Slope of a secant line example 1 | Taking derivatives | Differential Calculus | Khan Academy

TL;DR

This video explains how to find the slope of a secant line for a curve with the equation y = ln(x), using the points P(e,1) and Q(x, ln(x)).

Transcript

A curve has the equation y equals the natural log of x, and passes through the points P equals e comma 1. And Q is equal to x natural log of x. Write an expression in x that gives the slope of the secant joining P and Q. So I think I'm going to need my little scratch pad for this one right over here. So this is the same question over again. Now let... Read More

Key Insights

• ☺️ The natural logarithm of x approaches negative infinity as x approaches 0.
• ❣️ The curve y = ln(x) passes through the points (e, 1) and (x, ln(x)).
• 💱 To find the slope of the secant line, the change in y is divided by the change in x.
• ❣️ The slope of the secant line equation for y = ln(x) is (ln(x) - 1) / (x - e).
• 😥 Point P(e, 1) and an arbitrary point Q(x, ln(x)) are used to find the slope of the secant line.
• ❣️ The graph of y = ln(x) has a horizontal asymptote at y = -∞ as x approaches 0.
• 🫥 The slope of the secant line represents the average rate of change between two points on the curve.

Questions & Answers

Q: What is the equation of the curve discussed in the video?

The curve has the equation y = ln(x).

Q: What are the coordinates of point P?

Point P has the coordinates (e, 1).

Q: What is the change in x between points P and Q?

The change in x is x - e.

Q: How is the slope of the secant line calculated?

The slope of the secant line is calculated by taking the change in y (ln(x) - 1) divided by the change in x (x - e).

Summary & Key Takeaways

• The video discusses finding the slope of the secant line of a curve with the equation y = ln(x) using the points P(e,1) and Q(x, ln(x)).

• The natural logarithm of x approaches negative infinity as x gets smaller, and the curve passes through the points P and Q.

• To find the slope of the secant line, the change in y (ln(x) - 1) is divided by the change in x (x - e).