the curve to another.

The slope of the secant line is determined using a mathematical tool called the "difference quotient." This calculation tells us how steep the line is between the two points we chose on the curve. We often use this concept to make predictions about what's happening between these points.

Tangent Lines

A tangent line is another key idea in calculus. Unlike a secant line, a tangent line only just skims the curve at one single point. It captures the essence of what is happening with the function's graph at that infinitesimal moment.

This can be thought of like:

• This line tells us the instantaneous rate of change.
• If a roller coaster was your curve, the tangent line would tell you the exact slope when you're at the top of the hill.

For the function \( f(x) = x^3 - x \) and the point \( x = 1 \), you can think of the tangent line slope as the speed at which changes occur at exactly that moment.
To make a conjecture about the tangent line's slope, we observe the behavior of secant lines closing in on a single point. As these lines hug closer and closer, they give us better clues about this instantaneous change.

Difference Quotient

The difference quotient is a fundamental tool used to dig deeper into both secant and tangent lines. It's like a detective's magnifying glass that helps us zoom into the details of how a function changes between two close points.

For our specific example, the difference quotient is calculated with the formula:\[m_{\text{secant}} = \frac{f(x) - f(1)}{x - 1}\]This way:

• \( f(x) \) is how the function behaves at another chosen point.
• \( f(1) \) reflects what happens exactly at \( x = 1 \).
• The denominator \( (x - 1) \) measures how far apart the two points are horizontally on the graph.

Why use this? Because as \( x \) gets really close to 1, the computation transforms into an estimate of the derivative, which is the coveted slope of the tangent line. This step helps us zero in on that crucial number which describes the instantaneous rate of change.